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601 lines
15 KiB
HTML
601 lines
15 KiB
HTML
<!DOCTYPE html>
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.det {
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border: none;
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border-right: 2px solid black;
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display: inline-grid;
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align-content: space-evenly;
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}
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.txt {
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display: inline-block;
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padding: 3px 3px;
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font: 23px Verdana, Geneva, sans-serif;
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color: goldenrod;
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text-align: center;
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min-width: 25px;
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 class="center">Transformations and Matrices</h1>
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<p>A <a href="matrix-introduction.html">matrix</a> can do geometric transformations!</p>
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<p>Have a play with this 2D transformation app:</p>
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<script src="images/matrix-transform.js" type="text/javascript"></script>
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<script type="text/javascript">matrixtransformMain();</script>
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<p> </p>
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<p>Matrices can also transform from 3D to 2D (very useful for computer graphics), do 3D transformations and much much more.</p>
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<!-- css grid
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<div class="matG">
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<div>item 1</div>
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<div>item 2</div>
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<div>3</div>
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<div>long item 4</div>
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</div>
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-->
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<h2>The Mathematics</h2>
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<p>For each [x,y] point that makes up the shape we do this <a href="matrix-multiplying.html">matrix multiplication</a>:</p>
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<div style="text-align: center;">
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<div class="mat">
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<div class="cols2">
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<div>a</div>
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<div>b</div>
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<div>c</div>
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<div>d</div>
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</div>
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</div>
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<div class="mat">
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<div class="cols1">
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<div>x</div>
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<div>y</div>
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</div>
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</div>
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<div class="txt">=</div>
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<div class="mat">
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<div class="cols1">
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<div>ax + by</div>
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<div>cx + dy</div>
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</div>
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</div>
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</div>
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<!-- [a,b~c,d][x~y] = [ax + by~cx + dy] -->
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<!-- [a,b~c,d][x~y] = [ax + by~cx + dy] -->
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<p>When the transformation matrix [a,b,c,d] is the <a href="matrix-types.html">Identity Matrix</a> (the matrix equivalent of "1") the [x,y] values are not changed:</p>
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<div style="text-align: center;">
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<div class="mat">
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<div class="cols2">
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<div>1</div>
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<div>0</div>
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<div>0</div>
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<div>1</div>
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</div>
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</div>
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<div class="mat">
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<div class="cols1">
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<div>x</div>
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<div>y</div>
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</div>
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</div>
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<div class="txt">=</div>
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<div class="mat">
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<div class="cols1">
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<div>1x + 0y</div>
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<div>0x + 1y</div>
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</div>
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</div>
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<div class="txt">=</div>
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<div class="mat">
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<div class="cols1">
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<div>x</div>
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<div>y</div>
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</div>
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</div>
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</div>
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<!-- [1,0~0,1][x~y] = [1x + 0y~0x + 1y] = [x~y] -->
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<!-- [1,0~0,1][x~y] = [1x + 0y~0x + 1y] = [x~y] -->
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<p>Changing the "b" value leads to a "shear" transformation (try it above):</p>
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<div style="text-align: center;">
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<div class="mat">
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<div class="cols2">
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<div>1</div>
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<div>0.8</div>
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<div>0</div>
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<div>1</div>
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</div>
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</div>
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<div class="mat">
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<div class="cols1">
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<div>x</div>
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<div>y</div>
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</div>
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</div>
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<div class="txt">=</div>
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<div class="mat">
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<div class="cols1">
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<div>1x + 0.8y</div>
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<div>0x + 1y</div>
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</div>
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</div>
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<div class="txt">=</div>
