import * as glMatrix from "./common.js";
import * as quat from "./quat.js";
import * as mat4 from "./mat4.js";
/**
* Dual Quaternion<br>
* Format: [real, dual]<br>
* Quaternion format: XYZW<br>
* Make sure to have normalized dual quaternions, otherwise the functions may not work as intended.<br>
* @module quat2
*/
/**
* Creates a new identity dual quat
*
* @returns {quat2} a new dual quaternion [real -> rotation, dual -> translation]
*/
export function create() {
let dq = new glMatrix.ARRAY_TYPE(8);
dq[0] = 0;
dq[1] = 0;
dq[2] = 0;
dq[3] = 1;
dq[4] = 0;
dq[5] = 0;
dq[6] = 0;
dq[7] = 0;
return dq;
}
/**
* Creates a new quat initialized with values from an existing quaternion
*
* @param {quat2} a dual quaternion to clone
* @returns {quat2} new dual quaternion
* @function
*/
export function clone(a) {
let dq = new glMatrix.ARRAY_TYPE(8);
dq[0] = a[0];
dq[1] = a[1];
dq[2] = a[2];
dq[3] = a[3];
dq[4] = a[4];
dq[5] = a[5];
dq[6] = a[6];
dq[7] = a[7];
return dq;
}
/**
* Creates a new dual quat initialized with the given values
*
* @param {Number} x1 X component
* @param {Number} y1 Y component
* @param {Number} z1 Z component
* @param {Number} w1 W component
* @param {Number} x2 X component
* @param {Number} y2 Y component
* @param {Number} z2 Z component
* @param {Number} w2 W component
* @returns {quat2} new dual quaternion
* @function
*/
export function fromValues(x1, y1, z1, w1, x2, y2, z2, w2) {
let dq = new glMatrix.ARRAY_TYPE(8);
dq[0] = x1;
dq[1] = y1;
dq[2] = z1;
dq[3] = w1;
dq[4] = x2;
dq[5] = y2;
dq[6] = z2;
dq[7] = w2;
return dq;
}
/**
* Creates a new dual quat from the given values (quat and translation)
*
* @param {Number} x1 X component
* @param {Number} y1 Y component
* @param {Number} z1 Z component
* @param {Number} w1 W component
* @param {Number} x2 X component (translation)
* @param {Number} y2 Y component (translation)
* @param {Number} z2 Z component (translation)
* @returns {quat2} new dual quaternion
* @function
*/
export function fromRotationTranslationValues(x1, y1, z1, w1, x2, y2, z2) {
let dq = new glMatrix.ARRAY_TYPE(8);
dq[0] = x1;
dq[1] = y1;
dq[2] = z1;
dq[3] = w1;
let ax = x2 * 0.5,
ay = y2 * 0.5,
az = z2 * 0.5;
dq[4] = ax * w1 + ay * z1 - az * y1;
dq[5] = ay * w1 + az * x1 - ax * z1;
dq[6] = az * w1 + ax * y1 - ay * x1;
dq[7] = -ax * x1 - ay * y1 - az * z1;
return dq;
}
/**
* Creates a dual quat from a quaternion and a translation
*
* @param {quat2} dual quaternion receiving operation result
* @param {quat} q quaternion
* @param {vec3} t tranlation vector
* @returns {quat2} dual quaternion receiving operation result
* @function
*/
export function fromRotationTranslation(out, q, t) {
let ax = t[0] * 0.5,
ay = t[1] * 0.5,
az = t[2] * 0.5,
bx = q[0],
by = q[1],
bz = q[2],
bw = q[3];
out[0] = bx;
out[1] = by;
out[2] = bz;
out[3] = bw;
out[4] = ax * bw + ay * bz - az * by;
out[5] = ay * bw + az * bx - ax * bz;
out[6] = az * bw + ax * by - ay * bx;
out[7] = -ax * bx - ay * by - az * bz;
return out;
}
/**
* Creates a dual quat from a translation
*
* @param {quat2} dual quaternion receiving operation result
* @param {vec3} t translation vector
* @returns {quat2} dual quaternion receiving operation result
* @function
*/
export function fromTranslation(out, t) {
out[0] = 0;
out[1] = 0;
out[2] = 0;
out[3] = 1;
out[4] = t[0] * 0.5;
