Convert Vector to Euler (Cardan) Angles

Kuroyume0161 · 2006-06-26T08:55:00

Okay, I give up. I've scoured the internet and my texts. And I find the lack of information on this subject to be scandalous. I have a vector - a 3D vector, ya know v:(x,y,z) which is from the origin. I need to get Euler (properly Cardan henceforth) angles that would rotate a point, say (1,0,0), to match that 3D vector. When I say Cardan angles, I mean explicitly one of these: XYZ, XZY, YXZ, YZX, ZXY, ZYX. The rotation order may be any one of these, so no locked into a particular rotation order methods. No!! I cannot use quaternions or matrices or axis-angle or any other form. (PERIOD!) The deformation which I am forced to do *requires* (!!) Cardan angles since each axial rotation has a weight. So, no, don't bequest me to use quaternions or matrices or anything else. I need to convert a vector to Cardan Angles - I don't care if it involves quaternions, matrices, or axis-angles in between. The start is a 3D vector. The end is Cardan angles that take an axially-aligned vector and rotate it so as to match that 3D vector. And I know that there are two inevitable solutions (static frame and rotating frame). In this case, it should be safe to say that the frame is static - about the world coordinate system. I've tried Shoemake's approach first converting the 3D vector into an axis-angle version (three-points - origin, axial vector, 3D vector), then to matrix. The results have not been correct. For this approach, definitely need more information to fill in the process. This is a left-handed coordinate system with pre-concatenation of matrices, i.e.: Matrix m = RotZ() * RotY() * RotX() for an XYZ rotation matrix. Anyone smart enough to figure that out? Thanks, Robert

Math and Physics Programming

Started by Kuroyume0161 June 22, 2006 06:29 AM

10 comments, last by alvaro 17 years, 10 months ago

Kuroyume0161

100

Author

June 24, 2006 05:58 AM

Think that I have it figured out. And my suspicions were correct.

Taking the vector rVec and determining its main axis direction (if two values are the same, any of the two main axes will probably work similarly) tells me which axis to consider as the 'Roll' axis. Both vectors, rVec and eVec, are rotated so that their main axes (X or Y) are now the Z-axis. With that, I can calculate using the standard DirectionVector-to-EulerAngles maths. Then I just assign the Euler vector angles to the appropriate x,y,z. Now the rotations are 'minimal' and the 'Roll' axis as determined for the vector is always 0.0.

	// e- is the conformeE, r- is the conformeR	// - Make both endpoints local to world origin	Vector	eVec =	bc->GetVector(IPP_ENDPOINT) - bc->GetVector(IPP_ORIGIN);	BaseContainer*	bbc =	base->GetDataInstance();	Vector	rVec =	bbc->GetVector(IPP_ENDPOINT) - bbc->GetVector(IPP_ORIGIN);	// - If equivalent or either is zero vector (no direction), no more processing required	if (VectorEqual(eVec, rVec, EPSILONL))	return;	if (VectorEqual(eVec, zVec, EPSILONL))	return;	if (VectorEqual(rVec, zVec, EPSILONL))	return;	// - Determine main direction axis for vector	UCHAR	axis;	Vector	ar =	Vector(Abs(rVec.x), Abs(rVec.y), Abs(rVec.z));	// -- X	if ((ar.x > ar.y) && (ar.x > ar.z))	{		Matrix	align =	MatrixRotX(RAD_90) * MatrixRotZ(RAD_90) * MatrixScale(Vector(1.0));		// new y is old x, new z is old y, new x is old z		rVec *=	align;		eVec *=	align;		axis =	0;	}	// -- Y	else if (ar.y > ar.z)	{		// new x is old y, new z is old x, new y is old z		Matrix	align =	MatrixRotY(-RAD_90) * MatrixRotZ(-RAD_90) * MatrixScale(Vector(1.0));		rVec *=	align;		eVec *=	align;		axis =	1;	}	// -- Z	else	axis =	2;	// - Calculate angular offset between eVec/rVec Vectors in Rotation Order	// -- Calculate Euler angles of Conformer	Real	r;	Vector	eAngle;	Vector	rAngle;	r =	Sqrt(rVec.x*rVec.x + rVec.z*rVec.z);	rAngle.z =	0.0f;														// roll	rAngle.y =	(r < EPSILONS) ? 0.0f : Support::Angle(rVec.z, rVec.x);		// yaw	rAngle.x =	-Support::Angle(rVec.y, r);									// pitch	// -- Concatenate Rotation-order specific matrix	Matrix	mat =	MatrixMove(zVec) * MatrixRotX(rAngle.x) * MatrixRotY(rAngle.y) * MatrixRotZ(rAngle.z) * MatrixScale(Vector(1.0));	// -- Calculate Euler angles of Conformee	// Note that eVec is placed in the coordinate system where rVec is unrotated	eVec *=		!mat;	r =		Sqrt(eVec.x*eVec.x + eVec.z*eVec.z);	// These are the rotation angles from Conformee to Conformer Endpoint	eAngle.z =	0.0f;														// roll	eAngle.y =	(r < EPSILONS) ? 0.0f : -Support::Angle(eVec.z, eVec.x);	// yaw	eAngle.x =	Support::Angle(eVec.y, r);									// pitch	// Reorder rotations to avoid roll rotations	Vector	rotVec;	if (axis == 0)	{		rotVec.x =	eAngle.z;		rotVec.y =	eAngle.x;		rotVec.z =	eAngle.y;	}	else if (axis == 1)	{		rotVec.x =	eAngle.y;		rotVec.y =	eAngle.z;		rotVec.z =	eAngle.x;	}	else	{		rotVec.x =	eAngle.x;		rotVec.y =	eAngle.y;		rotVec.z =	eAngle.z;	}

Tests so far have all been positive. There is a likelyhood of finding an exception to this - a vector with equal non-zero values like v(10,10,10) would prove an interesting situation - but it should default to standard Z-axis Roll for better or worse.

Thanks,
Robert