Quaternions. Mike Bailey. A Useful Concept: Spherical Linear Interpolation. sin(1 t) sin t

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1 1 Quaternions This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License Mike Bailey mjb@cs.oregonstate.edu Quaternions.pptx A Useful Concept: Spherical Linear Interpolation sin(1 t) sin t Q'( t) P Q sin sin 0. t 1. where: + + sin 1 1

2 A Review of Complex Numbers 3 z x iy r[cos i sin ] re i z z x y x y ( i )( i ) 1 ) 1 1 x 1 (i.e., multiply the r s and add the θ s) i r1e i r e i( r1r e Since we are adding the θ s, we see that complex multiplication is really just rotation by θ if r = 1. Complex conjugate: Complex inverse: * i i x iy z 1 1 e r z re Note: (z)(z*) = r (z)(z -1 ) = 1. If r = 1., then z* = z -1 Quaternion Background 4 Discovered by Sir William Hamilton, 1843, while on a walk in Dublin. Legend says that he was so excited that he took out a knife and carved the equation into the stone of a bridge. Good thing spray-paint hadn t been invented yet Quaternions have 4 elements, one real and three complex: q q q q qqqq qq Q i j k (,,, ) (, ) By definition, 1 3 ii jj kk 1 ij k, jk i, ki j ji k, kj i, ik j 4 ijk 1 And, by definition, we always force Q 1. by making q q q q

3 Quaternion Multiplication 5 p p p p q q q q PQ ( i j k )( i j k ) p0q0 pq 1 1 pq p3q3 p1q0 p0q1 pq3 p3q PQ pq0 p0q p3q1 pq 1 3 p3q0 p0q3 pq 1 pq1 qp 0 p0q p q i j k Performing Rotations with Quaternions 6 A Quaternion can record a rotation transformation by an angle about an axis like this:,, where: q0 cos( ) q sin( ) nˆ Concatenated Quaternion Rotations are handled like this: R (,, )*, ) R 1 s rotation takes effect first, followed by R s A Quaternion can represent a point P like this: P Q(0, p) (0, p, p, p ) x y z A rotated point, P by one rotation is: P = * * A rotated point, P by multiple rotations is: P = 1 * * 1 3

4 Converting from a Quaternion to a Matrix 7 q q q q qq qq qq qq RQ ( ) qq qq q q q q qq qq qq 1 3qq 0 qq 3qq 0 1 q0q1qq3 From the GLM Notes: The Rotation Matrix for an Angle (θ) about an Arbitrary Axis (Ax, Ay, Az) 8 AA x x cos 1AA x x AA x y cos AA x y sinaz AA x z cos AA x z sina y M AA y x cos AA y x sinaz AA y y cos 1AA y y AA y z cos AA y z sina x AA z x cos AA z x sinay AA z y cos AA z y sinax AA z z cos 1 AA z z For this to be correct, A must be a unit vector 4

5 Converting from a Quaternion to a Matrix 9 q q q q qq qq qq qq RQ ( ) qq qq q q q q qq qq qq 1 3qq 0 qq 3qq 0 1 q0q1qq3 R = where: Note that the sum of the Trace (diagonal) elements is: Converting from a Matrix to a Quaternion 10 q q q q qq qq qq qq RQ ( ) qq qq q q q q qq qq qq 1 3qq 0 qq 3qq 0 1 q0q1qq3 Note that the sum of the Trace (diagonal) elements is: Letting t Trace( R) 1cos, then: 1 cos ( 1) t sin 1cos 5

6 Notes from Converting from a Quaternion to a Matrix 11 R= = R = Converting from a Matrix to a Quaternion 1 0 nzsin nysin 0 c b T R R nzsin 0 nxsin c 0 a nysin nxsin 0 b a 0 If we let a b c d a b c, then nˆ,, d d d q0 cos( ) q sin( ) nˆ 6

