/// <summary>
/// Calculates value and derivative of the position of the sphere with respect to whichDeriv
/// </summary>
float CalculateErrorDerivative()
{
DualVector3 thisYBasis = optimizer.vec("yBasis"), thisZBasis = optimizer.vec("zBasis");
thisYBasis = thisYBasis.Normalize(); thisZBasis = thisZBasis.Normalize();
DualVector3 thisXBasis = thisYBasis.Cross(thisZBasis);
thisYBasis = thisZBasis.Cross(thisXBasis);
//THIS IS A FULLY DIFFERENTIABLE RIGID TRANSFORM - NO SINGULARITIES
//Also Single Covers the space (when non-uniform scaling is disabled)
DualVector3 scale = optimizer.vec("scale");
DualMatrix4x3 thisTransform = new DualMatrix4x3(thisXBasis * scale.x, thisYBasis * scale.y, thisZBasis * scale.z, optimizer.vec("position"));
//Sum the Squared Errors
DualNumber error = new DualNumber();
for (int i = 0; i < icoVerts.Length; i++)
{
DualVector3 thisPoint = thisTransform * icoVerts[i]; DualVector3 basePoint = baseTransformMatrix * icoVerts[i];
if (drawDebug)
{
Debug.DrawLine(thisPoint, basePoint);
}
error += (thisPoint - basePoint).SqrMagnitude();
}
//Divide by Number of Squared Errors
error /= icoVerts.Length;
return(error.Derivative);
}
void Update()
{
baseTransformMatrix = new DualMatrix4x3(baseTransform.position, baseTransform.rotation, baseTransform.localScale);
Vector3 positionGradient = Vector3.zero;
if (fitPosition)
{
//Calculate derivative of the error wrt x, y, and z
positionGradient.x = SphereFittingError(0, true).Derivative;
positionGradient.y = SphereFittingError(1, false).Derivative;
positionGradient.z = SphereFittingError(2, false).Derivative;
}
Vector3 yBasisGradient = Vector3.zero;
Vector3 zBasisGradient = Vector3.zero;
if (fitRotation)
{
//Calculate derivative of the error wrt yx, yy, and yz
yBasisGradient.x = SphereFittingError(3, false).Derivative;
yBasisGradient.y = SphereFittingError(4, false).Derivative;
yBasisGradient.z = SphereFittingError(5, false).Derivative;
//Calculate derivative of the error wrt zx, zy, and zz
zBasisGradient.x = SphereFittingError(6, false).Derivative;
zBasisGradient.y = SphereFittingError(7, false).Derivative;
zBasisGradient.z = SphereFittingError(8, false).Derivative;
}
Vector3 scaleGradient = Vector3.zero;
if (fitScale)
{
//Calculate derivative of the error wrt sx, sy, and sz
scaleGradient.x = SphereFittingError(9, false).Derivative;
scaleGradient.y = SphereFittingError(10, false).Derivative;
scaleGradient.z = SphereFittingError(11, false).Derivative;
}
//Apply the System
transform.position -= positionGradient * alpha;
Vector3 forwardBasis = transform.forward - (zBasisGradient * alpha);
Vector3 upBasis = transform.up - (yBasisGradient * alpha);
transform.rotation = Quaternion.LookRotation(forwardBasis, upBasis);
transform.localScale -= scaleGradient * alpha;
}
/// <summary>
/// Calculates value and derivative of the position of the sphere with respect to whichDeriv
/// </summary>
DualNumber SphereFittingError(int whichDeriv, bool drawDebug)
{
DualVector3 thisPosition = new DualVector3(
whichDeriv == 0 ? DualNumber.Variable(transform.position.x) : DualNumber.Constant(transform.position.x),
whichDeriv == 1 ? DualNumber.Variable(transform.position.y) : DualNumber.Constant(transform.position.y),
whichDeriv == 2 ? DualNumber.Variable(transform.position.z) : DualNumber.Constant(transform.position.z));
DualVector3 thisYBasis = new DualVector3(
whichDeriv == 3 ? DualNumber.Variable(transform.up.x) : DualNumber.Constant(transform.up.x),
whichDeriv == 4 ? DualNumber.Variable(transform.up.y) : DualNumber.Constant(transform.up.y),
whichDeriv == 5 ? DualNumber.Variable(transform.up.z) : DualNumber.Constant(transform.up.z));
DualVector3 thisZBasis = new DualVector3(
whichDeriv == 6 ? DualNumber.Variable(transform.forward.x) : DualNumber.Constant(transform.forward.x),
whichDeriv == 7 ? DualNumber.Variable(transform.forward.y) : DualNumber.Constant(transform.forward.y),
whichDeriv == 8 ? DualNumber.Variable(transform.forward.z) : DualNumber.Constant(transform.forward.z));
DualVector3 thisScale = new DualVector3(
whichDeriv == 9 ? DualNumber.Variable(transform.localScale.x) : DualNumber.Constant(transform.localScale.x),
whichDeriv == 10 ? DualNumber.Variable(transform.localScale.y) : DualNumber.Constant(transform.localScale.y),
whichDeriv == 11 ? DualNumber.Variable(transform.localScale.z) : DualNumber.Constant(transform.localScale.z));
thisYBasis = thisYBasis.Normalize(); thisZBasis = thisZBasis.Normalize();
DualVector3 thisXBasis = thisYBasis.Cross(thisZBasis);
thisYBasis = thisZBasis.Cross(thisXBasis);
//THIS IS A FULLY DIFFERENTIABLE RIGID TRANSFORM - NO SINGULARITIES
DualMatrix4x3 thisTransform = new DualMatrix4x3(thisXBasis * thisScale.x, thisYBasis * thisScale.y, thisZBasis * thisScale.z, thisPosition);
//Sum the Squared Errors
DualNumber error = new DualNumber();
for (int i = 0; i < icoVerts.Length; i++)
{
DualVector3 thisPoint = thisTransform * icoVerts[i]; DualVector3 basePoint = baseTransformMatrix * icoVerts[i];
if (drawDebug)
{
Debug.DrawLine(thisPoint, basePoint);
}
error += (thisPoint - basePoint).SqrMagnitude();
}
//Divide by Number of Squared Errors
error /= icoVerts.Length;
return(error);
}
void Update()
{
baseTransformMatrix = new DualMatrix4x3(baseTransform.position, baseTransform.rotation, baseTransform.localScale);
//Set up the variables to optimize
optimizer.Add(transform.position, "position");
optimizer.Add(transform.up, "yBasis");
optimizer.Add(transform.forward, "zBasis");
optimizer.Add(transform.localScale, "scale");
optimizer.CalculateErrorDerivative = CalculateErrorDerivative;
//Calculate and Apply the Instantaneous Gradients at this point in time
transform.position -= optimizer.CalcErrorGradient("position") * alpha;
Vector3 forwardBasis = transform.forward - (optimizer.CalcErrorGradient("zBasis") * alpha);
Vector3 upBasis = transform.up - (optimizer.CalcErrorGradient("yBasis") * alpha);
transform.rotation = Quaternion.LookRotation(forwardBasis, upBasis);
transform.localScale -= optimizer.CalcErrorGradient("scale") * alpha;
}