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added sliders to sketches that should have one, improved lazy loading

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Pomax
2020-08-21 23:39:36 -07:00
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docs/chapters/explanation/content.en-GB.md
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@@ -39,9 +39,11 @@ Multiple functions, but only one variable. If we change the value for <i>t</i>,
There we go. <i>x</i>/<i>y</i> coordinates, linked through some mystery value <i>t</i>.
So, parametric curves don't define a <i>y</i> coordinate in terms of an <i>x</i> coordinate, like normal functions do, but they instead link the values to a "control" variable. If we vary the value of <i>t</i>, then with every change we get <strong>two</strong> values, which we can use as (<i>x</i>,<i>y</i>) coordinates in a graph. The above set of functions, for instance, generates points on a circle: We can range <i>t</i> from negative to positive infinity, and the resulting (<i>x</i>,<i>y</i>) coordinates will always lie on a circle with radius 1 around the origin (0,0). If we plot it for <i>t</i> from 0 to 5, we get this (use your up and down arrow keys to change the plot end value):
So, parametric curves don't define a <i>y</i> coordinate in terms of an <i>x</i> coordinate, like normal functions do, but they instead link the values to a "control" variable. If we vary the value of <i>t</i>, then with every change we get <strong>two</strong> values, which we can use as (<i>x</i>,<i>y</i>) coordinates in a graph. The above set of functions, for instance, generates points on a circle: We can range <i>t</i> from negative to positive infinity, and the resulting (<i>x</i>,<i>y</i>) coordinates will always lie on a circle with radius 1 around the origin (0,0). If we plot it for <i>t</i> from 0 to 5, we get this:
<graphics-element title="A (partial) circle: x=sin(t), y=cos(t)" src="./circle.js"></graphics-element>
<graphics-element title="A (partial) circle: x=sin(t), y=cos(t)" src="./circle.js">
<input type="range" min="0" max="10" step="0.1" value="5" class="slide-control">
</graphics-element>
Bézier curves are just one out of the many classes of parametric functions, and are characterised by using the same base function for all of the output values. In the example we saw above, the <i>x</i> and <i>y</i> values were generated by different functions (one uses a sine, the other a cosine); but Bézier curves use the "binomial polynomial" for both the <i>x</i> and <i>y</i> outputs. So what are binomial polynomials?