I have a 3D car which follows a predefined 3D Bezier path. I want the car's front wheels' rotation to match the car's changing direction.
I had the idea to match the wheel's orientation to the derivative of the path's direction (3D vector), aka the 2nd degree derivative of the Bezier path.
For some reason, this barely works. At some point it seems to work fine, while at others the wheel spins like hell. I noted that the 2nd degree derivative changes even when the Bezier path is a straight line: AFAIK in this case it should be 0.
So, my 1st question is if my idea to match the wheel's rotation to the 2nd degree is the way to go. If yes, my 2nd question is what on earth is going wrong?
Here is my Bezier 3D curve code:
package fanlib.math {
import flash.geom.Vector3D;
public class BezierCubic3D
{
public const anchor1:Vector3D = new Vector3D();
public const anchor2:Vector3D = new Vector3D();
public const control1:Vector3D = new Vector3D();
public const control2:Vector3D = new Vector3D();
/**
* Gets values from both 'getPointAt' and 'getDirectionAt'
*/
public const result:Vector3D = new Vector3D();
private const previous:Vector3D = new Vector3D(); // temporary (optimization)
// normalization aka arc-parameterization
public var arcLengths:Vector.<Number> = new Vector.<Number>;
public var steps:Number = 100;
private var _length:Number;
public function BezierCubic3D()
{
}
/**
* To get a point between anchor1 and anchor2, pass value [0...1]
* @param t
*/
public function getPointAt(t:Number):Vector3D {
const t2:Number = t*t;
const t3:Number = t*t2;
const threeT:Number = 3*t;
const threeT2:Number = 3*t2;
result.x = getPointAxisAt(anchor1.x, anchor2.x, control1.x, control2.x, t3, threeT, threeT2);
result.y = getPointAxisAt(anchor1.y, anchor2.y, control1.y, control2.y, t3, threeT, threeT2);
result.z = getPointAxisAt(anchor1.z, anchor2.z, control1.z, control2.z, t3, threeT, threeT2);
return result;
}
public function getPointAxisAt(a1:Number,a2:Number,c1:Number,c2:Number, t3:Number, threeT:Number, threeT2:Number):Number {
return t3 * (a2+3*(c1-c2)-a1) +
threeT2 * (a1-2*c1+c2) +
threeT * (c1-a1) +
a1;
}
/**
* @param t
* @return Un-normalized Vector3D!
*/
public function getDirectionAt(t:Number):Vector3D {
const threeT2:Number = 3 * t * t;
const sixT:Number = 6 * t;
result.x = getDirAxisAt(anchor1.x, anchor2.x, control1.x, control2.x, threeT2, sixT);
result.y = getDirAxisAt(anchor1.y, anchor2.y, control1.y, control2.y, threeT2, sixT);
result.z = getDirAxisAt(anchor1.z, anchor2.z, control1.z, control2.z, threeT2, sixT);
return result;
}
public function getDirAxisAt(a1:Number,a2:Number,c1:Number,c2:Number, threeT2:Number, sixT:Number):Number {
return threeT2 * (a2+3*(c1-c2)-a1) +
sixT * (a1-2*c1+c2) +
3 * (c1-a1);
}
public function getDirectionDerivativeAt(t:Number):Vector3D {
const sixT:Number = 6 * t;
result.x = getDirDerAxisAt(anchor1.x, anchor2.x, control1.x, control2.x, sixT);
result.y = getDirDerAxisAt(anchor1.y, anchor2.y, control1.y, control2.y, sixT);
result.z = getDirDerAxisAt(anchor1.z, anchor2.z, control1.z, control2.z, sixT);
return result;
}
public function getDirDerAxisAt(a1:Number,a2:Number,c1:Number,c2:Number, sixT:Number):Number {
return sixT * (a2+3*(c1-c2)-a1) +
6 * (a1-2*c1+c2);
}
/**
* Call this after any change to defining points and before accessing normalized points of curve.
*/
public function recalc():void {
arcLengths.length = steps + 1;
arcLengths[0] = 0;
const step:Number = 1 / steps;
previous.copyFrom(getPointAt(0));
_length = 0;
for (var i:int = 1; i <= steps; ++i) {
_length += Vector3D.distance(getPointAt(i * step), previous);
arcLengths[i] = _length;
previous.copyFrom(result);
}
}
/**
* 'recalc' must have already been called if any changes were made to any of the defining points
* @param u
* @return u normalized/converted to t
*/
public function normalizeT(u:Number):Number {
var targetLength:Number = u * arcLengths[steps];
var low:int = 0,
high:int = steps,
index:int; // TODO : have a look-up table of starting low/high indices for each step!
while (low < high) {
index = low + ((high - low) >>> 1);
if (arcLengths[index] < targetLength) {
low = index + 1;
} else {
high = index;
}
}
if (this.arcLengths[index] > targetLength) {
--index;
}
var lengthBefore:Number = arcLengths[index];
if (lengthBefore === targetLength) {
return index / steps;
} else {
return (index + (targetLength - lengthBefore) / (arcLengths[index + 1] - lengthBefore)) / steps;
}
}
public function getNormalizedPointAt(u:Number):Vector3D {
return getPointAt(normalizeT(u));
}
/**
* "Normalized" goes for t, not the return Vector3D!!!
* @param u
* @return Un-normalized Vector3D!
*/
public function getNormalizedDirectionAt(u:Number):Vector3D {
return getDirectionAt(normalizeT(u));
}
public function getNormalizedDirectionDerivativeAt(u:Number):Vector3D {
return getDirectionDerivativeAt(normalizeT(u));
}
public function get length():Number
{
return _length;
}
}
}
And here is the code that applies the 2nd degree derivative orientation to the car's wheels:
const dirDer:Vector3D = bezier.getDirectionDerivativeAt(time);
dirDer.negate(); // negate vector's values; for some reason, this gives better results
for each (wheel in dirWheels) {
wheel.setRotation(0,0,0); // must nullify before below line
const localDirDer:Vector3D = wheel.globalToLocalVector(dirDer); // convert dirDer vector to wheel's local axis; wheel translation does NOT affect conversion
wheel.setOrientation(localDirDer); // orients the object in a specific direction; Up-vector's default value = (0,1,0)
}
I even tried (to no avail):
for each (wheel in dirWheels) {
const localDirDer:Vector3D = wheel.parent.globalToLocalVector(dirDer); // convert dirDer vector to wheel's local axis; wheel translation does NOT affect conversion
wheel.setOrientation(localDirDer); // orients the object in a specific direction; Up-vector's default value = (0,1,0)
}
One clear example that something is wrong: even when the car is on a straight line, the wheel originally is non-rotated (as it should), but after the car passes the center point of the line, the wheel rotates 180 degrees!
