CS5620 Intro to Computer Graphics

Transcription

1 CS56 and Quaternions Piar s Luo Jr. A New Dimension - Time 3 4 Principles of Traditional Specifing Anticipation Suash/Stretch Secondar Action 5 6 C. Gotsman, G. Elber,. Ben-Chen Page

2 CS56 Keframes anual Tweening Automaticall fill in motion between these points in time 7 Tweening = Interpolation 8 VRL Eamples in VRL Fred Floops DEF PI PositionInterpolator { ke [.5 ] t i kevalue [ 5, ((t i ) (t i ) (t i )) 5, 5 DEF OI OrientationInterpolator { ke [ ] kevalue [, , t i (n (t i ) n (t i ) n (t i ) (t i )) uaternion rotation without uaternions rotation with uaternions 9 Three Degrees of Freedom Orientation and Euler Angles The orientation of an 3D object ma be described b three Euler angles: successive rotations (relative to an initial orientation) around three aes: -roll: rotation around ais. -roll: rotation around ais. -roll: rotation around ais. The order of rotation application is important, as rotations do not commute. Gimball An obvious wa to interpolate two orientations is to interpolate their Euler angles. C. Gotsman, G. Elber,. Ben-Chen Page

3 CS56 Rotation b Euler Angles Gimbal Lock R(,, ) R R R R ( ) cos sin sin cos R cos sin ( ) R cos ( ) sin sin cos sin cos cos cos sin cos cos sin sin sin sin cos sin cos sin cos cos cos sin sin sin cos sin sin sin cos sin cos sin cos cos When this is: sin cos cos sin sin sin cos cos cos cos sin sin cos sin sin cos where = +, so one degree of freedom is lost. sin cos cos sin 3 4 General Rotations Euler, 756 Theorem: (Euler): In 3D, an displacement of a rigid bod such that a point on the rigid bod remains fied, is euivalent to a single rotation about some ais that runs through the fied point. v n v Gimbal Lock Euivalent: In 3D space, an two Cartesian coordinate sstems with a common origin are related b a rotation about some fied ais. Conclusion: An 3D orientation ma be described b four parameters: angle and unit ais n=(n,n,n ). 5 6 R( v) R( v v ) General Rotations (cont d) n R( v ) R( v ) v v cos ( nv)sin ( nv) n ( v ( n v) n)cos ( nv)sin v cos n( nv)( cos ) ( nv)sin v v v Rv ( ) n v v ( nv) n v v v R( v ) v cos ( nv )sin R( v ) v v cos ( nv)sin Hamilton 843 Quaternions Definition: A uaternion is a uadruple =[s,v], where s is a scalar, and v a three-dimensional vector. The uaternions form a non-commutative group under the multiplication rule: Euivalent to: [ s, v ] [ s, v ] [ s s v v, s v s v v v ] s iv jv kv where i j k, ij k, ji k 7 8 C. Gotsman, G. Elber,. Ben-Chen Page 3 3

4 CS56 Quaternions (cont d) Special Cases The conjugate of [ s,v] is [ s,-v] The inverse of [ s,v] is: Corollar: - Scalars: [c,(,,)] Comple numbers: [,(,,)] 3D Vectors: [,(,,)] The norm of [ s,v] is s v v v [ s, v ] [ s, v ] [ s s v v, s v s v v v ] 9 Rotating with Quaternions Rotation b around unit direction n ma be represented b the unit uaternion right hand coordinate sstem Eamples The 3D vector [,v] is rotated b to: R v v v Since () [cos, nsin ] R ( v) [cos, nsin ] [, v] [cos, nsin ] [, v(cos sin ) n( nv)sin ( nv)cos sin ] [, vcos n( nv)( cos ) ( nv)sin ] n (,,), n (,,), / n (,,), / (,,,) (/,,, / ) (/,/,,) (cos, n sin ) ( w,,, ) Unit Quaternions The uaternions used for rotation have onl three degrees of freedom. The all lie on the surface of a unit sphere in 4D space, forming a subgroup. Theorem: A vector v is invariant under rotations around an ais through v. Proof: The rotation operator is =[s,cv] such that =. v [ s, cv] [, v] [ s, cv] [ s, cv] [ cv v, sv] [, s v c ( v v) v] ( s c ( v v))[, v] v v Theorem: Rotation b and - in the opposite direction are euivalent. Proof: [cos, nsin ] [cos, nsin ] Theorem: Rotation b and then b (around n) is euivalent to rotation b +. ( ( v ) ) ( ) v ( ) Proof: ( ) v( ) [cos, nsin ] [cos, nsin ] [cos cos sin sin, n(sin cos cos sin ] [cos, nsin ] 3 Quaternion to Rotation atri (column vectors) Rotation b unit-length uaternion: [cos, n sin ] [ w,,, ] R ([, v]) [, v] v cos n( n v)( cos ) ( n v)sin The first row of the euivalent rotation matri: R ([, v ]) v co s n ( n v n v n v )( co s ) ( n v n v ) sin 3 cos n ( cos ) n n ( cos ) n sin n n ( cos ) n sin 4 C. Gotsman, G. Elber,. Ben-Chen Page 4 4

5 CS56 Quaternion to Rotation atri (column vectors) [cos, n sin ] [ w,,, ] w w cos n ( cos ) sin n (sin ) ( n n n )sin n (sin ) ( n n )sin w w w w 5 Rotation atri to Quaternion Given rotation matri The euivalent uaternion is = [w,,, ] w w 4 4( ) 4w 33 w 33 4 w 3 3 4w w 3 3 4w w 4w w w w w 6 right hand coordinate sstem Eamples Prove Using Quaternions and atrices The uaternion i=(,,,) is rotation b around the ais. The uaternion j=(,,,) is rotation b around the ais. The uaternion k=(,,,) is rotation b around the ais. n (,,), (,,,) n (,,), / (/,,, / ) n (,,), / (/,/,,) (cos, n sin ) ( w,,, ) 7 8 in VRL DEF PI PositionInterpolator { ke [.5 ] t i kevalue [ 5, ((t i ) (t i ) (t i )) 5, 5 DEF OI OrientationInterpolator { ke [ ] kevalue [, , t i (n (t i ) n (t i ) n (t i ) (t i )) uaternion Geodesics A great circle is a circle on the surface of a sphere that has the same circumference as the sphere, dividing the sphere into two eual hemispheres. A great circle is a spherical geodesic the shortest path between two points on a sphere. rotation without uaternions rotation with uaternions 9 3 C. Gotsman, G. Elber,. Ben-Chen Page 5 5

6 CS56 Interpolating Two Orientations Orientations ma be interpolated b interpolating their respective uaternions. Quaternions on a 4D sphere ma be interpolated along a geodesic (the shortest path between two points on a sphere). For an u[,], writing (u) = (u) +(u), and solving the following sstem of euations for (u) and (u): cos,, cos( u ), cos cos( u ) cos u cos cos ( cos ) cos cos, ( u) ( u) ( u) ( u) ( u), cos( u ), ( u) ( u) ( u), u (u) u[,] () () sin u sin( u) ( u) ( u) sin sin Spherical Linear Interpolation (SLERP) sin( u) sin( u) ( u) u [,] sin sin 3 Euivalent to: ( u) ( ) u 3 Interpolating ultiple Orientations As with linear interpolation between multiple positions, spherical linear interpolation between multiple orientations generates discontinuities in the derivative at the interpolated points. This ma be solved b more elaborate interpolation schemes, which are not shortest path, similarl to splines for positions. v v 3 v v 33 C. Gotsman, G. Elber,. Ben-Chen Page 6 6