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<!doctype html>
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<meta name="description" content="ES1 course reader, Chapter 03: drawing shadows by hand. Light, shade and shadow, four casting rules, the 45° convention, plans, elevations, axonometric and perspective construction, fine-liner technique, and a short self-test.">
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<title>Drawing shadows · ES1 Course Reader</title>
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<div class="mobile-bar"><a href="#top" class="wordmark"><img class="course-logo" src="./assets/course-logo.png" alt="ES1" width="372" height="123"><span>Course reader</span></a><div class="mobile-actions"><button type="button" class="theme-toggle" data-theme-toggle aria-label="Switch to dark mode" aria-pressed="false"><svg viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="1.6" aria-hidden="true"><circle cx="12" cy="12" r="8"/><path d="M12 4a8 8 0 0 1 0 16Z" fill="currentColor" stroke="none"/></svg></button><button id="menu-toggle" aria-expanded="false" aria-controls="sidebar">Contents <span aria-hidden="true">☰</span></button></div></div>
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<p class="sidebar-intro">Environmental systems<br>as a design material.</p>
<div class="chapter-label">THE CHAPTERS</div><div class="course-chapters"><a href="./site-climate.html"><span>01</span>Site + climate</a><a href="./index.html"><span>02</span>Solar geometry</a><a href="./shadow-drawing.html" aria-current="page"><span>03</span>Drawing shadows</a><a href="./sun-paths-shading.html"><span>04</span>Sun paths and shading</a><a href="./solar-radiation.html"><span>05</span>Solar radiation</a><a href="./hdr-photography.html"><span>06</span>HDR photography</a><a href="./heat-transfer.html"><span>07</span>Heat transfer</a><a href="./heat-balance.html"><span>08</span>Heat balance</a><a href="./psychrometrics.html"><span>09</span>Psychrometrics</a></div>
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<a href="#vocabulary" class="active"><span>01</span>Light, shade, shadow</a>
<a href="#rules"><span>02</span>Casting shadows</a>
<a href="#convention"><span>03</span>Real or 45° sun</a>
<a href="#plans"><span>04</span>Plans</a>
<a href="#elevations"><span>05</span>Elevations, sections</a>
<a href="#axonometric"><span>06</span>Axonometric</a>
<a href="#perspective"><span>07</span>Perspective</a>
<a href="#technique"><span>08</span>Fine-liner technique</a>
<a href="#quiz"><span>09</span>Mini quiz</a>
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<div class="sidebar-bottom"><button type="button" class="theme-toggle" data-theme-toggle aria-label="Switch to dark mode" aria-pressed="false"><span data-theme-label>Dark mode</span><svg viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="1.6" aria-hidden="true"><circle cx="12" cy="12" r="8"/><path d="M12 4a8 8 0 0 1 0 16Z" fill="currentColor" stroke="none"/></svg></button><div class="reading-label"><span>Reading position</span><span id="progress-number">0%</span></div><div class="progress-track"><div id="reading-progress"></div></div><a href="./index.html#author">About the instructor <span aria-hidden="true">↗</span></a><a href="#figures">Figure files (SVG) <span aria-hidden="true">↗</span></a><a href="#references">Sources and model notes <span aria-hidden="true">↗</span></a><p>Environmental Systems I<br>Cornell AAP</p></div>
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<div class="topline"><span>ENVIRONMENTAL SYSTEMS I</span><span>PLAN · ELEVATION · SECTION · AXONOMETRIC · PERSPECTIVE</span></div>
<div class="hero-heading"><div><p class="eyebrow">CHAPTER 03</p><h1>Drawing shadows<span class="title-dot">.</span></h1></div></div>
<p class="hero-lead">Shadows give a drawing depth. They show how far a balcony projects in an elevation, how tall a block is in a plan, and the shape of a building in a perspective. Constructed correctly, they also show where the sun falls. Hand construction uses the angles from <a href="./index.html">Chapter 02</a>: <b class="sym alt">ALT</b> and <b class="sym azi">AZI</b> for plans and three-dimensional views, and <b class="sym azi">HSA</b> and <b class="sym third">VSA</b> for elevations and sections.</p>
<div class="hero-bottom"><p>The constructions use the sun angles of Chapter 02, and equation numbers continue from it. Shadow length (Eq. 22) and the profile angle (Eq. 16–17) are derived in <a href="./sun-paths-shading.html">Chapter 04</a> and used here as given. Rays are parallel, so every shadow is built from the shadows of a few points joined by straight lines. Each figure can be saved as editable SVG. A short quiz closes the chapter.</p>
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<section id="vocabulary" class="reader-section">
<div class="section-heading"><span class="section-number">01</span><div><p class="eyebrow">WHAT TO CONSTRUCT</p><h2>Light, shade and shadow</h2></div></div>
<p class="section-lead">Rays are parallel. Every shadow is therefore built from the shadows of a few points, joined by straight lines. Construct lightly in pencil; ink only the result.</p>
<dl class="defs">
<div><dt>Light</dt><dd>A surface facing the sun. Leave it white.</dd></div>
<div><dt>Shade</dt><dd>A surface turned away from the sun (self-shading). It is darker than a lit surface but lighter than a shadow, because it still receives sky and reflected light.</dd></div>
<div><dt>Shadow</dt><dd>Part of a lit surface where the sun is blocked by another object or another part of the same building. A shadow can only fall on a surface that would otherwise be lit.</dd></div>
<div><dt>Casting edge</dt><dd>The edge between light and shade on the object. Only casting edges need to be constructed.</dd></div>
<div><dt>Shadow line</dt><dd>The shadow of a casting edge: the outline of the shadow.</dd></div>
</dl>
<!-- FIG:sketch-vocabulary_START -->
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stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1.5"/></marker><marker id="skv-arrow-third" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1.5"/></marker><marker id="skv-arrow-muted" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#607181, #aabccb)" stroke-width="1.5"/></marker><marker id="skv-arrow-red" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#b13d4c, #ff9daa)" stroke-width="1.5"/></marker></defs><g font-family="Helvetica Neue, Helvetica, Arial, sans-serif"><g id="skv-shadows-ground"><path d="M221.03,274.23 Q222.17,273.08 223.32,271.93 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#8bcfff)" stroke="none"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="613.49" y="151.22" font-size="12" style="--fs:12" fill="light-dark(#2479ad, #8bcfff)" font-weight="650">sun ray</text></g><g id="skv-labels"><path d="M188,106 Q348.19,153.17 508.38,200.35" fill="none" stroke="light-dark(#607181, #aabccb)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><circle cx="508.38" cy="200.35" r="1.8" fill="light-dark(#607181, #aabccb)" stroke="none"/><text x="20" y="110" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="700" letter-spacing="0.6">SHADE</text><text x="20" y="125" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">surface turned away</text><text x="20" y="139" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">from the sun</text><path d="M188,246 Q270.52,269.5 353.04,293" fill="none" stroke="light-dark(#607181, #aabccb)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><circle cx="353.04" cy="293" r="1.8" fill="light-dark(#607181, #aabccb)" stroke="none"/><text x="20" y="250" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="700" letter-spacing="0.6">SHADOW</text><text x="20" y="265" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">light blocked by</text><text x="20" y="279" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">another object</text><path d="M188,386 Q295.55,363.74 403.11,341.47" fill="none" stroke="light-dark(#607181, #aabccb)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><circle cx="403.11" cy="341.47" r="1.8" fill="light-dark(#607181, #aabccb)" stroke="none"/><text x="20" y="390" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="700" letter-spacing="0.6">SHADOW LINE</text><text x="20" y="405" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">shadow of the casting edge,</text><text x="20" y="419" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">along the bearing AZI + 180°</text><path d="M794,86 Q708.58,97.06 623.16,108.13" fill="none" stroke="light-dark(#607181, #aabccb)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><circle cx="623.16" cy="108.13" r="1.8" fill="light-dark(#607181, #aabccb)" stroke="none"/><text x="800" y="90" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="700" letter-spacing="0.6">LIGHT</text><text x="800" y="105" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">faces the sun:</text><text x="800" y="119" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">leave it white</text><path d="M794,296 Q712,270.2 630.01,244.41" fill="none" stroke="light-dark(#607181, #aabccb)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><circle cx="630.01" cy="244.41" r="1.8" 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fill="light-dark(#607181, #aabccb)">and shade</text></g><g id="skv-value-key"><path d="M218.62,470 Q242.35,469.89 266.08,470 M264,468.77 Q263.87,485.01 264,501.25 M265.79,500 Q241.66,500 217.53,500 M220,501.12 Q219.87,484.52 220,467.91" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text x="272" y="490" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">Light</text><path d="M312.12,472.77 Q313.44,471.45 314.76,470.13 M312.19,483.3 Q318.6,476.89 325.02,470.48 M312.24,493.86 Q324.01,482.11 335.76,470.34 M316.83,499.88 Q331.64,485.14 346.38,470.32 M327.42,499.9 Q341.58,485.78 355.7,471.62 M337.99,499.93 Q346.98,490.97 355.93,481.99 M348.64,499.89 Q352.12,496.41 355.6,492.93" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M309.63,470 Q334.15,470.01 358.67,470 M356,467.92 Q356.09,484.77 356,501.63 M358.53,500 Q333.99,500.01 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stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M402.82,470 Q426.01,470.01 449.21,470 M448,467.58 Q447.89,484.96 448,502.35 M450.13,500 Q426.47,500.09 402.81,500 M404,502.54 Q404.07,485.25 404,467.97" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text x="456" y="490" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">Shadow</text><text x="510" y="490" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">Value order: shadow darkest, shade lighter, light surfaces white.</text></g></g></svg></div><figcaption><span><strong>Figure 1.</strong> Light, shade and shadow · Hand drawing</span><a href="./assets/diagrams/solar/sketch-vocabulary.svg" download>Editable SVG ↓</a></figcaption></figure>