out[5] = t[1] * 0.5;
out[6] = t[2] * 0.5;
out[7] = 0;
return out;
}
/**
* Creates a dual quat from a quaternion
*
* @param {quat2} dual quaternion receiving operation result
* @param {quat} q the quaternion
* @returns {quat2} dual quaternion receiving operation result
* @function
*/
export function fromRotation(out, q) {
out[0] = q[0];
out[1] = q[1];
out[2] = q[2];
out[3] = q[3];
out[4] = 0;
out[5] = 0;
out[6] = 0;
out[7] = 0;
return out;
}
/**
* Creates a new dual quat from a matrix (4x4)
*
* @param {quat2} out the dual quaternion
* @param {mat4} a the matrix
* @returns {quat2} dual quat receiving operation result
* @function
*/
export function fromMat4(out, a) {
//TODO Optimize this
let outer = quat.create();
mat4.getRotation(outer, a);
let t = new glMatrix.ARRAY_TYPE(3);
mat4.getTranslation(t, a);
fromRotationTranslation(out, outer, t);
return out;
}
/**
* Copy the values from one dual quat to another
*
* @param {quat2} out the receiving dual quaternion
* @param {quat2} a the source dual quaternion
* @returns {quat2} out
* @function
*/
export function copy(out, a) {
out[0] = a[0];
out[1] = a[1];
out[2] = a[2];
out[3] = a[3];
out[4] = a[4];
out[5] = a[5];
out[6] = a[6];
out[7] = a[7];
return out;
}
/**
* Set a dual quat to the identity dual quaternion
*
* @param {quat2} out the receiving quaternion
* @returns {quat2} out
*/
export function identity(out) {
out[0] = 0;
out[1] = 0;
out[2] = 0;
out[3] = 1;
out[4] = 0;
out[5] = 0;
out[6] = 0;
out[7] = 0;
return out;
}
/**
* Set the components of a dual quat to the given values
*
* @param {quat2} out the receiving quaternion
* @param {Number} x1 X component
* @param {Number} y1 Y component
* @param {Number} z1 Z component
* @param {Number} w1 W component
* @param {Number} x2 X component
* @param {Number} y2 Y component
* @param {Number} z2 Z component
* @param {Number} w2 W component
* @returns {quat2} out
* @function
*/
export function set(out, x1, y1, z1, w1, x2, y2, z2, w2) {
out[0] = x1;
out[1] = y1;
out[2] = z1;
out[3] = w1;
out[4] = x2;
out[5] = y2;
out[6] = z2;
out[7] = w2;
return out;
}
/**
* Gets the real part of a dual quat
* @param {quat} out real part
* @param {quat2} a Dual Quaternion
* @return {quat} real part
*/
export const getReal = quat.copy;
/**
* Gets the dual part of a dual quat
* @param {quat} out dual part
* @param {quat2} a Dual Quaternion
* @return {quat} dual part
*/
export function getDual(out, a) {
out[0] = a[4];
out[1] = a[5];
out[2] = a[6];
out[3] = a[7];
return out;
}
/**
* Set the real component of a dual quat to the given quaternion
*
* @param {quat2} out the receiving quaternion
* @param {quat} q a quaternion representing the real part
* @returns {quat2} out
* @function
*/
export const setReal = quat.copy;
/**
* Set the dual component of a dual quat to the given quaternion
*
* @param {quat2} out the receiving quaternion
* @param {quat} q a quaternion representing the dual part
* @returns {quat2} out
* @function
*/
export function setDual(out, q) {
out[4] = q[0];
out[5] = q[1];
out[6] = q[2];
out[7] = q[3];
return out;
}
/**
* Gets the translation of a normalized dual quat
* @param {vec3} out translation
* @param {quat2} a Dual Quaternion to be decomposed
* @return {vec3} translation
*/
export function getTranslation(out, a) {
let ax = a[4],
ay = a[5],
az = a[6],
aw = a[7],
