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<!DOCTYPE html><html><head><meta http-equiv=Content-Type content="text/html; charset=utf-8"><title>H3DU.Hypotrochoid</title><meta name="viewport" content="width=device-width"><link rel=stylesheet type="text/css" href="/style.css"></head><body> <div class="header">
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<h1>H3DU.Hypotrochoid</h1>
<p><a href="index.html">Back to documentation index.</a></p>
<p><a name='H3DU.Hypotrochoid'></a></p>
<h3>H3DU.Hypotrochoid(outerRadius, innerRadius, distFromInnerCenter, [rotationDegrees])</h3>
<p><strong>Augments:</strong> <a href="H3DU.Curve.html">H3DU.Curve</a></p>
<p>A <a href="H3DU.Curve.html">curve evaluator object</a> for a curve drawn by a circle that rolls along the inside
of another circle, whose position is fixed, with a center of (0,0).</p>
<p>The following curves can be generated with this class (in the following
descriptions, O = <code>outerRadius</code>, R means <code>innerRadius</code>,
and D = <code>distFromInnerCenter</code>).<ul>
<li>Hypocycloid: D = R (hypotrochoid touching the fixed circle).</li>
<li>Curtate hypocycloid: D < R (hypotrochoid not touching the fixed circle).</li>
<li>Prolate hypocycloid: D > R (hypotrochoid crossing the fixed circle).</li>
<li>Circle: O = R*2; the circle will have radius R - D.</li>
<li>Ellipse: O = R*2; the ellipse (unrotated) will have width abs(R+D)*2
and height abs(R-D)*2.</li>
<li>Line segment with length O*2: O = R*2; D = R.</li>
<li>Deltoid: O = R*3; D = R.</li>
<li>Astroid: O = R*4; D = R.</li>
<li>N-pointed hypocycloid: O = R * N; D = R.</li></ul></p>
<p>This class is considered a supplementary class to the
Public Domain HTML 3D Library and is not considered part of that
library.</p>
<p>To use this class, you must include the script "extras/evaluators.js"; the
class is not included in the "h3du_min.js" file which makes up
the HTML 3D Library. Example:</p>
<pre><script type="text/javascript" src="extras/evaluators.js"></script>
</pre>
<h4>Parameters</h4>
<ul>
<li><code>outerRadius</code> (Type: number)<br>Radius of the circle whose position is fixed.</li>
<li><code>innerRadius</code> (Type: number)<br>Radius of the rolling circle. A hypocycloid results when distFromInnerCenter=innerRadius.</li>
<li><code>distFromInnerCenter</code> (Type: number)<br>Distance from the center of the rolling circle to the drawing pen.</li>
<li><code>rotationDegrees</code> (Type: number) (optional)<br>Starting angle of the curve from the positive x-axis toward the positive y-axis, in degrees. Default is 0.</li>
</ul>
<h3>Methods</h3>
<ul>
<li><a href="#H3DU.Hypotrochoid_accel">accel</a><br>Finds an approximate acceleration vector at the specified u-coordinate of this curve.</li>
<li><a href="#H3DU.Hypotrochoid_arcLength">arcLength</a><br>Finds an approximate arc length (distance) between the start of this
curve and the point at the specified u-coordinate of this curve.</li>
<li><a href="#H3DU.Hypotrochoid_changeEnds">changeEnds</a><br>Creates a curve evaluator object for a curve that is generated using
the same formula as this one (and uses the same u-coordinates),
but has a different set of end points.</li>
<li><a href="#H3DU.Hypotrochoid_endPoints">endPoints</a><br>Gets the endpoints of this curve.</li>
<li><a href="#H3DU.Hypotrochoid_evaluate">evaluate</a><br>Finds the coordinates of a point on the curve from the specified u-coordinate.</li>
<li><a href="#H3DU.Hypotrochoid_fitRange">fitRange</a><br>Creates a curve evaluator object for a curve that follows the same
path as this one but has its u-coordinates remapped to fit the specified range.</li>
<li><a href="#H3DU.Hypotrochoid_getLength">getLength</a><br>Convenience method for getting the total length of this curve.</li>
<li><a href="#H3DU.Hypotrochoid_getPoints">getPoints</a><br>Gets an array of positions on the curve at fixed intervals
of u-coordinates.</li>
<li><a href="#H3DU.Hypotrochoid_getPointsAsObjects">getPointsAsObjects</a><br>Gets an array of positions on the curve at fixed intervals
of u-coordinates.</li>
<li><a href="#H3DU.Hypotrochoid_jerk">jerk</a><br>Finds an approximate jerk vector at the specified u-coordinate of this curve.</li>
<li><a href="#H3DU.Hypotrochoid_normal">normal</a><br>Finds an approximate principal normal vector at the specified u-coordinate of this curve.</li>
<li><a href="#H3DU.Hypotrochoid.rose">rose</a><br>Creates a <a href="H3DU.Curve.html">curve evaluator object</a> for a rose, a special
form of hypotrochoid.</li>
<li><a href="#H3DU.Hypotrochoid_scaleTo">scaleTo</a><br>Creates a modified version of this curve so that it
fits the specified radius.</li>
<li><a href="#H3DU.Hypotrochoid_tangent">tangent</a><br>Convenience method for finding an approximate tangent vector of this curve at the specified u-coordinate.</li>
<li><a href="#H3DU.Hypotrochoid_toArcLengthParam">toArcLengthParam</a><br>Creates a curve evaluator object for a curve that follows the same
path as this one but has its u-coordinates remapped to
an <i>arc length parameterization</i>.</li>
<li><a href="#H3DU.Hypotrochoid_velocity">velocity</a><br>Finds an approximate velocity vector at the specified u-coordinate of this curve.</li>
</ul>
<p><a name='H3DU.Hypotrochoid_accel'></a></p>
<h3>H3DU.Hypotrochoid#accel(u)</h3>
<p>Finds an approximate acceleration vector at the specified u-coordinate of this curve.
