47 lines
2.8 KiB
HTML
47 lines
2.8 KiB
HTML
<!doctype html>
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<title>Townengine Wiki : Trigonometry</title>
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<link rel="stylesheet" href="style.css">
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<h1 style="margin-bottom:0in">G1. Trigonometry<span style="float:right">{twn}</span></h1>
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<a href="index.html">Go back</a>
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<blockquote>
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<p>Truth is, most games are played in continuous space, where geometry matters.
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Math classes turn out to be not so useless after all.
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But even if you don't know your numbers well, you can compose functions by their meaning, and not number logic.
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This page will try to help you with that, providing examples.
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</blockquote>
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<p><strong>the Hell are Vectors</strong>
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<blockquote>
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<p>There are distinct meanings they posses: <i>position in space</i>, <i>travel between points</i> and <i>direction</i>.
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Often you need to juggle those representation around.
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<p>For example, player and enemy positions by themselves don't tell you anything about their spatial relation.
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For that you need to compute the <i>difference between them</i>, which is plain per component subtraction.
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Resulted direction of travel depends on order of inputs, - it will point to position on the right side, the one you subtract with, but not from.
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<p>It's important to understand that after subtraction the travel vector doesn't have relation with original positions,
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it is effectively in a <i>subspace</i>, where (0,0,0) is at position you subtracted with to get it.
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Operation is reversible by means of addition, resulting in position in space once again.
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<p>I often forget what the direction is pointed at but here's a quick way to remind yourself: take two numbers, 10 and 0, and try subtracting them in different order.
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Sign of result will tell you which part is pointed at, +10 -> direction is to the 10, -10 -> direction is to the 0. (Draw it in your head)
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<p>Sometimes direction is more important than the travel, for which you need to drop the length of it.
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This is called <i>normalization</i>, where result is called a <i>normal</i> (You probably heard those terms).
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This is achieved by dividing every component by the length (magnitude) of vector itself.
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Length is computed as <i>sqrt(x*x, y*y, z*z)</i>. This effectively creates a vector of length 1.
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As every direction sized the same, when graphed they trace a circle.
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</blockquote>
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<p><strong>Center between two points</strong>
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<blockquote>
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<p><b>Method one. </b>Add two positions and divide them by two: (a + b) / 2. This is cheap, but harder to remember.
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<p><b>Method two. </b>Find the travel between points, divide it by two and then translate the origin: (a + (a - b) / 2)
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</blockquote>
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</body>
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</html>
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