<!-- FIG:sketch-vocabulary_END -->
</section>
<section id="rules" class="reader-section">
<div class="section-heading"><span class="section-number">02</span><div><p class="eyebrow">FOUR RULES · EQ. 22</p><h2>Casting shadows</h2></div></div>
<p class="section-lead">Four rules cover nearly all of an architectural drawing. With them, most shadow outlines are copied rather than constructed.</p>
<ol class="facts">
<li><strong>Vertical edge on the ground.</strong> The shadow runs along the sun's bearing, <b class="sym azi">AZI</b> + 180°, with length <em>L</em> = H / tan <b class="sym alt">ALT</b> (Eq. 22). All vertical edges cast parallel shadows, and their lengths are proportional to their heights.</li>
<li><strong>Vertical edge meets a wall.</strong> The shadow turns at the base of the wall and climbs it vertically until it meets the ray through the top.</li>
<li><strong>Edge parallel to a surface.</strong> Its shadow on that surface is parallel to the edge and equally long. A roof edge's shadow on the ground is a copy of the edge, moved along one ray.</li>
<li><strong>Edge meeting a surface.</strong> Its shadow on that surface starts at the meeting point. Edges perpendicular to a wall all cast parallel shadows on it.</li>
</ol>
<!-- FIG:sketch-rules_START -->
<figure class="fig" id="fig-sketch-rules"><div class="fig-plate"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 1000 350" class="diagram" role="img" aria-label="Four rules for casting shadows by hand"><title>Four rules for casting shadows by hand</title><defs><marker id="skr-arrow-ink" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5"/></marker><marker id="skr-arrow-alt" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5"/></marker><marker id="skr-arrow-azi" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" 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fill="light-dark(#2479ad, #8bcfff)" font-weight="700" letter-spacing="1.2">RULE 1</text><text x="24" y="48" font-size="14" style="--fs:14" fill="light-dark(#26333e, #edf4fa)" font-weight="650">Vertical edge on the ground</text><path d="M73.2,228.58 Q151.92,209.74 230.85,191.83 M229.03,196.21 Q199.84,164.79 170.53,133.48 M176.9,134.22 Q98.42,151.82 20.25,170.75 M21.94,167.66 Q51.4,199.56 81.18,231.18" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M84.86,172.25 Q86.18,170.93 87.5,169.61 M85.34,174.31 Q87.39,172.27 89.43,170.22 M87.38,174.82 Q89.4,172.81 91.41,170.79 M89.33,175.41 Q91.32,173.41 93.32,171.42 M91.24,176.05 Q93.31,173.96 95.4,171.89 M93.29,176.54 Q95.31,174.51 97.35,172.48 M95.19,177.19 Q97.27,175.12 99.33,173.04 M97.19,177.73 Q99.19,175.73 101.19,173.74 M99.14,178.33 Q101.21,176.25 103.29,174.18 M101.2,178.82 Q103.21,176.79 105.23,174.78 M103.11,179.45 Q105.11,177.45 107.11,175.45 M105.07,180.04 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M174.7,186.77 Q176.77,184.71 178.84,182.63 M176.76,187.25 Q178.81,185.21 180.85,183.17 M178.66,187.9 Q180.69,185.89 182.7,183.86 M180.63,188.48 Q182.68,186.42 184.74,184.36 M182.65,189.01 Q184.71,186.94 186.77,184.88 M184.6,189.6 Q186.61,187.59 188.63,185.57 M186.6,190.15 Q188.39,188.35 190.19,186.56 M188.61,190.69 Q188.84,190.46 189.06,190.23" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M190.38,185.9 Q176.36,182.02 162.42,177.82 M162.42,177.82 Q161.72,180.33 160.97,182.82 M160.97,182.82 Q174.92,186.97 188.94,190.89 M188.94,190.89 Q189.66,188.4 190.38,185.9" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M189.66,188.4 Q189.62,175.81 189.66,163.21" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><path d="M207.21,152.48 Q184.45,166.4 161.69,180.32" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#skr-arrow-alt)"/><circle cx="161.69" cy="180.32" r="2.2" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><text x="24" y="296" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">The shadow runs along the bearing,</text><text x="24" y="311" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">AZI + 180°. All vertical edges cast</text><text x="24" y="326" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">parallel shadows, L = H / tan ALT.</text></g><g id="skr-rule-2"><text x="274" y="30" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)" font-weight="700" letter-spacing="1.2">RULE 2</text><text x="274" y="48" font-size="14" style="--fs:14" fill="light-dark(#26333e, #edf4fa)" font-weight="650">Vertical edge meets a wall</text><path d="M326.12,243.52 Q402.89,225.09 479.89,207.67 M479.27,212.09 Q448.29,178.89 417.33,145.67 M424.19,148.2 Q348.03,166.19 271.77,183.74 M270.52,179.49 Q301.48,212.18 331.93,245.33" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M244.71,178.42 Q246.44,176.68 248.19,174.95 M248.57,179.37 Q251.61,176.32 254.66,173.28 M251.85,180.89 Q256.33,176.41 260.81,171.94 M255.84,181.72 Q261.43,176.12 267.03,170.53 M259.39,182.98 Q266.38,175.96 273.4,168.97 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stroke-linecap="round" stroke-linejoin="round"/><path d="M398.47,112.33 Q371.63,128.75 344.79,145.17" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#skr-arrow-alt)"/><circle cx="344.79" cy="145.17" r="2.2" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><text x="274" y="296" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">At the wall the shadow turns and</text><text x="274" y="311" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">climbs it vertically: it stays in the</text><text x="274" y="326" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">vertical plane of the rays.</text></g><g id="skr-rule-3"><text x="524" y="30" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)" font-weight="700" letter-spacing="1.2">RULE 3</text><text x="524" y="48" font-size="14" style="--fs:14" fill="light-dark(#26333e, #edf4fa)" font-weight="650">Edge parallel to the ground</text><path d="M576.2,221.51 Q652.81,204.04 729.24,185.83 M729.56,190.4 Q699,158.17 668.98,125.44 M674.39,126.16 Q597.78,144.64 520.9,161.95 M522.28,159.38 Q552.51,191.27 582.21,223.64" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M557.38,178.3 Q558.89,176.79 560.4,175.28 M558.28,179.94 Q560.32,177.89 562.37,175.85 M560.25,180.52 Q562.27,178.5 564.29,176.47 M562.22,181.09 Q564.28,179.04 566.34,176.97 M564.2,181.65 Q566.21,179.65 568.21,177.65 M566.11,182.29 Q568.16,180.24 570.21,178.19 M568.11,182.83 Q570.18,180.78 572.24,178.71 M570.07,183.43 Q572.15,181.34 574.24,179.25 M572.04,184 Q574.11,181.93 576.17,179.86 M574.03,184.56 Q576.08,182.5 578.13,180.45 M576.03,185.1 Q578.07,183.05 580.12,181.01 M578.08,185.59 Q580.09,183.58 582.1,181.58 M580,186.22 Q582.06,184.15 584.12,182.1 M581.92,186.84 Q583.95,184.82 585.97,182.8 M584,187.31 Q585.97,185.34 587.93,183.38 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Q595.23,168.45 598.64,165.04 M595.14,171.08 Q598.56,167.65 602,164.22 M598.5,170.26 Q601.92,166.85 605.33,163.43 M601.81,169.5 Q605.23,166.09 608.64,162.67 M605.2,168.65 Q608.57,165.3 611.92,161.94 M608.48,167.92 Q611.86,164.55 615.23,161.17 M611.77,167.18 Q615.17,163.78 618.57,160.38 M615.12,166.38 Q618.53,162.96 621.95,159.55 M618.4,165.64 Q621.84,162.18 625.29,158.75 M621.8,164.78 Q625.21,161.39 628.6,157.99 M625.08,164.05 Q627.79,161.35 630.49,158.64 M628.43,163.24 Q629.71,161.96 631,160.68" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M558.27,179.7 Q594.84,170.92 631.52,162.62 M631.52,162.62 Q630.94,160.09 630.34,157.56 M630.34,157.56 Q593.7,166.03 557.09,174.64 M557.09,174.64 Q557.68,177.17 558.27,179.7" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M608.21,191.75 Q608.14,169.01 608.21,146.26" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><path d="M681.45,174.68 Q681.57,151.93 681.45,129.18" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><path d="M608.21,146.26 Q644.91,138.06 681.45,129.18" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><path d="M627.71,134.33 Q592.69,155.75 557.68,177.17" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#skr-arrow-alt)"/><circle cx="557.68" cy="177.17" r="2.2" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><path d="M700.96,117.25 Q665.94,138.67 630.93,160.09" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#skr-arrow-alt)"/><circle cx="630.93" cy="160.09" r="2.2" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><text x="524" y="296" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">Its shadow on the ground is parallel</text><text x="524" y="311" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">and equally long: copy the edge,</text><text x="524" y="326" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">shifted by one ray.