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Quaternion.ts
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Quaternion.ts
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import { Vec3 } from '../math/Vec3'
/**
* A Quaternion describes a rotation in 3D space. The Quaternion is mathematically defined as Q = x*i + y*j + z*k + w, where (i,j,k) are imaginary basis vectors. (x,y,z) can be seen as a vector related to the axis of rotation, while the real multiplier, w, is related to the amount of rotation.
* @param x Multiplier of the imaginary basis vector i.
* @param y Multiplier of the imaginary basis vector j.
* @param z Multiplier of the imaginary basis vector k.
* @param w Multiplier of the real part.
* @see http://en.wikipedia.org/wiki/Quaternion
*/
export class Quaternion {
x: number
y: number
z: number
w: number
constructor(x = 0, y = 0, z = 0, w = 1) {
this.x = x
this.y = y
this.z = z
this.w = w
}
/**
* Set the value of the quaternion.
*/
set(x: number, y: number, z: number, w: number): Quaternion {
this.x = x
this.y = y
this.z = z
this.w = w
return this
}
/**
* Convert to a readable format
* @return "x,y,z,w"
*/
toString(): string {
return `${this.x},${this.y},${this.z},${this.w}`
}
/**
* Convert to an Array
* @return [x, y, z, w]
*/
toArray(): [number, number, number, number] {
return [this.x, this.y, this.z, this.w]
}
/**
* Set the quaternion components given an axis and an angle in radians.
*/
setFromAxisAngle(vector: Vec3, angle: number): Quaternion {
const s = Math.sin(angle * 0.5)
this.x = vector.x * s
this.y = vector.y * s
this.z = vector.z * s
this.w = Math.cos(angle * 0.5)
return this
}
/**
* Converts the quaternion to [ axis, angle ] representation.
* @param targetAxis A vector object to reuse for storing the axis.
* @return An array, first element is the axis and the second is the angle in radians.
*/
toAxisAngle(targetAxis = new Vec3()): [Vec3, number] {
this.normalize() // if w>1 acos and sqrt will produce errors, this cant happen if quaternion is normalised
const angle = 2 * Math.acos(this.w)
const s = Math.sqrt(1 - this.w * this.w) // assuming quaternion normalised then w is less than 1, so term always positive.
if (s < 0.001) {
// test to avoid divide by zero, s is always positive due to sqrt
// if s close to zero then direction of axis not important
targetAxis.x = this.x // if it is important that axis is normalised then replace with x=1; y=z=0;
targetAxis.y = this.y
targetAxis.z = this.z
} else {
targetAxis.x = this.x / s // normalise axis
targetAxis.y = this.y / s
targetAxis.z = this.z / s
}
return [targetAxis, angle]
}
/**
* Set the quaternion value given two vectors. The resulting rotation will be the needed rotation to rotate u to v.
*/
setFromVectors(u: Vec3, v: Vec3): Quaternion {
if (u.isAntiparallelTo(v)) {
const t1 = sfv_t1
const t2 = sfv_t2
u.tangents(t1, t2)
this.setFromAxisAngle(t1, Math.PI)
} else {
const a = u.cross(v)
this.x = a.x
this.y = a.y
this.z = a.z
this.w = Math.sqrt(u.length() ** 2 * v.length() ** 2) + u.dot(v)
this.normalize()
}
return this
}
/**
* Multiply the quaternion with an other quaternion.
*/
mult(quat: Quaternion, target = new Quaternion()): Quaternion {
const ax = this.x
const ay = this.y
const az = this.z
const aw = this.w
const bx = quat.x
const by = quat.y
const bz = quat.z
const bw = quat.w
target.x = ax * bw + aw * bx + ay * bz - az * by
target.y = ay * bw + aw * by + az * bx - ax * bz
target.z = az * bw + aw * bz + ax * by - ay * bx
target.w = aw * bw - ax * bx - ay * by - az * bz
return target
}
/**
* Get the inverse quaternion rotation.
