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127 lines
6.6 KiB
HTML
127 lines
6.6 KiB
HTML
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<title>Steradian</title>
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<h1 align="center">Steradian</h1>
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<p align="center"><b>A steradian is used to measure "solid angles"</b></p>
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<p>A steradian is related to the surface area of a sphere <br />
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in the same way a <a href="radians.html">radian</a> is related to the circumference of a circle:</p>
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<table border="0">
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<tr>
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<td align="right">A <b>Radian</b> "cuts out" a length of a circle's circumference equal to the radius.</td>
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<td align="right"> </td>
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<td align="center"><img src="images/radian-circle.svg" alt="radian definition" /></td>
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</tr>
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<tr>
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<td align="right">A <b>Steradian</b> "cuts out" an area of a sphere<br />
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equal to (radius)<sup>2</sup>.</td>
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<td align="right"> </td>
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<td align="center"><img src="images/steradian-definition.svg" alt="steradian area = r^2" /></td>
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</tr>
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</table>
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<p> </p>
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<div class="words"> The SI Unit abbreviation is <b>sr</b></div><div class="words"> The name steradian is made up from the Greek <i>stereos</i> for "solid" and radian. </div>
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<h2>Sphere vs Steradian</h2>
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<ul>
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<li>The surface area of a <a href="sphere.html">sphere</a> is 4<font face="Times New Roman, Times, serif"><font size="+1">π</font></font>r<sup>2</sup>,</li>
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<li>The surface area of a steradian is just r<sup>2.</sup></li>
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</ul>
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<p>So a sphere measures 4<font face="Times New Roman, Times, serif" size="+1">π</font> steradians, or about 12.57 steradians. Likewise a steradian is 1/12.57, or about 8% of a sphere.</p>
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<p>And because we measure an angle, it doesn't matter what size the sphere is, it will always measure 4<font face="Times New Roman, Times, serif" size="+1">π</font> steradians.</p>
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<div class="example">
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<h3 style="float:right; margin: 0 0 5px 10px;"><img src="images/steradian-4pi.svg" alt="sphere area = 4pi, steradian area = 1" /></h3>
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<h3><b>Example:</b> The "unit sphere":</h3>
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<ul>
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<li>has a radius of 1</li>
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<li>has a surface area of 4<font face="Times New Roman, Times, serif"><font size="+1">π</font></font>, </li>
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<li>a steradian "cuts out" an area of <b>1</b>.</li>
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</ul><p> </p>
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</div>
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<h2>Radiant Intensity</h2>
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<p>Radiant intensity (how brightly something shines) can be measured in watts per steradian (W/sr).</p>
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<div class="example">
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<h3><b>Example:</b> You measure the light coming from a powerful globe. <br>
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<br>
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Your sensor is 50mm × 50mm in size, and if you hold it 2m away it measures 0.1 Watts. <br>
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<br>
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What is the radiant intensity in <b>W/sr</b> (Watts per steradian)?</h3>
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<div class="indent50px">
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<p><b>Answer</b>: At 2m, one steradian cuts through 2×2 = 4 m<sup>2</sup> of the sphere.</p>
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<p>And because the sensor is relatively small, its flat surface area is approximately the area of sphere that it occupies. So 0.05 × 0.05 = 0.0025m<sup>2</sup>.</p>
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<p>So, one steradian receives about 0.1 W × (4m<sup>2</sup>/0.0025m<sup>2</sup>) = <b>160 W/sr</b>.</p>
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</div>
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</div>
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<h2>In Degrees </h2>
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<p>Because we can convert from radians to degrees we can also convert from steradians to "square degrees":</p>
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<div class="indent50px">
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<p>A radian is 180/<font face="Times New Roman, Times, serif" size="+1">π </font>degrees, or about 57.296°.</p>
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<p>A steradian is (180/<font face="Times New Roman, Times, serif" size="+1">π</font>)<sup>2</sup> square degrees or about 3282.8 square degrees.</p>
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</div><p> </p>
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<div class="related">
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<a href="radians.html">Radian</a> <a href="index.html">Geometry Index</a></div>
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