</text></g><g id="skr-rule-4"><text x="774" y="30" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)" font-weight="700" letter-spacing="1.2">RULE 4</text><text x="774" y="48" font-size="14" style="--fs:14" fill="light-dark(#26333e, #edf4fa)" font-weight="650">Edge perpendicular to a wall</text><path d="M825.06,237.23 Q902.56,219.67 979.84,201.15 M978.33,204.54 Q948.61,173.14 919.35,141.3 M925.86,143.03 Q847.31,161.77 768.56,179.7 M772.26,176.57 Q801.79,208.24 831.33,239.9" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M735.68,182.71 Q735.86,182.53 736.03,182.36 M737.83,185.37 Q739.95,183.25 742.06,181.14 M741.57,186.44 Q745.13,182.87 748.7,179.31 M745.5,187.31 Q750,182.79 754.52,178.3 M748.95,188.67 Q754.96,182.65 760.98,176.64 M752.69,189.74 Q759.96,182.48 767.23,175.21 M756.52,190.72 Q764.99,182.21 773.5,173.74 M760.34,191.71 Q770.21,181.83 780.09,171.96 M763.81,193.04 Q774.83,182 785.88,170.98 M767.54,194.12 Q779.85,181.83 792.15,169.52 M771.22,195.26 Q784.91,181.52 798.64,167.83 M775.43,195.85 Q790.14,181.17 804.83,166.45 M779.11,196.98 Q795.26,180.91 811.32,164.77 M782.67,198.23 Q799.97,180.96 817.24,163.66 M786.56,199.15 Q805.16,180.61 823.69,162.01 M790.21,200.3 Q810.01,180.45 829.86,160.65 M793.88,201.45 Q815.02,180.23 836.23,159.09 M797.61,202.52 Q820.1,180.05 842.57,157.56 M801.55,203.39 Q825.19,179.65 848.92,156.02 M805.07,204.68 Q830.12,179.56 855.23,154.52 M808.78,205.77 Q835.2,179.48 861.49,153.07 M812.37,206.99 Q838.64,180.71 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stroke-linejoin="round"/><path d="M737.39,185.5 Q775.98,196.93 814.73,207.83 M813.29,207.84 Q877.23,192.51 941.35,177.98 M941.07,178.37 Q939.35,176.52 937.62,174.67 M938.25,175.05 Q899.84,163.66 861.27,152.82 M862.36,152.78 Q798.28,167.62 734.24,182.65 M734.57,182.08 Q736.51,184.18 738.48,186.27" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><polygon points="814.1,207.65 811.15,204.48 811.15,135.87 814.1,139.03" fill="light-dark(#ffffff, #202124)" stroke="none"/><path d="M811.55,143.04 Q812.71,141.88 813.86,140.73 M811.59,153.61 Q812.71,152.48 813.84,151.36 M811.21,164.6 Q812.56,163.24 813.92,161.89 M811.41,175 Q812.58,173.84 813.74,172.67 M811.55,185.47 Q812.77,184.26 813.98,183.04 M811.52,196.11 Q812.74,194.89 813.95,193.67 M812.64,205.59 Q813.32,204.92 814,204.24" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><polygon points="814.1,207.65 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d="M872.02,148.76 Q874.34,146.43 876.66,144.12 M868.46,154.86 Q873.4,149.9 878.36,144.97 M865.03,160.84 Q872.09,153.78 879.15,146.72 M861.54,166.88 Q868.54,159.85 875.57,152.84 M858.14,172.82 Q865.12,165.83 872.11,158.86 M854.57,178.94 Q861.62,171.92 868.64,164.87 M851.81,184.25 Q858.53,177.54 865.23,170.82 M853.52,185.08 Q857.59,181.02 861.65,176.95 M855.14,186.01 Q856.69,184.45 858.25,182.9" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M875.21,143.14 Q863.3,163.6 851.55,184.15 M851.55,184.15 Q853.8,185.45 856.05,186.75 M856.05,186.75 Q867.99,166.31 879.71,145.74 M879.71,145.74 Q877.46,144.45 875.21,143.14" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M877.46,144.44 Q885.43,152.8 893.22,161.34" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><path 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stroke-linejoin="round"/><path d="M839.45,169.45 Q844.92,175.21 850.28,181.07" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><path d="M872.78,167.31 Q847.98,182.47 823.18,197.64" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#skr-arrow-alt)"/><circle cx="823.18" cy="197.64" r="2.2" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><text x="774" y="296" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">Its shadow starts where it touches</text><text x="774" y="311" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">the wall. Edges perpendicular to the</text><text x="774" y="326" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">wall cast parallel shadows on it.</text></g></g></svg></div><figcaption><span><strong>Figure 2.</strong> Four rules for casting shadows · Hand drawing</span><a href="./assets/diagrams/solar/sketch-rules.svg" download>Editable SVG ↓</a></figcaption></figure>
<!-- FIG:sketch-rules_END -->
</section>
<section id="convention" class="reader-section">
<div class="section-heading"><span class="section-number">03</span><div><p class="eyebrow">DESIGN STUDY OR PRESENTATION · EQ. 12</p><h2>Real sun or 45° convention</h2></div></div>
<p class="section-lead">For a design study, use the real sun for the site, date and hour: <b class="sym alt">ALT</b> and <b class="sym azi">AZI</b> from Eq. 7–8, the <a href="./sun-paths-shading.html#sunpath">sun-path diagram</a>, or the sky hemisphere in <a href="./index.html#position">Chapter 02</a>. For presentation drawings with no particular time, architects use a convention. The rays run along the body diagonal of a cube, from upper left front to lower right rear. They appear at 45° in both plan and elevation, so no angle needs to be measured.</p>
<div class="eq" id="eq-12"><span class="eq-math"><span class="eq-line"><span class="eq-label">45° convention, true altitude</span><math><mi class="sym alt">ALT</mi><mo>=</mo><mrow><mi>arctan</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mrow><mo>(</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mo>)</mo></mrow></mrow><mo>=</mo><mn>35.3°</mn></math></span><span class="eq-line"><span class="eq-label">elevation</span><math><mtext>offset</mtext><mo>=</mo><mtext>depth</mtext></math><span class="eq-note">down and right</span></span><span class="eq-line"><span class="eq-label">plan</span><math><mtext>offset</mtext><mo>=</mo><mtext>height</mtext></math><span class="eq-note">right and up the sheet</span></span></span><span class="eq-n">(12)</span></div>
<!-- FIG:sketch-convention_START -->
<figure class="fig" id="fig-sketch-convention"><div class="fig-plate"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 990 380" class="diagram" role="img" aria-label="The 45° convention: sun along the cube diagonal"><title>The 45° convention: sun along the cube diagonal</title><defs><marker id="skc-arrow-ink" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5"/></marker><marker id="skc-arrow-alt" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5"/></marker><marker id="skc-arrow-azi" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1.5"/></marker><marker id="skc-arrow-third" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1.5"/></marker><marker id="skc-arrow-muted" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#607181, #aabccb)" stroke-width="1.5"/></marker><marker id="skc-arrow-red" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#b13d4c, #ff9daa)" stroke-width="1.5"/></marker></defs><g font-family="Helvetica Neue, Helvetica, Arial, sans-serif"><g id="skc-cube"><text x="30" y="34" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" font-weight="700" letter-spacing="1.2">A · CUBE DIAGONAL</text><path d="M130.35,295.4 Q93.17,264.87 56.46,233.78 M59.49,240.18 Q59.79,164.36 59.49,88.53 M57.68,90.63 Q95,121.27 131.84,152.48 M128.67,146.47 Q128.53,222 128.67,297.53" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><path d="M126.3,294.47 Q200.39,279.42 274.61,265.05 M270.51,269.48 Q270.3,193.9 270.51,118.31 M273.55,121.09 Q199,135.28 124.67,150.62 M128.67,146.95 Q129.17,222.48 128.67,298.01" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><path d="M125.26,150.51 Q198.99,135.14 273.01,121.2 M273.6,124.26 Q236.63,93.41 199.64,62.59 M204.26,63.42 Q130.86,77.43 57.67,92.5 M56.76,89.86 Q93.83,120.21 130.34,151.23" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><path d="M201.33,208.17 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stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#skc-arrow-alt)"/><circle cx="201.33" cy="208.17" r="2.6" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="104.5" y="131.25" font-size="12" style="--fs:12" fill="light-dark(#2479ad, #8bcfff)" text-anchor="end" font-weight="650">sun ray</text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="171" y="273.08" font-size="12" style="--fs:12" fill="light-dark(#007c9a, #6bd8f0)" font-weight="650">plan: 45°</text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="147.59" y="233.85" font-size="12" style="--fs:12" fill="light-dark(#6a4fb3, #c3b2ff)" font-weight="650">elevation: 45°</text><text x="30" y="332" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">Rays run along the body diagonal of a cube:</text><text x="30" y="347" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">from upper left front to lower right rear.</text><text x="30" y="362" font-size="12" style="--fs:12" fill="light-dark(#2479ad, #8bcfff)" font-weight="650">True altitude = arctan(1/√2) = 35.3°.