The implementation in <a href="H3DU.Curve.html">H3DU.Curve</a> calls the evaluator's <code>accel</code>
method if it implements it; otherwise, does a numerical differentiation using
the velocity vector.</p>
<p>The <b>acceleration</b> of a curve is a vector which is the second-order derivative of the curve's position at the specified coordinate. The vector returned by this method <i>should not</i> be "normalized" to a <a href="tutorial-glmath.html">unit vector</a>.</p>
<h4>Parameters</h4>
<ul>
<li><code>u</code> (Type: number)<br>u-coordinate of a point on the curve.</li>
</ul>
<h4>Return Value</h4>
<p>An array describing an acceleration vector. It should have at least as many
elements as the number of dimensions of the underlying curve. (Type: Array.<number>)</p>
<p><a name='H3DU.Hypotrochoid_arcLength'></a></p>
<h3>H3DU.Hypotrochoid#arcLength(u)</h3>
<p>Finds an approximate arc length (distance) between the start of this
curve and the point at the specified u-coordinate of this curve.</p>
<h4>Parameters</h4>
<ul>
<li><code>u</code> (Type: number)<br>u-coordinate of a point on the curve.</li>
</ul>
<h4>Return Value</h4>
<p>The approximate arc length of this curve at the specified u-coordinate. (Type: Array.<number>)</p>
<p><a name='H3DU.Hypotrochoid_changeEnds'></a></p>
<h3>H3DU.Hypotrochoid#changeEnds(ep1, ep2)</h3>
<p>Creates a curve evaluator object for a curve that is generated using
the same formula as this one (and uses the same u-coordinates),
but has a different set of end points.
For example, this method can be used to shrink the path of a curve
from [0, π] to [0, π/8].</p>
<p>Note, however, that in general, shrinking
the range of a curve will not shrink the length of a curve
in the same proportion, unless the curve's path runs at
constant speed with respect to time. For example, shrinking the range of a curve
from [0, 1] to [0, 0.5] will not generally result in a curve that's exactly half as
long as the original curve.</p>
<p>For some curves, this method can
also be used to grow the path of the curve.</p>
<h4>Parameters</h4>
<ul>
<li><code>ep1</code> (Type: number)<br>New start point of the curve.</li>
<li><code>ep2</code> (Type: number)<br>New end point of the curve.</li>
</ul>
<h4>Return Value</h4>
<p>Return value. (Type: <a href="H3DU.Curve.html">H3DU.Curve</a>)</p>
<p><a name='H3DU.Hypotrochoid_endPoints'></a></p>
<h3>H3DU.Hypotrochoid#endPoints()</h3>
<p>Gets the endpoints of this curve.
For this curve evaluator object, the curve
starts at 0 and ends at π*2.</p>
<h4>Return Value</h4>
<p>An array containing the two
endpoints of the curve. The first number is the start of the curve,
and the second number is the end of the curve. (Type: Array.<number>)</p>
<p><a name='H3DU.Hypotrochoid_evaluate'></a></p>
<h3>H3DU.Hypotrochoid#evaluate(u)</h3>
<p>Finds the coordinates of a point on the curve from the specified u-coordinate.</p>
<h4>Parameters</h4>
<ul>
<li><code>u</code> (Type: number)<br>u-coordinate.</li>
</ul>
<h4>Return Value</h4>
<p>A 3-element array specifying a 3D point.