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<div class="mat">
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<div class="cols1">
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<div>x+0.8y</div>
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<div>y</div>
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</div>
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</div>
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<div class="txt"></div>
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</div>
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<!-- [1,0.8~0,1][x~y] = [1x + 0.8y~0x + 1y] = [x+0.8y~y] -->
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<p>And this one will do a diagonal "flip" about the x=y line (try it also):</p>
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<div style="text-align: center;">
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<div class="mat">
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<div class="cols2">
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<div>0</div>
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||
<div>1</div>
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<div>1</div>
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<div>0</div>
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</div>
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</div>
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<div class="mat">
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<div class="cols1">
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<div>x</div>
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<div>y</div>
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</div>
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</div>
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<div class="txt">=</div>
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<div class="mat">
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<div class="cols1">
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<div>0x + 1y</div>
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<div>1x + 0y</div>
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</div>
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</div>
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<div class="txt">=</div>
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<div class="mat">
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<div class="cols1">
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<div>y</div>
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<div>x</div>
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</div>
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</div>
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</div>
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<!-- [0,1~1,0][x~y] = [0x + 1y~1x + 0y] = [y~x] -->
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<!-- [0,1~1,0][x~y] = [0x + 1y~1x + 0y] = [y~x] -->
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<p>What more can you discover? </p>
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<h2>Many Transformations at Once</h2>
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<p>We can "chain" transformations by <a href="matrix-multiplying.html">multiplying matrices</a>. </p>
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<div class="example">
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<h3>Example: a diagonal flip followed by a horizontal shear:</h3>
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<p>Be careful! The transforms go right-to-left, so it is <b>shear × flip = "flip then shear"</b> </p>
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<div style="text-align: center;">
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<div class="mat">
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<div class="cols2">
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||
<div>1</div>
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||
<div>0.8</div>
|
||
<div>0</div>
|
||
<div>1</div>
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||
</div>
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||
</div>
|
||
<div class="mat">
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||
<div class="cols2">
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||
<div>0</div>
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||
<div>1</div>
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||
<div>1</div>
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||
<div>0</div>
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||
</div>
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||
</div>
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<div class="txt">=</div>
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||
<div class="mat">
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<div class="cols2">
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||
<div>0.8</div>
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||
<div>1</div>
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||
<div>1</div>
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<div>0</div>
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</div>
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</div>
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<div class="txt"></div>
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</div>
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<!-- [1,0.8~0,1][0,1~1,0] = [0.8,1~1,0] -->
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<p>And we can then use that result in a transform:</p>
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<div style="text-align: center;">
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<div class="mat">
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<div class="cols2">
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<div>0.8</div>
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<div>1</div>
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||
<div>1</div>
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||
<div>0</div>
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||
</div>
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||
</div>
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||
<div class="mat">
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<div class="cols1">
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<div>x</div>
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<div>y</div>
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</div>
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</div>
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<div class="txt">=</div>
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<div class="mat">
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<div class="cols1">
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<div>0.8x+y</div>
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<div>x</div>
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</div>
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</div>
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</div>
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<!-- [0.8,1~1,0][x~y] = [0.8x+y~x] -->
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<p>It does these two transforms in one go:</p>
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<p class="center"><img src="images/transform-flip.gif" alt="First transform" height="156" width="160"> <span class="large">then</span> <img src="images/transform-flip-shear.gif" alt="Second transform" height="153" width="220"></p>
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</div>
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Now what if we <b>change the order</b> of those two transformations?
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<div class="example">
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<h3>Example: a horizontal shear followed by a diagonal flip:</h3>
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<p>Remember the order goes <b>flip × shear = "shear then flip"</b></p>
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<div style="text-align: center;">
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<div class="mat">
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<div class="cols2">
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<div>0</div>
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<div>1</div>
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<div>1</div>