</text></g><g id="skc-plan"><text x="330" y="34" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" font-weight="700" letter-spacing="1.2">B · PLAN</text><path d="M400.46,190.96 Q430.78,160.61 461.13,130.3 M400.32,195.92 Q433.08,163.04 465.96,130.27 M400.33,200.72 Q435.4,165.57 470.55,130.49 M400.45,205.39 Q438.16,167.71 475.84,130.01 M400.36,210.3 Q440.4,170.09 480.61,130.05 M400.33,215.14 Q442.84,172.83 485.15,130.32 M400.48,219.79 Q445.19,175.07 489.91,130.36 M400.45,224.63 Q447.69,177.63 494.69,130.39 M400.11,229.78 Q449.86,180.09 499.55,130.34 M400.47,234.23 Q452.47,182.06 504.64,130.06 M400.47,239.04 Q454.84,184.72 509.17,130.34 M400.39,243.92 Q457.37,187.15 514.14,130.17 M400.38,248.74 Q459.83,189.61 518.96,130.16 M404.29,249.65 Q463.98,190.12 523.51,130.42 M408.78,249.97 Q468.5,190.02 528.45,130.29 M413.82,249.72 Q473.69,190.02 533.39,130.16 M418.62,249.74 Q478.56,190.08 538.22,130.14 M423.41,249.76 Q483.14,189.83 543.07,130.09 M428.38,249.59 Q488.17,190.13 547.63,130.34 M433.07,249.72 Q491.34,191.52 549.53,133.25 M438.01,249.58 Q494.07,193.67 549.98,137.61 M442.51,249.89 Q496.22,196.41 549.71,142.69 M447.55,249.66 Q498.63,198.39 549.9,147.31 M452.15,249.87 Q500.89,201.2 549.56,152.46 M457.14,249.68 Q503.7,203.38 550,156.83 M461.86,249.77 Q505.95,205.76 549.95,161.68 M466.54,249.91 Q508.3,208.26 549.94,166.5 M471.34,249.91 Q510.59,210.59 549.91,171.34 M476.2,249.86 Q512.94,213.21 549.59,176.47 M480.94,249.93 Q515.2,215.55 549.58,181.29 M486.02,249.66 Q521.6,214 557.26,178.41 M526.19,214.29 Q543.81,196.58 561.53,178.95 M526.26,219.03 Q543.91,201.41 561.53,183.76 M526.41,223.69 Q544.13,206.05 561.76,188.34 M526.13,228.78 Q543.83,211.07 561.54,193.36 M526.47,233.25 Q544.22,215.51 561.96,197.76 M526.18,238.35 Q544.04,220.56 561.83,202.69 M526.45,242.89 Q544.12,225.22 561.78,207.55 M526.21,247.93 Q543.92,230.25 561.59,212.55 M526.37,252.58 Q544.09,234.95 561.73,217.22 M526.41,257.35 Q544.18,239.56 561.97,221.79 M526,262.56 Q543.73,244.75 561.53,227.03 M526.43,266.94 Q543.99,249.39 561.55,231.83 M526.41,271.77 Q544.01,254.22 561.56,236.62 M526.39,276.6 Q543.97,259 561.57,241.42 M526.18,281.62 Q543.98,263.83 561.77,246.03 M526.4,286.2 Q544.19,268.48 561.92,250.69 M526.2,291.22 Q543.99,273.49 561.72,255.69 M526.26,295.97 Q543.94,278.35 561.56,260.66 M529.05,297.98 Q545.37,281.61 561.75,265.28" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M399.63,250 Q445.2,250.27 490.78,250 M489.56,250.44 Q507.93,232.13 526.24,213.76 M550,184.4 Q550,156.93 550,129.45 M550.59,130 Q504.86,130.2 459.13,130 M460.4,129.6 Q430.22,160.1 399.73,190.27 M400,189.34 Q400.15,219.9 400,250.47 M525.17,298 Q528.92,298.02 532.67,298 M531.56,298.44 Q547,282.92 562.52,267.48 M562,268.53 Q561.79,222.97 562,177.41 M562.35,178 Q558.75,178.01 555.15,178 M556.42,177.58 Q552.93,181.07 549.44,184.56 M526,213.46 Q525.55,255.99 526,298.53" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><polygon points="400,250 490,250 490,190 400,190" fill="light-dark(#ffffff, #202124)" stroke="none"/><path d="M397.48,250 Q445.53,249.89 493.59,250 M490,253.41 Q489.98,220.65 490,187.9 M492.99,190 Q445.08,190.35 397.18,190 M400,186.72 Q400.12,220.35 400,253.99" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><polygon points="526,298 532,298 532,208 526,208" fill="light-dark(#ffffff, #202124)" stroke="none"/><path d="M523.41,298 Q528.67,298.02 533.94,298 M532,301.21 Q531.56,252.6 532,204 M536.34,208 Q529.35,208.01 522.37,208 M526,205.09 Q525.79,252.86 526,300.62" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><path d="M490,190 Q520.08,160.08 550,130" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><circle cx="550" cy="130" r="2.4" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><path d="M490,190 Q520,190 550,190" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><path d="M550,190 Q550,160 550,130" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="520" y="205" font-size="12" style="--fs:12" fill="light-dark(#2479ad, #8bcfff)" text-anchor="middle" font-weight="700">H</text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="558" y="164" font-size="12" style="--fs:12" fill="light-dark(#2479ad, #8bcfff)" font-weight="700">H</text><text x="445" y="224" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle">+2.0</text><text x="529" y="313" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle">+1.0</text><path d="M340,304 Q354,290 368,276" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#skc-arrow-alt)"/><text x="336" y="318" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)" font-weight="650">sun</text><text x="330" y="332" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">A point at height H casts its shadow</text><text x="330" y="347" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">H to the right and H up the sheet.</text><text x="330" y="362" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">Read H from the spot height: no angle needed.</text></g><g id="skc-elevation"><text x="650" y="34" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" font-weight="700" letter-spacing="1.2">C · ELEVATION</text><polygon points="690,300 948.4,300 948.4,75.6 690,75.6" fill="light-dark(#ffffff, #202124)" stroke="none"/><path d="M714.01,189.95 Q715.05,188.92 716.07,187.89 M714.22,194.56 Q717.41,191.36 720.6,188.17 M716.71,196.87 Q721.02,192.58 725.31,188.27 M718.99,199.4 Q724.71,193.66 730.45,187.94 M721.66,201.54 Q728.47,194.7 735.32,187.88 M724.08,203.93 Q731.91,196.09 739.74,188.26 M726.27,206.54 Q735.39,197.39 744.54,188.27 M728.75,208.88 Q739.04,198.58 749.33,188.29 M731.23,211.2 Q742.86,199.63 754.43,188.01 M733.66,213.58 Q746.33,200.91 759,188.24 M735.82,216.23 Q749.88,202.21 763.9,188.15 M738.24,218.61 Q753.5,203.42 768.7,188.16 M740.57,221.1 Q757.08,204.53 773.66,188.01 M742.94,223.53 Q760.71,205.73 778.51,187.96 M853.38,113.09 Q855.05,111.42 856.73,109.75 M745.39,225.89 Q764.33,206.87 783.36,187.92 M853.59,117.69 Q857.41,113.89 861.21,110.07 M747.73,228.36 Q767.8,208.27 787.89,188.2 M853.48,122.61 Q859.9,116.16 866.36,109.73 M752.5,228.39 Q772.51,208.27 792.64,188.26 M853.41,127.49 Q862.31,118.57 871.23,109.67 M757.35,228.35 Q777.62,208.13 797.84,187.86 M853.64,132.06 Q864.64,121.05 875.65,110.05 M762.04,228.48 Q782.3,208.12 802.65,187.86 M853.58,136.93 Q867.19,123.29 880.84,109.68 M766.75,228.58 Q786.92,208.37 807.12,188.2 M853.36,141.96 Q860.93,134.4 868.49,126.83 M870.77,124.56 Q878.19,117.17 885.57,109.75 M771.92,228.21 Q791.2,208.84 810.57,189.57 M853.6,146.53 Q860.96,139.15 868.34,131.79 M875.41,124.72 Q882.83,117.28 890.28,109.86 M776.79,228.15 Q794.84,210.05 812.94,192 M853.46,151.48 Q860.94,143.96 868.47,136.47 M880.31,124.63 Q887.8,117.14 895.29,109.65 M781.39,228.36 Q798.38,211.35 815.39,194.36 M853.59,156.16 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Q910.66,219.3 918.09,211.86 M865.79,268.97 Q867.13,267.63 868.47,266.29 M908.22,226.54 Q914.48,220.27 920.76,214 M913.07,226.5 Q917.07,222.49 921.09,218.49 M917.61,226.77 Q919.16,225.22 920.71,223.67" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M713.2,194 Q730.7,211.69 748.4,229.2 M747.44,228.6 Q795.42,228.32 843.39,228.6 M843,229.43 Q842.98,225.38 843,221.33 M843.59,222.39 Q825.9,204.94 808.46,187.26 M809.65,187.8 Q761.51,187.94 713.36,187.8 M713.8,187.4 Q713.83,191.36 713.8,195.33" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M851.87,269.4 Q887.81,269.42 923.75,269.4 M921.2,272.14 Q920.97,240.49 921.2,208.84 M922.86,211.6 Q886.8,211.6 850.74,211.6 M853.2,208.78 Q853.4,240.41 853.2,272.04" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><path d="M887.2,269.4 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font-size="12" style="--fs:12" fill="light-dark(#2479ad, #8bcfff)" text-anchor="middle" font-weight="700">d</text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="849" y="215.6" font-size="12" style="--fs:12" fill="light-dark(#2479ad, #8bcfff)" font-weight="700">d</text><path d="M669.6,65.4 Q683.2,79 696.8,92.6" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#skc-arrow-alt)"/><text x="665.6" y="61.4" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)" font-weight="650">sun</text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="761.4" y="173.2" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle">balcony, projects d</text><text x="650" y="332" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">Anything projecting d from the wall casts</text><text x="650" y="347" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">its shadow d down and d to the right.</text><text x="650" y="362" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">Window reveals of depth r: shadow r wide.</text></g></g></svg></div><figcaption><span><strong>Figure 3.</strong> The 45° convention · Hand drawing</span><a href="./assets/diagrams/solar/sketch-convention.svg" download>Editable SVG ↓</a></figcaption></figure>
<!-- FIG:sketch-convention_END -->
<div class="callout"><span>CONVENTION</span>
<p class="measure">The 45° sun is a drawing device for showing depth. It belongs to no particular date or hour, and it says nothing about whether a space actually receives sun. When a drawing has to show sun access, overshadowing or the performance of a shading device, use the real sun and state the date and time on the sheet.</p>
</div>
</section>
<section id="plans" class="reader-section">
<div class="section-heading"><span class="section-number">04</span><div><p class="eyebrow">PLAN · SHADOW SCALE</p><h2>Plans</h2></div></div>
<ol class="facts">
<li>Draw a north arrow and a sun arrow. The shadow bearing is <b class="sym azi">AZI</b> + 180°.</li>
<li>Build a shadow scale at drawing scale: one vertical pole, marked in meters, and parallel rays at <b class="sym alt">ALT</b>. Every height's shadow length can then be read off with dividers. Alternatively, multiply heights by the factor 1 / tan <b class="sym alt">ALT</b> from the table below.</li>
<li>From each corner of each roof outline, lay off <em>L</em> along the bearing.</li>
<li>Join the tips with lines parallel to the roof edges (rule 3). Then draw the shadows of the vertical casting edges from the building corners to the tips (rule 1).</li>
<li>Where a shadow falls on a lower roof, use only the height above that roof. Hatch the shadow and leave the roofs white.</li>
</ol>
<!-- FIG:sketch-plan_START -->