*/
inverse(target = new Quaternion()): Quaternion {
const x = this.x
const y = this.y
const z = this.z
const w = this.w
this.conjugate(target)
const inorm2 = 1 / (x * x + y * y + z * z + w * w)
target.x *= inorm2
target.y *= inorm2
target.z *= inorm2
target.w *= inorm2
return target
}
/**
* Get the quaternion conjugate
*/
conjugate(target = new Quaternion()): Quaternion {
target.x = -this.x
target.y = -this.y
target.z = -this.z
target.w = this.w
return target
}
/**
* Normalize the quaternion. Note that this changes the values of the quaternion.
*/
normalize(): Quaternion {
let l = Math.sqrt(this.x * this.x + this.y * this.y + this.z * this.z + this.w * this.w)
if (l === 0) {
this.x = 0
this.y = 0
this.z = 0
this.w = 0
} else {
l = 1 / l
this.x *= l
this.y *= l
this.z *= l
this.w *= l
}
return this
}
/**
* Approximation of quaternion normalization. Works best when quat is already almost-normalized.
* @author unphased, https://github.com/unphased
*/
normalizeFast(): Quaternion {
const f = (3.0 - (this.x * this.x + this.y * this.y + this.z * this.z + this.w * this.w)) / 2.0
if (f === 0) {
this.x = 0
this.y = 0
this.z = 0
this.w = 0
} else {
this.x *= f
this.y *= f
this.z *= f
this.w *= f
}
return this
}
/**
* Multiply the quaternion by a vector
*/
vmult(v: Vec3, target = new Vec3()): Vec3 {
const x = v.x
const y = v.y
const z = v.z
const qx = this.x
const qy = this.y
const qz = this.z
const qw = this.w
// q*v
const ix = qw * x + qy * z - qz * y
const iy = qw * y + qz * x - qx * z
const iz = qw * z + qx * y - qy * x
const iw = -qx * x - qy * y - qz * z
target.x = ix * qw + iw * -qx + iy * -qz - iz * -qy
target.y = iy * qw + iw * -qy + iz * -qx - ix * -qz
target.z = iz * qw + iw * -qz + ix * -qy - iy * -qx
return target
}
/**
* Copies value of source to this quaternion.
* @return this
*/
copy(quat: Quaternion): Quaternion {
this.x = quat.x
this.y = quat.y
this.z = quat.z
this.w = quat.w
return this
}
/**
* Convert the quaternion to euler angle representation. Order: YZX, as this page describes: https://www.euclideanspace.com/maths/standards/index.htm
* @param order Three-character string, defaults to "YZX"
*/
toEuler(target: Vec3, order = 'YZX'): void {
let heading
let attitude
let bank
const x = this.x
const y = this.y
const z = this.z
const w = this.w
switch (order) {
case 'YZX':
const test = x * y + z * w
if (test > 0.499) {
// singularity at north pole
heading = 2 * Math.atan2(x, w)
attitude = Math.PI / 2
bank = 0
}
if (test < -0.499) {
// singularity at south pole
heading = -2 * Math.atan2(x, w)
attitude = -Math.PI / 2
bank = 0
}
if (heading === undefined) {
const sqx = x * x
const sqy = y * y
const sqz = z * z
heading = Math.atan2(2 * y * w - 2 * x * z, 1 - 2 * sqy - 2 * sqz) // Heading
attitude = Math.asin(2 * test) // attitude
bank = Math.atan2(2 * x * w - 2 * y * z, 1 - 2 * sqx - 2 * sqz) // bank
}
break
default:
throw new Error(`Euler order ${order} not supported yet.`)
}
target.y = heading
target.z = attitude as number
target.x = bank as number
}
/**
* Set the quaternion components given Euler angle representation.
*
* @param order The order to apply angles: 'XYZ' or 'YXZ' or any other combination.