7 Quaternions in GLM 13 #include "glm/vec.hpp" #include "glm/vec3.hpp" #include "glm/mat4x4.hpp" #include "glm/gtc/matrix_transform.hpp" #include "glm/gtc/matrix_inverse.hpp" #include gtc/quaternion.hpp #include glm/gtx/quaternion.hpp glm::quat rot1 = glm::angleaxis( glm::radians(45.f), glm::vec3( 0.707f, 0.707f, 0. ) ); glm::quat rot = glm::angleaxis( glm::radians(90.f), glm::vec3( 1., 0., 0.,) ); glm::quat combinedrots = rot * rot1; glm::vec4 v = glm::vec4( 1., 1., 1., 1. ); glm::vec4 vp = combinedrots * v; glm::mat4 rotmatrix = glm::tomat4( combinedrots ); glm::vec4 vpp = rotmatrix * v; // same result as vp Converting from a Quaternion to OpenGL 14 glrotate3f( θ, n x, n y, n z ); glm::rotate( glm::quat const & q, θ R, glm::vec3( n x, n y, n z ) ); 7

8 A Useful Concept: Spherical Linear Interpolation, where P and Q are Quaternions 15 sin(1 t) sin t Q'( t) P Q sin sin 0. t 1. where: sin 1 My Code (Quat.h, Quat.cpp) 16 Rotate Slerp( float t, const Quat& p, const Quat& q ) { float angr; // angle between p and q in radians float c, s; // cosine and sine of the angle between p and q float cp, cq; // coefficients to multiply quaternions p and q Rotate r; // dot product to get the angle between p and q: c = p.s*q.s + p.vx*q.vx + p.vy*q.vy + p.vz*q.vz; angr = acos( c ); // sine of that angle: s = sin( angr ); // if the sine is 0., then p == q: if( s == 0. ) { r = p; return r; } // do spherical interpolation: cp = sin( (1.-t)*angr ) / s; cq = sin( t *angr ) / s; } r = cp*p + cq*q; return r; 8

9 int main( int argc, char *argv[ ] ) { Rotate r1 = Rotate( 45.*DR, 0., 0., 1. ); Rotate r = Rotate( 90.*DR, 0., 0., 1. ); Rotate r3 = Rotate( 90.*DR, 1., 0., 0. ); Rotate r4 = r * r1; Rotate r5 = r3 * r * r1; My Code (Quat.h, Quat.cpp) 17 fprintf( stderr, "r1 = %s\n", r1.tostring() ); fprintf( stderr, "r = %s\n", r.tostring() ); fprintf( stderr, "r*r1 = %s\n", r4.tostring() ); fprintf( stderr, "r3 = %s\n", r3.tostring() ); fprintf( stderr, "r3*r*r1 = %s\n", r5.tostring() ); fprintf( stderr, "\nr3*r*r1 matrix =\n" ); r5.printmatrix(); Point p1 = Point( 1., 1., 0. ); Point p = r4 * p1; fprintf( stderr, "Original point = %s\n", p1.tostring() ); fprintf( stderr, "Transformed point = %s\n", p.tostring() ); // try interpolating from r1 to r5: const int N = 10; float dt = 1. / (float)( N - 1 ); float t = 0.; for( int i = 0; i < N; i++, t += dt ) { Rotate r15 = Slerp( t, r1, r5 ); fprintf( stderr, "%d %5.3f %s\n", i, t, r15.tostring() ); } } My Code (Quat.h, Quat.cpp) 18 r1 = degrees about ( ) r = degrees about ( ) r*r1 = degrees about ( ) r3 = degrees about ( ) r3*r*r1 = degrees about ( ) r3*r*r1 matrix = Original point = Transformed point = degrees about ( ) degrees about ( ) degrees about ( ) degrees about ( ) degrees about ( ) degrees about ( ) degrees about ( ) degrees about ( ) degrees about ( ) degrees about ( ) 9

10 A Really Good (i.e., Complete) Reference 19 Andrew Hanson, Visualizing Quaternions, Morgan-Kaufmann,