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M390.93,268 Q319.1,267.3 247.28,268 M250,264.71 Q250.44,314 250,363.29" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><path d="M441.59,463.5 Q446.11,463.5 450.64,463.5 M448.95,467.8 Q449.09,406.05 448.95,344.29 M451.89,348.5 Q447.69,348.48 443.5,348.5 M445.5,345.49 Q445.17,406.59 445.5,467.68" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><path d="M535.96,312.7 A10.02,10.02 0 0 1 525.21,328.9 A9.71,9.71 0 0 1 507.42,336.72 A10.04,10.04 0 0 1 488.22,333.67 A9.64,9.64 0 0 1 473.71,320.73 A10.53,10.53 0 0 1 468.5,302 A10.25,10.25 0 0 1 474.25,283.43 A10.35,10.35 0 0 1 489.12,270.91 A10.19,10.19 0 0 1 508.4,268.43 A9.63,9.63 0 0 1 525.97,276.75 A10.43,10.43 0 0 1 536.24,293.26 A10.27,10.27 0 0 1 535.96,312.7 Z" fill="light-dark(#ffffff, #202124)" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><circle cx="503" cy="302.5" r="1.6" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="319" y="318" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle">+9.0</text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="238.5" y="221.4" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle">+3.0</text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="429.4" y="433" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle">+2.0</text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="503" y="350.2" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle">tree +9.0</text></g><g id="skp-construction"><path d="M254.45,365.25 Q199.33,300.27 144.18,235.31" fill="none" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><path d="M251.19,358.99 Q210.73,393.42 170.19,427.74" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><path d="M171.09,426.97 Q169.78,345.45 168.47,263.94" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><circle cx="168.47" cy="263.94" r="2.6" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><circle cx="171.09" cy="426.97" r="2" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><path d="M168.89,289.93 A26,26 0 0 0 185.3,283.76" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="182.56" y="305.37" font-size="12" style="--fs:12" fill="light-dark(#2479ad, #8bcfff)" text-anchor="middle" font-weight="650"></text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="138.47" y="309.94" font-size="12" style="--fs:12" fill="light-dark(#2479ad, #8bcfff)" text-anchor="end" font-weight="650">ALT 39.4°</text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="224.54" y="403.49" font-size="12" style="--fs:12" fill="light-dark(#2479ad, #8bcfff)" font-weight="650">H = 9 m, folded flat</text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="225.31" y="318.77" font-size="12" style="--fs:12" fill="light-dark(#007c9a, #6bd8f0)" font-weight="650">L = 11.0 m</text><path d="M470,104 Q470,87 470,70" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#skp-arrow-ink)"/><text x="470" y="62" font-size="14" style="--fs:14" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle" font-weight="700">N</text><path d="M463.53,412.38 Q451.88,398.65 440.24,384.93" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#skp-arrow-alt)"/><circle cx="470" cy="420" r="7" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><text x="460" y="446" font-size="12" style="--fs:12" fill="light-dark(#007c9a, #6bd8f0)" font-weight="650">AZI 139.7°</text></g><g id="skp-shadow-scale"><text x="560" y="34" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" font-weight="700" letter-spacing="1.2">SHADOW SCALE · DRAW ONCE, READ ALL HEIGHTS</text><path d="M560,360 Q740,360.05 920,360" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><path d="M900,360 Q900.09,308.25 900,256.5" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><path d="M900,348.5 Q902.5,348.5 905,348.5" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M900,337 Q902.5,337 905,337" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><text x="910" y="341" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">2 m</text><path d="M900,325.5 Q902.5,325.5 905,325.5" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><text x="910" y="329.5" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">3 m</text><path d="M900,314 Q902.5,314 905,314" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M900,302.5 Q902.5,302.5 905,302.5" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M900,291 Q902.5,291 905,291" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><text x="910" y="295" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">6 m</text><path d="M900,279.5 Q902.5,279.5 905,279.5" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M900,268 Q902.5,268 905,268" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M900,256.5 Q902.5,256.5 905,256.5" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><text x="910" y="260.5" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">9 m</text><path d="M900,337 Q886,348.5 872,360" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><circle cx="872" cy="360" r="2.3" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><path d="M872,364 Q872,373 872,382" fill="none" stroke="light-dark(#607181, #aabccb)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M872,380 Q886,380 900,380" fill="none" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="866" y="384" font-size="12" style="--fs:12" fill="light-dark(#007c9a, #6bd8f0)" text-anchor="end" font-weight="650">2 m → 2.4 m</text><path d="M900,325.5 Q879,342.75 858,360" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><circle cx="858" cy="360" r="2.3" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><path d="M858,364 Q858,382 858,400" fill="none" stroke="light-dark(#607181, #aabccb)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M858,398 Q879,398 900,398" fill="none" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="852" y="402" font-size="12" style="--fs:12" fill="light-dark(#007c9a, #6bd8f0)" text-anchor="end" font-weight="650">3 m → 3.7 m</text><path d="M900,256.5 Q837,308.25 774.01,360" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><circle cx="774.01" cy="360" r="2.3" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><path d="M774.01,364 Q774.01,391 774.01,418" fill="none" stroke="light-dark(#607181, #aabccb)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M774.01,416 Q837,416 900,416" fill="none" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="768.01" y="420" font-size="12" style="--fs:12" fill="light-dark(#007c9a, #6bd8f0)" text-anchor="end" font-weight="650">9 m → 11.0 m</text><path d="M804.01,360 A30,30 0 0 0 797.19,340.96" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="817.31" y="348.49" font-size="12" style="--fs:12" fill="light-dark(#2479ad, #8bcfff)" text-anchor="middle" font-weight="650">39.4°</text><text x="560" y="450" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">Each meter of height casts 1.22 m of shadow (1 / tan ALT).</text><text x="560" y="466" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">Take lengths off with dividers; lay them along the bearing.</text></g><text x="30" y="540" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">Shadow of block A (+9 m) falls partly on the roof of block B (+3 m): there the shadow only needs the height difference, 6 m × 1.22.</text></g></svg></div><figcaption><span><strong>Figure 4.</strong> Plan shadows at a real date and time, with a shadow scale · Hand drawing</span><a href="./assets/diagrams/solar/sketch-plan.svg" download>Editable SVG ↓</a></figcaption></figure>
<!-- FIG:sketch-plan_END -->
<!-- SKETCHTABLE_START -->
<div class="table-wrap"><table class="nomenclature compact"><caption>Ithaca, 42.4° N · shadow length per meter of height (1 / tan ALT) · shadow bearing (AZI + 180°) · solar time</caption><thead><tr><th scope="col">Date</th><th scope="col">08:00</th><th scope="col">10:00</th><th scope="col">12:00</th><th scope="col">14:00</th><th scope="col">16:00</th></tr></thead><tbody><tr><th scope="row">Mar 21 · Sep 21</th><td>2.56 · 292°</td><td>1.22 · 320°</td><td>0.93 · 0°</td><td>1.22 · 40°</td><td>2.56 · 68°</td></tr><tr><th scope="row">Jun 21</th><td>1.31 · 271°</td><td>0.61 · 298°</td><td>0.34 · 0°</td><td>0.61 · 62°</td><td>1.31 · 89°</td></tr><tr><th scope="row">Dec 21</th><td>14.17 · 307°</td><td>2.98 · 331°</td><td>2.23 · 0°</td><td>2.98 · 29°</td><td>14.17 · 53°</td></tr></tbody></table></div>
<!-- SKETCHTABLE_END -->
</section>
<section id="elevations" class="reader-section">
<div class="section-heading"><span class="section-number">05</span><div><p class="eyebrow">PLAN + SECTION → ELEVATION · EQ. 13</p><h2>Elevations and sections</h2></div></div>
<p class="section-lead">An elevation shows the facade straight on, so the sun appears in projection. In plan the rays make the horizontal shadow angle <b class="sym azi">HSA</b> with the facade normal. In section they make the vertical shadow angle <b class="sym third">VSA</b> with the horizontal (<a href="./sun-paths-shading.html#profile">Chapter 04, Section 03</a>, Eq. 16–17). A point that projects a distance <em>w</em> in front of the wall casts its shadow sideways and down by:</p>
<div class="eq" id="eq-13"><span class="eq-math"><span class="eq-line"><span class="eq-label">sideways</span><math><mrow><mi mathvariant="normal">Δ</mi><mi>u</mi></mrow><mo>=</mo><mi>w</mi><mo>·</mo><mrow><mi>tan</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi class="sym azi">HSA</mi></mrow></math></span><span class="eq-line"><span class="eq-label">down</span><math><mrow><mi mathvariant="normal">Δ</mi><mi>v</mi></mrow><mo>=</mo><mi>w</mi><mo>·</mo><mrow><mi>tan</mi><mo lspace="0" rspace="0.1667em">⁡</mo><mi class="sym third">VSA</mi></mrow></math><span class="eq-note">shadow on the wall only if |HSA| < 90°</span></span></span><span class="eq-n">(13)</span></div>