bx = -a[0],
by = -a[1],
bz = -a[2],
bw = a[3];
out[0] = (ax * bw + aw * bx + ay * bz - az * by) * 2;
out[1] = (ay * bw + aw * by + az * bx - ax * bz) * 2;
out[2] = (az * bw + aw * bz + ax * by - ay * bx) * 2;
return out;
}
/**
* Translates a dual quat by the given vector
*
* @param {quat2} out the receiving dual quaternion
* @param {quat2} a the dual quaternion to translate
* @param {vec3} v vector to translate by
* @returns {quat2} out
*/
export function translate(out, a, v) {
let ax1 = a[0],
ay1 = a[1],
az1 = a[2],
aw1 = a[3],
bx1 = v[0] * 0.5,
by1 = v[1] * 0.5,
bz1 = v[2] * 0.5,
ax2 = a[4],
ay2 = a[5],
az2 = a[6],
aw2 = a[7];
out[0] = ax1;
out[1] = ay1;
out[2] = az1;
out[3] = aw1;
out[4] = aw1 * bx1 + ay1 * bz1 - az1 * by1 + ax2;
out[5] = aw1 * by1 + az1 * bx1 - ax1 * bz1 + ay2;
out[6] = aw1 * bz1 + ax1 * by1 - ay1 * bx1 + az2;
out[7] = -ax1 * bx1 - ay1 * by1 - az1 * bz1 + aw2;
return out;
}
/**
* Rotates a dual quat around the X axis
*
* @param {quat2} out the receiving dual quaternion
* @param {quat2} a the dual quaternion to rotate
* @param {number} rad how far should the rotation be
* @returns {quat2} out
*/
export function rotateX(out, a, rad) {
let bx = -a[0],
by = -a[1],
bz = -a[2],
bw = a[3],
ax = a[4],
ay = a[5],
az = a[6],
aw = a[7],
ax1 = ax * bw + aw * bx + ay * bz - az * by,
ay1 = ay * bw + aw * by + az * bx - ax * bz,
az1 = az * bw + aw * bz + ax * by - ay * bx,
aw1 = aw * bw - ax * bx - ay * by - az * bz;
quat.rotateX(out, a, rad);
bx = out[0];
by = out[1];
bz = out[2];
bw = out[3];
out[4] = ax1 * bw + aw1 * bx + ay1 * bz - az1 * by;
out[5] = ay1 * bw + aw1 * by + az1 * bx - ax1 * bz;
out[6] = az1 * bw + aw1 * bz + ax1 * by - ay1 * bx;
out[7] = aw1 * bw - ax1 * bx - ay1 * by - az1 * bz;
return out;
}
/**
* Rotates a dual quat around the Y axis
*
* @param {quat2} out the receiving dual quaternion
* @param {quat2} a the dual quaternion to rotate
* @param {number} rad how far should the rotation be
* @returns {quat2} out
*/
export function rotateY(out, a, rad) {
let bx = -a[0],
by = -a[1],
bz = -a[2],
bw = a[3],
ax = a[4],
ay = a[5],
az = a[6],
aw = a[7],
ax1 = ax * bw + aw * bx + ay * bz - az * by,
ay1 = ay * bw + aw * by + az * bx - ax * bz,
az1 = az * bw + aw * bz + ax * by - ay * bx,
aw1 = aw * bw - ax * bx - ay * by - az * bz;
quat.rotateY(out, a, rad);
bx = out[0];
by = out[1];
bz = out[2];
bw = out[3];
out[4] = ax1 * bw + aw1 * bx + ay1 * bz - az1 * by;
out[5] = ay1 * bw + aw1 * by + az1 * bx - ax1 * bz;
out[6] = az1 * bw + aw1 * bz + ax1 * by - ay1 * bx;
out[7] = aw1 * bw - ax1 * bx - ay1 * by - az1 * bz;
return out;
}
/**
* Rotates a dual quat around the Z axis
*
* @param {quat2} out the receiving dual quaternion
* @param {quat2} a the dual quaternion to rotate
* @param {number} rad how far should the rotation be
* @returns {quat2} out
*/
export function rotateZ(out, a, rad) {
let bx = -a[0],
by = -a[1],
bz = -a[2],
bw = a[3],
ax = a[4],
ay = a[5],
az = a[6],
aw = a[7],
ax1 = ax * bw + aw * bx + ay * bz - az * by,
ay1 = ay * bw + aw * by + az * bx - ax * bz,
az1 = az * bw + aw * bz + ax * by - ay * bx,
aw1 = aw * bw - ax * bx - ay * by - az * bz;
quat.rotateZ(out, a, rad);
bx = out[0];
by = out[1];
bz = out[2];
bw = out[3];
out[4] = ax1 * bw + aw1 * bx + ay1 * bz - az1 * by;
out[5] = ay1 * bw + aw1 * by + az1 * bx - ax1 * bz;
