# H3DU.Hypotrochoid

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Augments: H3DU.Curve

A curve evaluator object 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).

The following curves can be generated with this class (in the following descriptions, O = outerRadius, R means innerRadius, and D = distFromRollerCenter).

• Hypocycloid: D = R (hypotrochoid touching the fixed circle).
• Curtate hypocycloid: D < R (hypotrochoid not touching the fixed circle).
• Prolate hypocycloid: D > R (hypotrochoid crossing the fixed circle).
• Circle: O = R*2; the circle will have radius R - D.
• Ellipse: O = R*2; the ellipse (unrotated) will have width abs(R+D)*2 and height abs(R-D)*2.
• Line segment with length O*2: O = R*2; D = R.
• Deltoid: O = R*3; D = R.
• Astroid: O = R*4; D = R.
• N-pointed hypocycloid: O = R * N; D = R.

This class is considered a supplementary class to the Public Domain HTML 3D Library and is not considered part of that library.

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:

<script type="text/javascript" src="extras/evaluators.js"></script>

#### Parameters

Radius of the circle whose position is fixed.
• distFromInnerCenter (Type: number)
Distance from the center of the rolling circle to the drawing pen.
• rotationDegrees (Type: number) (optional)
Starting angle of the curve from the positive X axis toward the positive Y axis, in degrees. Default is 0.

### Methods

• accel
Finds an approximate acceleration vector at the given U coordinate of this curve.
• arcLength
Finds an approximate arc length (distance) between the start of this curve and the point at the given U coordinate of this curve.
• changeEnds
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.
• endPoints
Gets the endpoints of this curve.
• evaluate
Finds the coordinates of a point on the curve from the given U coordinate.
• fitRange
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 given range.
• getLength
Convenience method for getting the total length of this curve.
• getPoints
Gets an array of positions on the curve at fixed intervals of U coordinates.
• jerk
Finds an approximate jerk vector at the given U coordinate of this curve.
• normal
Finds an approximate principal normal vector at the given U coordinate of this curve.
• rose
Creates a curve evaluator object for a rose, a special form of hypotrochoid.
• scaleTo
Creates a modified version of this curve so that it fits the given radius.
• tangent
Convenience method for finding an approximate tangent vector of this curve at the given U coordinate.
• toArcLengthParam
Creates a curve evaluator object for a curve that follows the same path as this one but has its U coordinates remapped to an arc length parameterization.
• velocity
Finds an approximate velocity vector at the given U coordinate of this curve.

### H3DU.Hypotrochoid#accel(u)

Finds an approximate acceleration vector at the given U coordinate of this curve. The implementation in H3DU.Curve calls the evaluator's accel method if it implements it; otherwise, does a numerical differentiation using the velocity vector.

The acceleration of a curve is a vector which is the second-order derivative of the curve's position at the given coordinate. The vector returned by this method should not be "normalized" to a unit vector.

#### Parameters

• u (Type: number)
U coordinate of a point on the curve.

#### Return Value

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>)

### H3DU.Hypotrochoid#arcLength(u)

Finds an approximate arc length (distance) between the start of this curve and the point at the given U coordinate of this curve.

#### Parameters

• u (Type: number)
U coordinate of a point on the curve.

#### Return Value

The approximate arc length of this curve at the given U coordinate. (Type: Array.<number>)

### H3DU.Hypotrochoid#changeEnds(ep1, ep2)

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].

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.

#### Parameters

• ep1 (Type: number)
New start point of the curve.
• ep2 (Type: number)
New end point of the curve.

#### Return Value

Return value. (Type: H3DU.Curve)

### H3DU.Hypotrochoid#endPoints()

Gets the endpoints of this curve. For this curve evaluator object, the curve starts at 0 and ends at π*2.

#### Return Value

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>)

### H3DU.Hypotrochoid#evaluate(u)

Finds the coordinates of a point on the curve from the given U coordinate.

#### Parameters

• u (Type: number)
U coordinate.

#### Return Value

A 3-element array specifying a 3D point. Only the X and Y coordinates can be other than 0. (Type: Array.<number>)

### H3DU.Hypotrochoid#fitRange(ep1, ep2)

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 given 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.

#### Parameters

• ep1 (Type: number)
New value to use as the start point of the curve.
• ep2 (Type: number)
New value to use as the end point of the curve.

#### Return Value

Return value. (Type: H3DU.Curve)

### H3DU.Hypotrochoid#getLength()

Convenience method for getting the total length of this curve.

#### Return Value

The distance from the start of the curve to its end. (Type: number)

### H3DU.Hypotrochoid#getPoints(count)

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.

#### Parameters

• count (Type: number)
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.

#### Return Value

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>>)

### H3DU.Hypotrochoid#jerk(u)

Finds an approximate jerk vector at the given U coordinate of this curve. The implementation in H3DU.Curve calls the evaluator's jerk method if it implements it; otherwise, does a numerical differentiation using the acceleration vector.

The jerk of a curve is a vector which is the third-order derivative of the curve's position at the given coordinate. The vector returned by this method should not be "normalized" to a unit vector.

#### Parameters

• u (Type: number)
U coordinate of a point on the curve.

#### Return Value

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>)

### H3DU.Hypotrochoid#normal(u)

Finds an approximate principal normal vector at the given U coordinate of this curve. The implementation in H3DU.Curve calls the evaluator's normal method if it implements it; otherwise, does a numerical differentiation using the velocity vector.

The principal normal of a curve is the derivative of the "normalized" velocity vector divided by that derivative's length. The normal returned by this method should be "normalized" to a unit vector. (Compare with H3DU.Surface#gradient.)

#### Parameters

• u (Type: number)
U coordinate of a point on the curve.

#### Return Value

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>)

### (static) H3DU.Hypotrochoid.rose(n, distFromInnerCenter, [rotationDegrees])

Creates a curve evaluator object for a rose, a special form of hypotrochoid.

#### Parameters

• n (Type: number)
Parameter that determines the petal form of the rose. For example, the rose is symmetrical if this number is even.
• distFromInnerCenter (Type: number)
Distance from the center of the rolling circle to the drawing pen.
• rotationDegrees (Type: number) (optional)
Starting angle of the curve from the positive X axis toward the positive Y axis, in degrees. Default is 0.

#### Return Value

The resulting curve evaluator object. (Type: H3DU.Hypotrochoid)

Creates a modified version of this curve so that it fits the given radius.

#### Return Value

Return value. (Type: H3DU.Hypotrochoid)

### H3DU.Hypotrochoid#tangent(u)

Convenience method for finding an approximate tangent vector of this curve at the given U coordinate. The tangent vector is the same as the velocity vector, but "normalized" to a unit vector.

#### Parameters

• u (Type: number)
U coordinate of a point on the curve.

#### Return Value

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>)

### H3DU.Hypotrochoid#toArcLengthParam()

Creates a curve evaluator object for a curve that follows the same path as this one but has its U coordinates remapped to an arc length parameterization. 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.

The end points of the curve (obtained by calling the endPoints method) will be (0, N), where N is the distance to the end of the curve from its start.

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.

#### Return Value

Return value. (Type: H3DU.Curve)

### H3DU.Hypotrochoid#velocity(u)

Finds an approximate velocity vector at the given U coordinate of this curve. The implementation in H3DU.Curve calls the evaluator's velocity method if it implements it; otherwise, does a numerical differentiation using the position (from the evaluate method).

The velocity of a curve is a vector which is the derivative of the curve's position at the given coordinate. The vector returned by this method should not be "normalized" to a unit vector.

#### Parameters

• u (Type: number)
U coordinate of a point on the curve.

#### Return Value

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>)

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