<p class="measure">Draw the plan above the elevation and a section beside it. Rays in the plan give the sideways shift and rays in the section give the drop. Project both into the elevation; where the projection lines cross is the shadow of that point. For the 45° convention, HSA = 45° and VSA = 45°, so both offsets equal <em>w</em>.</p>
<!-- FIG:sketch-elevation_START -->
<figure class="fig" id="fig-sketch-elevation"><div class="fig-plate"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 1000 580" class="diagram" role="img" aria-label="Elevation shadows constructed from plan and section"><title>Elevation shadows constructed from plan and section</title><defs><marker id="ske-arrow-ink" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5"/></marker><marker id="ske-arrow-alt" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5"/></marker><marker id="ske-arrow-azi" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" 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stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="219.81" y="239.15" font-size="12" style="--fs:12" fill="light-dark(#007c9a, #6bd8f0)" text-anchor="middle" font-weight="650"></text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="225.6" y="247.6" font-size="12" style="--fs:12" fill="light-dark(#007c9a, #6bd8f0)" text-anchor="end" font-weight="650">HSA 40.3°</text><path d="M562.8,282 Q540.8,282.08 518.8,282" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-dasharray="2 3" stroke-linecap="round" stroke-linejoin="round"/><path d="M545.11,262.94 A26,26 0 0 0 536.8,282" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="526.14" y="270.01" font-size="12" style="--fs:12" fill="light-dark(#6a4fb3, #c3b2ff)" text-anchor="middle" font-weight="650"></text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="532.8" y="248" font-size="12" style="--fs:12" fill="light-dark(#6a4fb3, #c3b2ff)" text-anchor="end" font-weight="650">VSA 47.1°</text></g><g id="ske-notes"><text x="700" y="116" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" font-weight="700" letter-spacing="1.2">ITHACA · MAR 21 · 14:00 SOLAR</text><text x="700" y="134" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">ALT 39.4° · AZI 220.3° · facade faces south</text><text x="700" y="180" font-size="16" style="--fs:16" fill="light-dark(#2479ad, #8bcfff)" font-weight="700">1</text><text x="720" y="180" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">Plan: draw rays at HSA = AZI − 180° = 40.3°</text><text x="720" y="195" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">to the facade normal. Each gives the sideways</text><text x="720" y="210" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">shift w · tan HSA = 0.85 w.</text><text x="700" y="242" font-size="16" style="--fs:16" fill="light-dark(#2479ad, #8bcfff)" font-weight="700">2</text><text x="720" y="242" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">Section: draw rays at VSA = 47.1° (Eq. 17).</text><text x="720" y="257" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">Each gives the drop w · tan VSA = 1.08 w.</text><text x="700" y="304" font-size="16" style="--fs:16" fill="light-dark(#2479ad, #8bcfff)" font-weight="700">3</text><text x="720" y="304" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">Project both into the elevation. Where the</text><text x="720" y="319" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">lines cross is the shadow of that point.</text><text x="700" y="366" font-size="16" style="--fs:16" fill="light-dark(#2479ad, #8bcfff)" font-weight="700">4</text><text x="720" y="366" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">Join the points with lines parallel to the</text><text x="720" y="381" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">casting edges, then hatch.</text><text x="700" y="440" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">Point 1, balcony edge, w = 1.4 m:</text><text x="700" y="455" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">1.19 m across, 1.51 m down.</text><text x="700" y="470" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">Window reveals (w = 0.3 m) follow the same rule.</text></g></g></svg></div><figcaption><span><strong>Figure 5.</strong> Elevation shadows from plan and section · Hand drawing</span><a href="./assets/diagrams/solar/sketch-elevation.svg" download>Editable SVG ↓</a></figcaption></figure>
<!-- FIG:sketch-elevation_END -->
<p class="measure">A section through an opening uses the same <b class="sym third">VSA</b> ray. It shows how far down the facade an overhang's shadow reaches and where the sun patch falls on the floor inside (Lechner's sunbeam drawing). The interactive section in <a href="./sun-paths-shading.html#profile">Chapter 04, Section 03</a> performs this construction for any date.</p>
</section>
<section id="axonometric" class="reader-section">
<div class="section-heading"><span class="section-number">06</span><div><p class="eyebrow">PARALLEL PROJECTION</p><h2>Axonometric</h2></div></div>
<p class="section-lead">An axonometric keeps parallel lines parallel, so all rays are parallel and all bearing lines are parallel. Construct one shadow point properly: find its position in the plan with Eq. 22 and draw it in the axonometric. That fixes the direction of the ray and of the bearing line. Copy both directions with set squares for every other point.</p>
<!-- FIG:sketch-axo_START -->
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stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><path d="M216.19,181.82 Q340.95,242.66 465.72,303.5" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#ska-arrow-alt)"/><circle cx="465.72" cy="303.5" r="2.6" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="171.68" y="377.22" font-size="14" style="--fs:14" fill="light-dark(#2479ad, #8bcfff)" font-weight="700">1</text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="473.72" y="319.5" font-size="14" style="--fs:14" fill="light-dark(#2479ad, #8bcfff)" font-weight="700">2</text></g><g id="ska-notes"><text x="668" y="96" font-size="16" style="--fs:16" fill="light-dark(#2479ad, #8bcfff)" font-weight="700">1</text><text x="690" y="96" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">Find the shadow of one point in plan</text><text x="690" y="111" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">(Eq. 22; the 4 m pole casts 6.6 m) and draw</text><text x="690" y="126" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">that ray. All other rays are parallel to it.</text><text x="668" y="168" font-size="16" style="--fs:16" fill="light-dark(#2479ad, #8bcfff)" font-weight="700">2</text><text x="690" y="168" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">Through each base point draw a bearing line;</text><text x="690" y="183" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">through its top, a ray. Where they cross is</text><text x="690" y="198" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">the shadow point (1).</text><text x="668" y="240" font-size="16" style="--fs:16" fill="light-dark(#2479ad, #8bcfff)" font-weight="700">3</text><text x="690" y="240" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">If the bearing line meets a wall first, turn</text><text x="690" y="255" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">up the wall until it meets the ray (2).</text><text x="668" y="312" font-size="16" style="--fs:16" fill="light-dark(#2479ad, #8bcfff)" font-weight="700">4</text><text x="690" y="312" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">Connect the points. Edges parallel to the</text><text x="690" y="327" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">ground or to the wall cast parallel shadows:</text><text x="690" y="342" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">most of the outline is copied, not constructed.</text><text x="668" y="390" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">ALT 31.2° · AZI 235.7° · tower 12 m · block 4.5 m</text></g></g></svg></div><figcaption><span><strong>Figure 6.</strong> Axonometric shadow construction · Hand drawing</span><a href="./assets/diagrams/solar/sketch-axo.svg" download>Editable SVG ↓</a></figcaption></figure>
<!-- FIG:sketch-axo_END -->
</section>
<section id="perspective" class="reader-section">
<div class="section-heading"><span class="section-number">07</span><div><p class="eyebrow">VANISHING POINTS</p><h2>Perspective</h2></div></div>
<p class="section-lead">In a perspective, parallel lines converge. The sun's rays therefore converge to a vanishing point of their own, VP<sub>s</sub>. The shadows of vertical edges on the ground are horizontal lines, so they converge to a point on the horizon, VP<sub>sh</sub>, directly above or below VP<sub>s</sub>.</p>
<ol class="facts">
<li>Choose the sun's bearing by placing VP<sub>sh</sub> on the horizon.</li>
<li>Choose the altitude by placing VP<sub>s</sub> on the vertical through VP<sub>sh</sub>: below the horizon when the sun is behind you, above it when the sun is in front of you. The farther VP<sub>s</sub> is from the horizon, the higher the sun and the shorter the shadows.</li>
<li>For every vertical edge, draw a line from VP<sub>sh</sub> through its base and a line from VP<sub>s</sub> through its top. They cross at the shadow point.</li>
<li>Shadows of horizontal edges on the ground run to the same vanishing point as the edge itself (rule 3).</li>
</ol>
<!-- FIG:sketch-perspective_START -->