*
* See {@link https://www.mathworks.com/matlabcentral/fileexchange/20696-function-to-convert-between-dcm-euler-angles-quaternions-and-euler-vectors MathWorks} reference
*/
setFromEuler(x: number, y: number, z: number, order = 'XYZ'): Quaternion {
const c1 = Math.cos(x / 2)
const c2 = Math.cos(y / 2)
const c3 = Math.cos(z / 2)
const s1 = Math.sin(x / 2)
const s2 = Math.sin(y / 2)
const s3 = Math.sin(z / 2)
if (order === 'XYZ') {
this.x = s1 * c2 * c3 + c1 * s2 * s3
this.y = c1 * s2 * c3 - s1 * c2 * s3
this.z = c1 * c2 * s3 + s1 * s2 * c3
this.w = c1 * c2 * c3 - s1 * s2 * s3
} else if (order === 'YXZ') {
this.x = s1 * c2 * c3 + c1 * s2 * s3
this.y = c1 * s2 * c3 - s1 * c2 * s3
this.z = c1 * c2 * s3 - s1 * s2 * c3
this.w = c1 * c2 * c3 + s1 * s2 * s3
} else if (order === 'ZXY') {
this.x = s1 * c2 * c3 - c1 * s2 * s3
this.y = c1 * s2 * c3 + s1 * c2 * s3
this.z = c1 * c2 * s3 + s1 * s2 * c3
this.w = c1 * c2 * c3 - s1 * s2 * s3
} else if (order === 'ZYX') {
this.x = s1 * c2 * c3 - c1 * s2 * s3
this.y = c1 * s2 * c3 + s1 * c2 * s3
this.z = c1 * c2 * s3 - s1 * s2 * c3
this.w = c1 * c2 * c3 + s1 * s2 * s3
} else if (order === 'YZX') {
this.x = s1 * c2 * c3 + c1 * s2 * s3
this.y = c1 * s2 * c3 + s1 * c2 * s3
this.z = c1 * c2 * s3 - s1 * s2 * c3
this.w = c1 * c2 * c3 - s1 * s2 * s3
} else if (order === 'XZY') {
this.x = s1 * c2 * c3 - c1 * s2 * s3
this.y = c1 * s2 * c3 - s1 * c2 * s3
this.z = c1 * c2 * s3 + s1 * s2 * c3
this.w = c1 * c2 * c3 + s1 * s2 * s3
}
return this
}
clone(): Quaternion {
return new Quaternion(this.x, this.y, this.z, this.w)
}
/**
* Performs a spherical linear interpolation between two quat
*
* @param toQuat second operand
* @param t interpolation amount between the self quaternion and toQuat
* @param target A quaternion to store the result in. If not provided, a new one will be created.
* @returns {Quaternion} The "target" object
*/
slerp(toQuat: Quaternion, t: number, target = new Quaternion()): Quaternion {
const ax = this.x
const ay = this.y
const az = this.z
const aw = this.w
let bx = toQuat.x
let by = toQuat.y
let bz = toQuat.z
let bw = toQuat.w
let omega
let cosom
let sinom
let scale0
let scale1
// calc cosine
cosom = ax * bx + ay * by + az * bz + aw * bw
// adjust signs (if necessary)
if (cosom < 0.0) {
cosom = -cosom
bx = -bx
by = -by
bz = -bz
bw = -bw
}
// calculate coefficients
if (1.0 - cosom > 0.000001) {
// standard case (slerp)
omega = Math.acos(cosom)
sinom = Math.sin(omega)
scale0 = Math.sin((1.0 - t) * omega) / sinom
scale1 = Math.sin(t * omega) / sinom
} else {
// "from" and "to" quaternions are very close
// ... so we can do a linear interpolation
scale0 = 1.0 - t
scale1 = t
}
// calculate final values
target.x = scale0 * ax + scale1 * bx
target.y = scale0 * ay + scale1 * by
target.z = scale0 * az + scale1 * bz
target.w = scale0 * aw + scale1 * bw
return target
}
/**
* Rotate an absolute orientation quaternion given an angular velocity and a time step.
*/
integrate(angularVelocity: Vec3, dt: number, angularFactor: Vec3, target = new Quaternion()): Quaternion {
const ax = angularVelocity.x * angularFactor.x,
ay = angularVelocity.y * angularFactor.y,
az = angularVelocity.z * angularFactor.z,
bx = this.x,
by = this.y,
bz = this.z,
bw = this.w
const half_dt = dt * 0.5
target.x += half_dt * (ax * bw + ay * bz - az * by)
target.y += half_dt * (ay * bw + az * bx - ax * bz)
target.z += half_dt * (az * bw + ax * by - ay * bx)
target.w += half_dt * (-ax * bx - ay * by - az * bz)
return target
}
}
const sfv_t1 = new Vec3()
const sfv_t2 = new Vec3()