out[6] = az1 * bw + aw1 * bz + ax1 * by - ay1 * bx;
out[7] = aw1 * bw - ax1 * bx - ay1 * by - az1 * bz;
return out;
}
/**
* Rotates a dual quat by a given quaternion (a * q)
*
* @param {quat2} out the receiving dual quaternion
* @param {quat2} a the dual quaternion to rotate
* @param {quat} q quaternion to rotate by
* @returns {quat2} out
*/
export function rotateByQuatAppend(out, a, q) {
let qx = q[0],
qy = q[1],
qz = q[2],
qw = q[3],
ax = a[0],
ay = a[1],
az = a[2],
aw = a[3];
out[0] = ax * qw + aw * qx + ay * qz - az * qy;
out[1] = ay * qw + aw * qy + az * qx - ax * qz;
out[2] = az * qw + aw * qz + ax * qy - ay * qx;
out[3] = aw * qw - ax * qx - ay * qy - az * qz;
ax = a[4];
ay = a[5];
az = a[6];
aw = a[7];
out[4] = ax * qw + aw * qx + ay * qz - az * qy;
out[5] = ay * qw + aw * qy + az * qx - ax * qz;
out[6] = az * qw + aw * qz + ax * qy - ay * qx;
out[7] = aw * qw - ax * qx - ay * qy - az * qz;
return out;
}
/**
* Rotates a dual quat by a given quaternion (q * a)
*
* @param {quat2} out the receiving dual quaternion
* @param {quat} q quaternion to rotate by
* @param {quat2} a the dual quaternion to rotate
* @returns {quat2} out
*/
export function rotateByQuatPrepend(out, q, a) {
let qx = q[0],
qy = q[1],
qz = q[2],
qw = q[3],
bx = a[0],
by = a[1],
bz = a[2],
bw = a[3];
out[0] = qx * bw + qw * bx + qy * bz - qz * by;
out[1] = qy * bw + qw * by + qz * bx - qx * bz;
out[2] = qz * bw + qw * bz + qx * by - qy * bx;
out[3] = qw * bw - qx * bx - qy * by - qz * bz;
bx = a[4];
by = a[5];
bz = a[6];
bw = a[7];
out[4] = qx * bw + qw * bx + qy * bz - qz * by;
out[5] = qy * bw + qw * by + qz * bx - qx * bz;
out[6] = qz * bw + qw * bz + qx * by - qy * bx;
out[7] = qw * bw - qx * bx - qy * by - qz * bz;
return out;
}
/**
* Rotates a dual quat around a given axis. Does the normalisation automatically
*
* @param {quat2} out the receiving dual quaternion
* @param {quat2} a the dual quaternion to rotate
* @param {vec3} axis the axis to rotate around
* @param {Number} rad how far the rotation should be
* @returns {quat2} out
*/
export function rotateAroundAxis(out, a, axis, rad) {
//Special case for rad = 0
if (Math.abs(rad) < glMatrix.EPSILON) {
return copy(out, a);
}
let axisLength = Math.sqrt(axis[0] * axis[0] + axis[1] * axis[1] + axis[2] * axis[2]);
rad = rad * 0.5;
let s = Math.sin(rad);
let bx = s * axis[0] / axisLength;
let by = s * axis[1] / axisLength;
let bz = s * axis[2] / axisLength;
let bw = Math.cos(rad);
let ax1 = a[0],
ay1 = a[1],
az1 = a[2],
aw1 = a[3];
out[0] = ax1 * bw + aw1 * bx + ay1 * bz - az1 * by;
out[1] = ay1 * bw + aw1 * by + az1 * bx - ax1 * bz;
out[2] = az1 * bw + aw1 * bz + ax1 * by - ay1 * bx;
out[3] = aw1 * bw - ax1 * bx - ay1 * by - az1 * bz;
let ax = a[4],
ay = a[5],
az = a[6],
aw = a[7];
out[4] = ax * bw + aw * bx + ay * bz - az * by;
out[5] = ay * bw + aw * by + az * bx - ax * bz;
out[6] = az * bw + aw * bz + ax * by - ay * bx;
out[7] = aw * bw - ax * bx - ay * by - az * bz;
return out;
}
/**
* Adds two dual quat's
*
* @param {quat2} out the receiving dual quaternion
* @param {quat2} a the first operand
* @param {quat2} b the second operand
* @returns {quat2} out
* @function
*/
export function add(out, a, b) {
out[0] = a[0] + b[0];
out[1] = a[1] + b[1];
out[2] = a[2] + b[2];
out[3] = a[3] + b[3];
out[4] = a[4] + b[4];