<figure class="fig" id="fig-sketch-perspective"><div class="fig-plate"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 1000 1050" class="diagram" role="img" aria-label="Shadows in two-point perspective for three sun directions"><title>Shadows in two-point perspective for three sun directions</title><defs><marker id="skx-arrow-ink" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5"/></marker><marker id="skx-arrow-alt" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5"/></marker><marker id="skx-arrow-azi" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 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style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" font-weight="700" letter-spacing="1.2">A · SUN PARALLEL TO THE PICTURE PLANE</text><path d="M30,160 Q345,159.69 660,160" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text x="34" y="154" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)">HL · eye height</text><circle cx="630" cy="160" r="2.2" fill="light-dark(#607181, #aabccb)" stroke="none"/><text x="630" y="176" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">VPR</text><path d="M630,160 Q479.95,193.71 330,227.88" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M630,160 Q480.02,146 330,132.42" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><circle cx="30" cy="160" r="2.2" fill="light-dark(#607181, #aabccb)" stroke="none"/><text x="30" y="176" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">VPL</text><path d="M30,160 Q180.04,193.76 330,227.88" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M30,160 Q180.03,146.49 330,132.42" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><g id="skx-shadow-a"><path d="M275.06,215.39 Q276.13,214.32 277.2,213.25 M279.24,216.01 Q280.99,214.26 282.75,212.51 M283.01,217.05 Q285.73,214.33 288.45,211.61 M287.03,217.85 Q290.39,214.49 293.75,211.12 M291.1,218.58 Q295.49,214.18 299.88,209.8 M294.97,219.52 Q300.22,214.3 305.44,209.05 M298.58,220.72 Q304.64,214.64 310.72,208.58 M302.87,221.23 Q309.75,214.35 316.64,207.47 M306.75,222.17 Q314.61,214.27 322.51,206.41 M310.52,223.2 Q319.19,214.51 327.87,205.85 M314.71,223.82 Q323.92,214.57 333.17,205.36 M318.56,224.78 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Q435.6,218.33 445.07,208.87 M431.13,227.61 Q440.37,218.4 449.58,209.16 M435.7,227.85 Q444.66,218.88 453.63,209.92 M440.84,227.52 Q449.38,218.97 457.93,210.43 M445.63,227.53 Q454.14,219.03 462.64,210.53 M450.17,227.8 Q458.39,219.56 466.63,211.35 M455.21,227.57 Q463.18,219.59 471.15,211.63 M459.91,227.68 Q467.68,219.9 475.45,212.14 M466.74,225.66 Q473.19,219.25 479.6,212.8 M475.17,222.04 Q479.66,217.54 484.17,213.04 M483.07,218.94 Q485.65,216.36 488.24,213.78 M490.83,216 Q491.57,215.25 492.31,214.51" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M329.39,227.88 Q395.76,228.53 462.14,227.88 M460.99,228.04 Q478.47,221.2 495.82,214.02 M495.66,214.37 Q456.37,209.45 417,205.19 M418.35,205.25 Q373.81,205.52 329.28,205.25 M330.42,205.19 Q299.83,209.73 269.26,214.42 M269.38,214.17 Q299.88,221.06 330.37,227.97" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" 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fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><path d="M349.55,108.55 Q422.22,161.57 495.11,214.31" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><path d="M385.69,214.31 Q443.96,214.25 502.23,214.31" fill="none" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><circle cx="495.11" cy="214.31" r="2.2" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><path d="M229.55,108.55 Q302.28,161.49 375.11,214.31" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><path d="M265.21,214.31 Q323.47,214.22 381.73,214.31" fill="none" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><circle cx="375.11" cy="214.31" r="2.2" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><path d="M470.84,196.67 A30,30 0 0 0 465.11,214.31" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="445.66" y="202.24" font-size="12" style="--fs:12" fill="light-dark(#2479ad, #8bcfff)" text-anchor="middle" font-weight="650">ALT 36°</text><text x="720" y="110" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">Rays keep their true angle (ALT) and stay</text><text x="720" y="127" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">parallel. Shadows of vertical edges are</text><text x="720" y="144" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">horizontal. The quickest construction.</text><path d="M30,338 Q500,338 970,338" fill="none" stroke="light-dark(#ccdce7, #3f5262)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/></g><g id="skx-case-b"><text x="30" y="378" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" font-weight="700" letter-spacing="1.2">B · SUN BEHIND THE VIEWER</text><path d="M30,460 Q345,460.35 660,460" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text x="34" y="454" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)">HL · eye height</text><circle cx="630" cy="460" r="2.2" fill="light-dark(#607181, #aabccb)" stroke="none"/><text x="630" y="476" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">VPR</text><path d="M630,460 Q480.03,494.07 330,527.88" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-linecap="round" 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stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M458.57,515.41 Q458.83,489.44 458.57,463.46" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><circle cx="491.47" cy="500.61" r="2.2" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><path d="M581.73,460 Q581.84,531.27 581.73,602.54" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-dasharray="3 3" stroke-linecap="round" stroke-linejoin="round"/><circle cx="581.73" cy="460" r="3" fill="light-dark(#007c9a, #6bd8f0)" stroke="none"/><circle cx="581.73" cy="602.54" r="3" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="589.73" y="453" font-size="12" style="--fs:12" fill="light-dark(#007c9a, #6bd8f0)" font-weight="700">VPsh</text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="589.73" y="616.54" font-size="12" style="--fs:12" fill="light-dark(#2479ad, #8bcfff)" font-weight="700">VPs</text><text x="720" y="460" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">Lit faces toward you; shadows fall away and</text><text x="720" y="477" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">are partly hidden. Rays converge to VPs below</text><text x="720" y="494" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">the horizon; shadows of verticals converge to</text><text x="720" y="511" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">VPsh on it, vertically above VPs.</text><path d="M30,688 Q500,688 970,688" fill="none" stroke="light-dark(#ccdce7, #3f5262)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/></g><g id="skx-case-c"><text x="30" y="728" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" font-weight="700" letter-spacing="1.2">C · SUN IN FRONT OF THE VIEWER (BACKLIGHT)</text><path d="M30,900 Q345,900.09 660,900" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text x="34" y="894" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)">HL · eye height</text><circle cx="630" cy="900" r="2.2" fill="light-dark(#607181, #aabccb)" stroke="none"/><text x="630" y="916" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">VPR</text><path d="M630,900 Q480.04,919.41 330,938.18" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M630,900 Q479.94,871.68 330,842.72" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><circle cx="30" cy="900" r="2.2" fill="light-dark(#607181, #aabccb)" stroke="none"/><text x="30" y="916" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">VPL</text><path d="M30,900 Q179.96,919.37 330,938.18" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M30,900 Q180.04,871.58 330,842.72" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><g 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d="M444.52,899.04 Q284.4,927.71 124.36,956.84" fill="none" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><circle cx="131.64" cy="955.53" r="2.2" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><path d="M439.19,900 Q439.21,815.12 439.19,730.25" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-dasharray="3 3" stroke-linecap="round" stroke-linejoin="round"/><circle cx="439.19" cy="900" r="3" fill="light-dark(#007c9a, #6bd8f0)" stroke="none"/><circle cx="439.19" cy="730.25" r="3" fill="light-dark(#2479ad, #8bcfff)" stroke="none"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="447.19" y="893" font-size="12" style="--fs:12" fill="light-dark(#007c9a, #6bd8f0)" font-weight="700">VPsh</text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="447.19" y="724.25" font-size="12" style="--fs:12" fill="light-dark(#2479ad, #8bcfff)" font-weight="700">VPs = sun</text><circle cx="439.19" cy="730.25" r="9" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text x="720" y="810" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">Faces toward you are in shade; shadows</text><text x="720" y="827" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">come toward you. VPs is the sun itself,</text><text x="720" y="844" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)">above the horizon; VPsh is directly below it.</text></g></g></svg></div><figcaption><span><strong>Figure 7.</strong> Shadows in perspective · Hand drawing</span><a href="./assets/diagrams/solar/sketch-perspective.svg" download>Editable SVG ↓</a></figcaption></figure>
<!-- FIG:sketch-perspective_END -->
<p class="measure">When the sun is parallel to the picture plane, there are no sun vanishing points. Rays are drawn parallel at their true altitude and the shadows of vertical edges are horizontal. This is the fastest case for a sketch. With the sun behind you, the visible faces are lit and shadows fall away; with the sun in front, the visible faces are in shade and shadows come toward you.</p>
</section>
<section id="technique" class="reader-section">
