# Curve

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### new Curve(curve, [curveParam])

A curve evaluator object for a parametric curve.

A parametric curve is a curve whose points are based on a parametric curve function. A curve function takes a number (U) and returns a point (in 1 or more dimensions, but usually 2 or 3) that lies on the curve. For example, in 3 dimensions, a curve function has the following form:

F(u) = [ x(u), y(u), z(u) ]

where x(u) returns an X coordinate, y(u) a Y coordinate, and z(u) returns a Z coordinate.

Classes or JavaScript objects defining parametric curves should implement the `evaluate` method and, optionally, the other methods mentioned in the "curve" parameter below.

#### Parameters

• `curve` (Type: Object)
A curve evaluator object, which is an object that must contain an `evaluate` method and may contain an `endPoints`, `velocity`, `accel`, `jerk`, `normal`, and/or `arcLength` method, as described in the corresponding methods of this class.
• `curveParam` (Type: Object) (optional)
An object for reparameterizing a curve object. It implements a method named `endPoints`, which has the same meaning as Curve#endPoints and whose return value takes precedence over the given curve's `endPoints` method. It also implements a method named `getCoordinate(u)`, which converts a U coordinate in the old parameterization to a U coordinate in the new parameterization.

#### Examples

The following is a simple example of a parametric curve.

```var simpleCurve = new Curve({
"evaluate":function(u) {
return [Math.cos(u) * 1.5, Math.sin(u) * 0.8, 0];
},
"endPoints":function() {
return [-Math.PI, Math.PI];
}
});
```

The following function defines a parametric circle curve. It demonstrates how all methods defined for curve evaluator objects can be implemented.

```var circle=new Curve({"evaluate":function(u) {
"use strict";
return [Math.cos(u),Math.sin(u),0]
},
"velocity":function(u) {
return [-Math.sin(u),Math.cos(u),0]
},
"accel":function(u) {
return [-Math.cos(u),-Math.sin(u),0]
},
"jerk":function(u) {
return [Math.sin(u),-Math.cos(u),0]
},
"normal":function(u) {
// NOTE: The velocity vector will already be a
// unit vector, so we use the accel vector instead
return MathUtil.vec3normalize(this.accel(u));
},
"arcLength":function(u) {
return u;
},
"endPoints":function(u) {
return [0,Math.PiTimes2]
}
});
```

The following method starts a curve at a different offset and wraps the portion of the curve behind that offset at the end of the original curve. This is useful for offsetting the points retrieved with the getPoints method.

```function wrapAtOffset(curve, offset) {
"use strict";
var c=curve
if(offset!=0) {
var ep=curve.endPoints();
c=new Curve({
evaluate:function(u) {curves.evaluate(
u+offset>ep ? (u+offset)-ep : (u+offset))},
endPoints:function() {return ep;}
});
}
return c;
}
```

### 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
Returns the starting and ending U coordinates of this curve.
• evaluate
Finds the position of this curve at 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.
• getPointsAsObjects
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.
• tangent
Convenience method for finding an approximate tangent vector of this curve at the given U coordinate.
• toArcLengthParam
Gets 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.

### Curve#accel(u)

Finds an approximate acceleration vector at the given U coordinate of this curve. The implementation in 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>)

### Curve#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. The implementation in Curve calls the evaluator's `arcLength` method if it implements it; otherwise, calculates a numerical integral using the velocity vector.

The arc length function returns a number; if the curve is "smooth", this is the integral, from the starting point to `u`, of the length of the velocity vector.

#### 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: number)

### Curve#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.

For some curves, this method can also be used to grow the path of the 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: Curve)

### Curve#endPoints()

Returns the starting and ending U coordinates of this curve.

#### Return Value

A two-element array. The first element is the starting coordinate of the curve, and the second is its ending coordinate. Returns `[0, 1]` if the evaluator doesn't implement an `endPoints` method.

### Curve#evaluate(u)

Finds the position of this curve at the given U coordinate.

#### Parameters

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

#### Return Value

An array describing a position. It should have at least as many elements as the number of dimensions of the underlying curve. (Type: Array.<number>)

### Curve#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: Curve)

### Curve#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)

### Curve#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>> | Array.<Object>)

### Curve#getPointsAsObjects(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. 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.

#### 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>> | Array.<Object>)

#### Examples

The following example initializes a three.js BufferGeometry with the points retrieved by this method. This example requires the three.js library.

```var points=curve.getPointsAsObjects(50)
var buffer=new THREE.BufferGeometry()
.setFromPoints(points);
```

### Curve#jerk(u)

Finds an approximate jerk vector at the given U coordinate of this curve. The implementation in 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>)

### Curve#normal(u)

Finds an approximate principal normal vector at the given U coordinate of this curve. The implementation in 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 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>)

### Curve#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>)

### Curve#toArcLengthParam()

Gets 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. Returns this object if this curve already uses an arc length parameterization. (Type: Curve)

#### Examples

The following example uses the arc-length parameterization to generate, uniformly at random, a point that lies anywhere on a curve.

```var arclen = curve.toArcLengthParam();
var point = arclen.evaluate(Math.random()*arclen.getLength())
```

### Curve#velocity(u)

Finds an approximate velocity vector at the given U coordinate of this curve. The implementation in 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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