Only the x- and y-coordinates can be other than 0. (Type: Array.<number>)</p>
<p><a name='H3DU.Hypotrochoid_fitRange'></a></p>
<h3>H3DU.Hypotrochoid#fitRange(ep1, ep2)</h3>
<p>Creates a curve evaluator object for a curve that follows the same
path as this one but has its u-coordinates remapped to fit the specified range.
For example, this method can be used to shrink the range of u-coordinates
from [-π, π] to [0, 1] without shortening the path of the curve.
Here, -π now maps to 0, and π now maps to 1.</p>
<h4>Parameters</h4>
<ul>
<li><code>ep1</code> (Type: number)<br>New value to use as the start point of the curve.</li>
<li><code>ep2</code> (Type: number)<br>New value to use as the end point of the curve.</li>
</ul>
<h4>Return Value</h4>
<p>Return value. (Type: <a href="H3DU.Curve.html">H3DU.Curve</a>)</p>
<p><a name='H3DU.Hypotrochoid_getLength'></a></p>
<h3>H3DU.Hypotrochoid#getLength()</h3>
<p>Convenience method for getting the total length of this curve.</p>
<h4>Return Value</h4>
<p>The distance from the start of the curve to its end. (Type: number)</p>
<p><a name='H3DU.Hypotrochoid_getPoints'></a></p>
<h3>H3DU.Hypotrochoid#getPoints(count)</h3>
<p>Gets an array of positions on the curve at fixed intervals
of u-coordinates. Note that these positions will not generally be
evenly spaced along the curve unless the curve uses
an arc-length parameterization.</p>
<h4>Parameters</h4>
<ul>
<li><code>count</code> (Type: number)<br>Number of positions to generate. Throws an error if this number is 0. If this value is 1, returns an array containing the starting point of this curve.</li>
</ul>
<h4>Return Value</h4>
<p>An array of curve positions. The first
element will be the start of the curve. If "count" is 2 or greater, the last element
will be the end of the curve. (Type: Array.<Array.<number>> | Array.<Object>)</p>
<p><a name='H3DU.Hypotrochoid_getPointsAsObjects'></a></p>
<h3>H3DU.Hypotrochoid#getPointsAsObjects(count)</h3>
<p>Gets an array of positions on the curve at fixed intervals
of u-coordinates. Note that these positions will not generally be
evenly spaced along the curve unless the curve uses
an arc-length parameterization. The positions will be in the form of objects with
up to four properties: x, y, z, and w retrieve the first, second, third,
and fourth coordinate of each position, respectively.</p>
<h4>Parameters</h4>
<ul>
<li><code>count</code> (Type: number)<br>Number of positions to generate. Throws an error if this number is 0. If this value is 1, returns an array containing the starting point of this curve.</li>
</ul>
<h4>Return Value</h4>
<p>An array of curve positions. The first
element will be the start of the curve. If "count" is 2 or greater, the last element
will be the end of the curve. (Type: Array.<Array.<number>> | Array.<Object>)</p>
<h4>Example</h4>
<p>The following example initializes a three.js BufferGeometry with the points retrieved by this method. This example requires the three.js library.</p>
<pre>var points=curve.getPointsAsObjects(50)
var buffer=new THREE.BufferGeometry()
.setFromPoints(points);
</pre>
<p><a name='H3DU.Hypotrochoid_jerk'></a></p>
<h3>H3DU.Hypotrochoid#jerk(u)</h3>
<p>Finds an approximate jerk vector at the specified u-coordinate of this curve.
The implementation in <a href="H3DU.Curve.html">H3DU.Curve</a> calls the evaluator's <code>jerk</code>
method if it implements it; otherwise, does a numerical differentiation using
the acceleration vector.</p>
<p>The <b>jerk</b> of a curve is a vector which is the third-order derivative of the curve's position at the specified coordinate. The vector returned by this method <i>should not</i> be "normalized" to a <a href="tutorial-glmath.html">unit vector</a>.</p>
<h4>Parameters</h4>
<ul>
<li><code>u</code> (Type: number)<br>u-coordinate of a point on the curve.</li>
</ul>
<h4>Return Value</h4>
<p>An array describing a jerk vector. It should have at least as many
elements as the number of dimensions of the underlying curve. (Type: Array.<number>)</p>
<p><a name='H3DU.Hypotrochoid_normal'></a></p>
<h3>H3DU.Hypotrochoid#normal(u)</h3>
<p>Finds an approximate principal normal vector at the specified u-coordinate of this curve.