<div class="section-heading"><span class="section-number">08</span><div><p class="eyebrow">FINE-LINER</p><h2>Rendering with a fine-liner</h2></div></div>
<ul class="facts">
<li><strong>Pencil first.</strong> Draw rays, bearing lines and vanishing points lightly in pencil or non-photo blue. Ink only the outlines and the hatching, then erase.</li>
<li><strong>Line hierarchy.</strong> 0.7 for the ground line and cut elements, 0.5 for outlines against the background, 0.3 for edges between visible faces, 0.1 for hatching.</li>
<li><strong>Value order.</strong> Shadow darkest, shade lighter, lit faces white. Hatch shade with the same pen at about twice the spacing used for shadow. Glass and deep voids can go darker still.</li>
<li><strong>One direction.</strong> Hatch every shadow on the sheet in one direction, usually 45°, at constant spacing (about 1 mm at 1:100). Mixed directions read as different materials.</li>
<li><strong>Edges from strokes.</strong> Start each stroke on the shadow edge so the edge reads crisply. If an outline is needed, use the 0.1 pen. A heavy outline makes the shadow read as an object.</li>
</ul>
<!-- FIG:sketch-technique_START -->
<figure class="fig" id="fig-sketch-technique"><div class="fig-plate"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 1000 530" class="diagram" role="img" aria-label="Fine-liner technique for shade and shadow"><title>Fine-liner technique for shade and shadow</title><defs><marker id="skt-arrow-ink" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5"/></marker><marker id="skt-arrow-alt" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5"/></marker><marker id="skt-arrow-azi" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1.5"/></marker><marker id="skt-arrow-third" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1.5"/></marker><marker id="skt-arrow-muted" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#607181, #aabccb)" stroke-width="1.5"/></marker><marker id="skt-arrow-red" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#b13d4c, #ff9daa)" stroke-width="1.5"/></marker></defs><g font-family="Helvetica Neue, Helvetica, Arial, sans-serif"><text x="30" y="34" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" font-weight="700" letter-spacing="1.2">PEN WEIGHTS</text><path d="M30,64 Q100,64.39 170,64" fill="none" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><text x="186" y="68" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="650">pencil / 0.05</text><text x="270" y="68" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">construction: rays, bearings, VPs</text><path d="M30,90 Q100,89.76 170,90" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><text x="186" y="94" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="650">0.1</text><text x="270" y="94" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">hatching, shadow outlines, glazing</text><path d="M30,116 Q100,116.49 170,116" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text x="186" y="120" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="650">0.3</text><text x="270" y="120" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">edges between visible faces</text><path d="M30,142 Q100,142.36 170,142" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><text x="186" y="146" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="650">0.5</text><text x="270" y="146" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">outlines against the background</text><path d="M30,168 Q100,168.34 170,168" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><text x="186" y="172" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="650">0.7</text><text x="270" y="172" font-size="12" style="--fs:12" 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783.81,344.33" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M783.28,343.1 Q797.94,353.66 812.51,364.34 M810.25,360.63 Q810.28,365.21 810.25,369.79 M807.83,363.38 Q853.61,350.02 899.59,337.35" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><path d="M813.46,370.1 Q797.35,358.54 781.38,346.79 M784.64,351.46 Q784.63,345.79 784.64,340.12 M807.51,368.54 Q853.28,355.88 898.88,342.62 M897.04,347.23 Q897.02,340.81 897.04,334.4 M899.59,339.93 Q884.76,329.17 869.95,318.39 M875.2,318.4 Q828.53,332.13 781.6,344.95" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><text x="30" y="470" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">One hatch direction across the whole drawing.</text><text x="30" y="486" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">Shadow darker than shade; light faces stay white.</text><text x="30" y="502" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">Crisp shadow edges come from where strokes stop.</text><text x="520" y="470" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">Heavy outlines around shadows read as objects.</text><text x="520" y="486" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">Mixed hatch directions and a shade darker than</text><text x="520" y="502" font-size="12" style="--fs:12" fill="light-dark(#607181, #aabccb)">the shadow flatten the form.</text></g></svg></div><figcaption><span><strong>Figure 8.</strong> Fine-liner technique · Hand drawing</span><a href="./assets/diagrams/solar/sketch-technique.svg" download>Editable SVG ↓</a></figcaption></figure>
<!-- FIG:sketch-technique_END -->
<div class="callout"><span>CHECK THE SHEET</span>
<p class="measure">Do all shadows of vertical edges point the same way? Are their lengths proportional to the heights? Does every object that stands on the ground have a shadow that starts at its base? Is every shadow on a lit surface, never on one in shade? Are the shadows of horizontal edges parallel to those edges? Is the date and time, or "45° convention", written on the drawing?</p>
</div>
</section>
<section id="quiz" class="reader-section quiz-section">
<div class="section-heading"><span class="section-number">09</span><div><p class="eyebrow">8 QUESTIONS · CONSTRUCTION AND CONVENTION</p><h2>Mini quiz</h2></div></div>
<p class="section-lead">Check the rules before you draw. Calculation questions take a number; "New values" generates a new variant. Each answer shows the construction or equation behind it.</p>
<form id="quiz-form" novalidate><div id="quiz-questions"></div><div class="quiz-actions"><button class="button" type="submit">Check all answers</button><button id="reset-quiz" class="quiet-button" type="button">Clear</button><p id="quiz-score" role="status"></p></div></form>
<p class="caption">Practice only. Responses stay in this browser.</p>
</section>
<section id="figures" class="reader-section references-section">
<p class="eyebrow">EDITABLE VECTOR FILES</p><h2>Figure files</h2>
<p class="measure">Each figure is available as SVG with plain colors, named groups and live text, for editing in Illustrator, Inkscape or Keynote. The files are generated from the same code as the page (<code>./reader generate:solar</code>).</p>
<!-- FIGLIST_START -->
<ol class="fig-list"><li><a href="./assets/diagrams/solar/sketch-vocabulary.svg" download>sketch-vocabulary.svg</a> <span>· Fig. 1 · Light, shade and shadow · Hand drawing</span></li><li><a href="./assets/diagrams/solar/sketch-rules.svg" download>sketch-rules.svg</a> <span>· Fig. 2 · Four rules for casting shadows · Hand drawing</span></li><li><a href="./assets/diagrams/solar/sketch-convention.svg" download>sketch-convention.svg</a> <span>· Fig. 3 · The 45° convention · Hand drawing</span></li><li><a href="./assets/diagrams/solar/sketch-plan.svg" download>sketch-plan.svg</a> <span>· Fig. 4 · Plan shadows at a real date and time, with a shadow scale · Hand drawing</span></li><li><a href="./assets/diagrams/solar/sketch-elevation.svg" download>sketch-elevation.svg</a> <span>· Fig. 5 · Elevation shadows from plan and section · Hand drawing</span></li><li><a href="./assets/diagrams/solar/sketch-axo.svg" download>sketch-axo.svg</a> <span>· Fig. 6 · Axonometric shadow construction · Hand drawing</span></li><li><a href="./assets/diagrams/solar/sketch-perspective.svg" download>sketch-perspective.svg</a> <span>· Fig. 7 · Shadows in perspective · Hand drawing</span></li><li><a href="./assets/diagrams/solar/sketch-technique.svg" download>sketch-technique.svg</a> <span>· Fig. 8 · Fine-liner technique · Hand drawing</span></li></ol>
<!-- FIGLIST_END -->
</section>
<section id="references" class="reader-section references-section">
<p class="eyebrow">SOURCES</p><h2>Sources and model notes</h2>
<ol class="source-list">
<li>S. V. Szokolay, <em>Introduction to Architectural Science</em>, 3rd ed., Routledge, 2014. Sections 1.3.1, 1.4.1; Method sheets M.1.1 (solar geometry), M.1.2 (stereographic charts, shadow-angle protractor).<span>Abbreviations, Eq. 7–9, 16–18, 21; device masks.</span></li>
<li>N. Lechner, <em>Heating, Cooling, Lighting</em>, 4th ed., Wiley, 2015. Chapter 6 (solar geometry), 9.5–9.9 (shading periods, overhangs).<span>Solar window, sunbeam drawing, test dates, heliodons and sundials.</span></li>
</ol>
<details class="math-details"><summary>Model assumptions</summary><p>The figures assume parallel rays, opaque objects and level ground. Shadows are computed in 3D from the sun position of Chapter 02 (Eq. 7–8) and projected into each view. The shadow-factor table uses Ithaca, 42.4° N, with geometric altitude (no refraction); entries are omitted when ALT is below 3°.</p>
</details>
<p><a class="text-link" href="./sun-paths-shading.html">Continue to Chapter 04: Sun paths and shading ↗</a></p>
<div class="reader-end"><span>ES1 / ENVIRONMENTAL SYSTEMS I</span>
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</div>
</section>
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