out[5] = a[5] + b[5];
out[6] = a[6] + b[6];
out[7] = a[7] + b[7];
return out;
}
/**
* Multiplies two dual quat's
*
* @param {quat2} out the receiving dual quaternion
* @param {quat2} a the first operand
* @param {quat2} b the second operand
* @returns {quat2} out
*/
export function multiply(out, a, b) {
let ax0 = a[0],
ay0 = a[1],
az0 = a[2],
aw0 = a[3],
bx1 = b[4],
by1 = b[5],
bz1 = b[6],
bw1 = b[7],
ax1 = a[4],
ay1 = a[5],
az1 = a[6],
aw1 = a[7],
bx0 = b[0],
by0 = b[1],
bz0 = b[2],
bw0 = b[3];
out[0] = ax0 * bw0 + aw0 * bx0 + ay0 * bz0 - az0 * by0;
out[1] = ay0 * bw0 + aw0 * by0 + az0 * bx0 - ax0 * bz0;
out[2] = az0 * bw0 + aw0 * bz0 + ax0 * by0 - ay0 * bx0;
out[3] = aw0 * bw0 - ax0 * bx0 - ay0 * by0 - az0 * bz0;
out[4] = ax0 * bw1 + aw0 * bx1 + ay0 * bz1 - az0 * by1 + ax1 * bw0 + aw1 * bx0 + ay1 * bz0 - az1 * by0;
out[5] = ay0 * bw1 + aw0 * by1 + az0 * bx1 - ax0 * bz1 + ay1 * bw0 + aw1 * by0 + az1 * bx0 - ax1 * bz0;
out[6] = az0 * bw1 + aw0 * bz1 + ax0 * by1 - ay0 * bx1 + az1 * bw0 + aw1 * bz0 + ax1 * by0 - ay1 * bx0;
out[7] = aw0 * bw1 - ax0 * bx1 - ay0 * by1 - az0 * bz1 + aw1 * bw0 - ax1 * bx0 - ay1 * by0 - az1 * bz0;
return out;
}
/**
* Alias for {@link quat2.multiply}
* @function
*/
export const mul = multiply;
/**
* Scales a dual quat by a scalar number
*
* @param {quat2} out the receiving dual quat
* @param {quat2} a the dual quat to scale
* @param {Number} b amount to scale the dual quat by
* @returns {quat2} out
* @function
*/
export function scale(out, a, b) {
out[0] = a[0] * b;
out[1] = a[1] * b;
out[2] = a[2] * b;
out[3] = a[3] * b;
out[4] = a[4] * b;
out[5] = a[5] * b;
out[6] = a[6] * b;
out[7] = a[7] * b;
return out;
}
/**
* Calculates the dot product of two dual quat's (The dot product of the real parts)
*
* @param {quat2} a the first operand
* @param {quat2} b the second operand
* @returns {Number} dot product of a and b
* @function
*/
export const dot = quat.dot;
/**
* Performs a linear interpolation between two dual quats's
* NOTE: The resulting dual quaternions won't always be normalized (The error is most noticeable when t = 0.5)
*
* @param {quat2} out the receiving dual quat
* @param {quat2} a the first operand
* @param {quat2} b the second operand
* @param {Number} t interpolation amount, in the range [0-1], between the two inputs
* @returns {quat2} out
*/
export function lerp(out, a, b, t) {
let mt = 1 - t;
if (dot(a, b) < 0) t = -t;
out[0] = a[0] * mt + b[0] * t;
out[1] = a[1] * mt + b[1] * t;
out[2] = a[2] * mt + b[2] * t;
out[3] = a[3] * mt + b[3] * t;
out[4] = a[4] * mt + b[4] * t;
out[5] = a[5] * mt + b[5] * t;
out[6] = a[6] * mt + b[6] * t;
out[7] = a[7] * mt + b[7] * t;
return out;
}
/**
* Calculates the inverse of a dual quat. If they are normalized, conjugate is cheaper
*
* @param {quat2} out the receiving dual quaternion
* @param {quat2} a dual quat to calculate inverse of
* @returns {quat2} out
*/
export function invert(out, a) {
let sqlen = squaredLength(a);
out[0] = -a[0] / sqlen;
out[1] = -a[1] / sqlen;
out[2] = -a[2] / sqlen;
out[3] = a[3] / sqlen;
out[4] = -a[4] / sqlen;
out[5] = -a[5] / sqlen;
out[6] = -a[6] / sqlen;
out[7] = a[7] / sqlen;
return out;
}
/**
* Calculates the conjugate of a dual quat
* If the dual quaternion is normalized, this function is faster than quat2.inverse and produces the same result.