The implementation in <a href="H3DU.Curve.html">H3DU.Curve</a> calls the evaluator's <code>normal</code>
method if it implements it; otherwise, does a numerical differentiation using the velocity vector.</p>
<p>The <b>principal normal</b> of a curve is the derivative of the "normalized" velocity
vector divided by that derivative's length. The normal returned by this method
<i>should</i> be "normalized" to a <a href="tutorial-glmath.html">unit vector</a>. (Compare with <a href="H3DU.Surface.html#H3DU.Surface_gradient">H3DU.Surface#gradient</a>.)</p>
<h4>Parameters</h4>
<ul>
<li><code>u</code> (Type: number)<br>u-coordinate of a point on the curve.</li>
</ul>
<h4>Return Value</h4>
<p>An array describing a normal vector. It should have at least as many
elements as the number of dimensions of the underlying curve. (Type: Array.<number>)</p>
<p><a name='H3DU.Hypotrochoid.rose'></a></p>
<h3>(static) H3DU.Hypotrochoid.rose(n, distFromInnerCenter, [rotationDegrees])</h3>
<p>Creates a <a href="H3DU.Curve.html">curve evaluator object</a> for a rose, a special
form of hypotrochoid.</p>
<h4>Parameters</h4>
<ul>
<li><code>n</code> (Type: number)<br>Parameter that determines the petal form of the rose. For example, the rose is symmetrical if this number is even.</li>
<li><code>distFromInnerCenter</code> (Type: number)<br>Distance from the center of the rolling circle to the drawing pen.</li>
<li><code>rotationDegrees</code> (Type: number) (optional)<br>Starting angle of the curve from the positive x-axis toward the positive y-axis, in degrees. Default is 0.</li>
</ul>
<h4>Return Value</h4>
<p>The resulting curve evaluator object. (Type: <a href="H3DU.Hypotrochoid.html">H3DU.Hypotrochoid</a>)</p>
<p><a name='H3DU.Hypotrochoid_scaleTo'></a></p>
<h3>H3DU.Hypotrochoid#scaleTo(radius)</h3>
<p>Creates a modified version of this curve so that it
fits the specified radius.</p>
<h4>Parameters</h4>
<ul>
<li><code>radius</code> (Type: number)<br>Desired radius of the curve.</li>
</ul>
<h4>Return Value</h4>
<p>Return value. (Type: <a href="H3DU.Hypotrochoid.html">H3DU.Hypotrochoid</a>)</p>
<p><a name='H3DU.Hypotrochoid_tangent'></a></p>
<h3>H3DU.Hypotrochoid#tangent(u)</h3>
<p>Convenience method for finding an approximate tangent vector of this curve at the specified u-coordinate.
The <b>tangent vector</b> is the same as the velocity vector, but "normalized" to a unit vector.</p>
<h4>Parameters</h4>
<ul>
<li><code>u</code> (Type: number)<br>u-coordinate of a point on the curve.</li>
</ul>
<h4>Return Value</h4>
<p>An array describing a normal vector. It should have at least as many
elements as the number of dimensions of the underlying curve. (Type: Array.<number>)</p>
<p><a name='H3DU.Hypotrochoid_toArcLengthParam'></a></p>
<h3>H3DU.Hypotrochoid#toArcLengthParam()</h3>
<p>Creates a curve evaluator object for a curve that follows the same
path as this one but has its u-coordinates remapped to
an <i>arc length parameterization</i>. Arc length
parameterization allows for moving along a curve's path at a uniform
speed and for generating points which are spaced evenly along that
path -- both features are more difficult with most other kinds
of curve parameterization.</p>
<p>The <i>end points</i> of the curve (obtained by calling the <code>endPoints</code>
method) will be (0, N), where N is the distance to the end of the curve from its
start.</p>
<p>When converting to an arc length parameterization, the curve
should be continuous and have a speed greater than 0 at every
point on the curve. The arc length parameterization used in
this method is approximate.</p>
<h4>Return Value</h4>
<p>Return value. (Type: <a href="H3DU.Curve.html">H3DU.Curve</a>)</p>
<p><a name='H3DU.Hypotrochoid_velocity'></a></p>
<h3>H3DU.Hypotrochoid#velocity(u)</h3>
<p>Finds an approximate velocity vector at the specified u-coordinate of this curve.
The implementation in <a href="H3DU.Curve.html">H3DU.Curve</a> calls the evaluator's <code>velocity</code>
method if it implements it; otherwise, does a numerical differentiation using
the position (from the <code>evaluate</code> method).</p>
<p>The <b>velocity</b> of a curve is a vector which is the derivative of the curve's position at the specified coordinate. The vector returned by this method <i>should not</i> be "normalized" to a <a href="tutorial-glmath.html">unit vector</a>.</p>
<h4>Parameters</h4>
<ul>
<li><code>u</code> (Type: number)<br>u-coordinate of a point on the curve.</li>
</ul>
<h4>Return Value</h4>
<p>An array describing a velocity vector. It should have at least as many
elements as the number of dimensions of the underlying curve. (Type: Array.<number>)</p>
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