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<div>0</div>
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</div>
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</div>
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<div class="mat">
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<div class="cols2">
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<div>1</div>
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<div>0.8</div>
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<div>0</div>
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<div>1</div>
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</div>
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</div>
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<div class="txt">=</div>
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<div class="mat">
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<div class="cols2">
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<div>0</div>
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<div>1</div>
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<div>1</div>
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<div>0.8</div>
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</div>
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</div>
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<div class="txt"></div>
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</div>
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<!-- [0,1~1,0][1,0.8~0,1] = [0,1~1,0.8] -->
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<p>It is a different result!</p>
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<div style="text-align: center;">
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<div class="mat">
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<div class="cols2">
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<div>0</div>
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||
<div>1</div>
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<div>1</div>
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<div>0.8</div>
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</div>
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</div>
|
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<div class="mat">
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<div class="cols1">
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<div>x</div>
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<div>y</div>
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</div>
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</div>
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<div class="txt">=</div>
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<div class="mat">
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<div class="cols1">
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<div>y</div>
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<div>x+0.8y</div>
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</div>
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</div>
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</div>
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<!-- [0,1~1,0.8][x~y] = [y~x+0.8y] -->
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<p>It does these two transforms in one go:</p>
|
||
<p class="center"><img src="images/transform-shear.gif" alt="First transform" height="178" width="238"> <span class="large">then</span> <img src="images/transform-shear-flip.gif" alt="Second transform" height="246" width="190"></p>
|
||
</div>
|
||
<p>This stuff is powerful as we can do LOTS of transforms at once and really speed up calculations. VERY useful for computer graphics.</p>
|
||
<p> But we have to be careful what <b>order</b> we do the transforms in!</p>
|
||
<p>It also shows us why the <b>order of multiplying matrices</b> is important (unlike ordinary numbers which can be mulitiplied in any order, example 2×3=3×2).</p>
|
||
|
||
<h2>Transforms In Code</h2>
|
||
<p>Need to code this yourself? Here is how.</p>
|
||
<p>The letter F is just a list of coordinates:</p>
|
||
<pre>[3, 4], [3, 5], [0, 5], [0, 0], [1, 0], [1, 1.8], [2.5, 1.8], [2.5, 2.8], [1, 2.8], [1, 4]</pre>
|
||
<p>And we loop through those points, making new points using the 2×2 matrix "a,b,c,d":</p>
|
||
<pre> for (let i = 0; i < shape.pts.length; i++) {
|
||
let pt = shape.pts[i]
|
||
let x = a * pt[0] + b * pt[1]
|
||
let y = c * pt[0] + d * pt[1]
|
||
newPts.push({ x: x, y: y })
|
||
}
|
||
</pre>
|
||
<p>We then plot the original points and the transformed points so we can see both!</p>
|
||
<h2>Rotation</h2>
|
||
<p>This matrix does a rotation of <b>θ</b> about the origin (0,0):</p>
|
||
<div style="text-align: center;">
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div>cos(θ)</div>
|
||
<div>−sin(θ)</div>
|
||
<div>sin(θ)</div>
|
||
<div>cos(θ)</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- [cos(θ),-sin(θ)~sin(θ),cos(θ)] = 0 -->
|
||
<div class="example">
|
||
<h3>Example: Rotate by 30°</h3>
|
||
<p>Calculating to 4 decimal places:</p>
|
||
<div style="text-align: center;">
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div>cos(30°)</div>
|
||
<div>−sin(30°)</div>
|
||
<div>sin(30°)</div>
|
||
<div>cos(30°)</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div>0.8660</div>
|
||
<div>−0.5</div>
|
||
<div>0.5</div>
|
||
<div>0.8660</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- [cos(30°),-sin(30°)~sin(30°),cos(30°)] = [0.8660,-0.5~0.5,0.8660] -->
|
||
|
||
<!-- [cos(θ),-sin(θ)~sin(θ),cos(θ)] = 0 -->
|
||
<p>Try it in the app at top! </p>
|
||
<p>And let's try it on a point here:</p>
|
||
|
||
<div style="text-align: center;">
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div>0.8660</div>
|
||
<div>−0.5</div>
|
||
<div>0.5</div>
|
||
<div>0.8660</div>
|
||
</div>
|
||
</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div>3</div>
|
||
<div>1</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div>0.8660×3−0.5×1</div>
|
||
<div>0.5×3+0.8660×1</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div>2.098</div>
|
||
<div>2.366</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt"></div>
|
||
</div>
|
||
<!-- [0.8660,-0.5~0.5,0.8660][3~1] = [0.8660*3-0.5*1~0.5*3+0.8660*1] = [2.098~2.366] -->
|
||
|
||
|
||
<p>And we get this:</p>
|
||
<p class="center"><img src="images/matrix-trans-30deg.svg" alt="matrix transform 30deg"></p>
|
||
|
||
|
||
</div>
|
||
<p> </p>
|
||
<p> </p>
|
||
<div class="fun">
|
||
<h3>Footnote: Swapping Rows and Columns</h3>
|
||
<p>Matrices are flexible!</p>
|
||
<p>This shear transformation:</p>
|
||
|
||
<div style="text-align: center;">
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div>1</div>
|
||
<div>0.8</div>
|
||
<div>0</div>
|
||
<div>1</div>
|
||
</div>
|
||
</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div>x</div>
|
||
<div>y</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div>1x + 0.8y</div>
|
||
<div>0x + 1y</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols1">
|
||
<div>x+0.8y</div>
|
||
<div>y</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- [1,0.8~0,1][x~y] = [1x + 0.8y~0x + 1y] = [x+0.8y~y] -->
|
||
|
||
|
||
<p><i>Can</i> be done like this:</p>
|
||
<div style="text-align: center;">
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div>x</div>
|
||
<div>y</div>
|
||
</div>
|
||
</div>
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div>1</div>
|
||
<div>0</div>
|
||
<div>0.8</div>
|
||
<div>1</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div>1x + 0.8y</div>
|
||
<div>0x + 1y</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div>x + 0.8y</div>
|
||
<div>y</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- [x,y][1,0~0.8,1] = [1x + 0.8y,0x + 1y] = [x + 0.8y,y] -->
|
||
|
||
<p>The rows and columns are all swapped over (<a href="matrix-types.html">transposed</a>), and the order of multiplication is reversed, but it still works. Just so you know.</p>
|
||
</div>
|
||
|
||
|
||
<p> </p>
|
||
<div class="questions">
|
||
<script>getQ(9289, 9290, 9291, 9292, 9293, 9294, 9295, 9296, 9297, 9298, 15141, 15142, 15143, 15144, 15145, 15146, 17800, 17801, 17802, 17803);</script>
|
||
</div>
|
||
|
||
<div class="related">
|
||
<a href="matrix-introduction.html">Introduction to Matrices</a> <a href="matrix-multiplying.html">Matrix Multiplication</a> <a href="matrix-calculator.html">Matrix Calculator</a> <a href="index-2.html">Algebra 2 Index</a>
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