*
* @param {quat2} out the receiving quaternion
* @param {quat2} a quat to calculate conjugate of
* @returns {quat2} out
*/
export function conjugate(out, a) {
out[0] = -a[0];
out[1] = -a[1];
out[2] = -a[2];
out[3] = a[3];
out[4] = -a[4];
out[5] = -a[5];
out[6] = -a[6];
out[7] = a[7];
return out;
}
/**
* Calculates the length of a dual quat
*
* @param {quat2} a dual quat to calculate length of
* @returns {Number} length of a
* @function
*/
export const length = quat.length;
/**
* Alias for {@link quat2.length}
* @function
*/
export const len = length;
/**
* Calculates the squared length of a dual quat
*
* @param {quat2} a dual quat to calculate squared length of
* @returns {Number} squared length of a
* @function
*/
export const squaredLength = quat.squaredLength;
/**
* Alias for {@link quat2.squaredLength}
* @function
*/
export const sqrLen = squaredLength;
/**
* Normalize a dual quat
*
* @param {quat2} out the receiving dual quaternion
* @param {quat2} a dual quaternion to normalize
* @returns {quat2} out
* @function
*/
export function normalize(out, a) {
let magnitude = squaredLength(a);
if (magnitude > 0) {
magnitude = Math.sqrt(magnitude);
out[0] = a[0] / magnitude;
out[1] = a[1] / magnitude;
out[2] = a[2] / magnitude;
out[3] = a[3] / magnitude;
out[4] = a[4] / magnitude;
out[5] = a[5] / magnitude;
out[6] = a[6] / magnitude;
out[7] = a[7] / magnitude;
}
return out;
}
/**
* Returns a string representation of a dual quatenion
*
* @param {quat2} a dual quaternion to represent as a string
* @returns {String} string representation of the dual quat
*/
export function str(a) {
return 'quat2(' + a[0] + ', ' + a[1] + ', ' + a[2] + ', ' + a[3] + ', ' +
a[4] + ', ' + a[5] + ', ' + a[6] + ', ' + a[7] + ')';
}
/**
* Returns whether or not the dual quaternions have exactly the same elements in the same position (when compared with ===)
*
* @param {quat2} a the first dual quaternion.
* @param {quat2} b the second dual quaternion.
* @returns {Boolean} true if the dual quaternions are equal, false otherwise.
*/
export function exactEquals(a, b) {
return a[0] === b[0] && a[1] === b[1] && a[2] === b[2] && a[3] === b[3] &&
a[4] === b[4] && a[5] === b[5] && a[6] === b[6] && a[7] === b[7];
}
/**
* Returns whether or not the dual quaternions have approximately the same elements in the same position.
*
* @param {quat2} a the first dual quat.
* @param {quat2} b the second dual quat.
* @returns {Boolean} true if the dual quats are equal, false otherwise.
*/
export function equals(a, b) {
let a0 = a[0],
a1 = a[1],
a2 = a[2],
a3 = a[3],
a4 = a[4],
a5 = a[5],
a6 = a[6],
a7 = a[7];
let b0 = b[0],
b1 = b[1],
b2 = b[2],
b3 = b[3],
b4 = b[4],
b5 = b[5],
b6 = b[6],
b7 = b[7];
return (Math.abs(a0 - b0) <= glMatrix.EPSILON * Math.max(1.0, Math.abs(a0), Math.abs(b0)) &&
Math.abs(a1 - b1) <= glMatrix.EPSILON * Math.max(1.0, Math.abs(a1), Math.abs(b1)) &&
Math.abs(a2 - b2) <= glMatrix.EPSILON * Math.max(1.0, Math.abs(a2), Math.abs(b2)) &&
Math.abs(a3 - b3) <= glMatrix.EPSILON * Math.max(1.0, Math.abs(a3), Math.abs(b3)) &&
Math.abs(a4 - b4) <= glMatrix.EPSILON * Math.max(1.0, Math.abs(a4), Math.abs(b4)) &&
Math.abs(a5 - b5) <= glMatrix.EPSILON * Math.max(1.0, Math.abs(a5), Math.abs(b5)) &&
Math.abs(a6 - b6) <= glMatrix.EPSILON * Math.max(1.0, Math.abs(a6), Math.abs(b6)) &&
Math.abs(a7 - b7) <= glMatrix.EPSILON * Math.max(1.0, Math.abs(a7), Math.abs(b7)));
}