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Skinning

Skinning. CS418 Computer Graphics John C. Hart. Simple Inverse Kinematics. Given target point ( x , y ) in position space, what are the parameters ( q , f ) in configuration space that place the hand on the target point? Use Law of Cosines to find q d 2 = a 2 + b 2 – 2 ab cos q

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Skinning

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  1. Skinning CS418 Computer Graphics John C. Hart

  2. Simple Inverse Kinematics • Given target point (x,y) in position space,what are the parameters (q,f) in configuration space that place the hand on the target point? • Use Law of Cosines to find q d2 = a2 + b2 – 2abcosq cosq = (a2 + b2 – d2)/2ab cosq = (a2 + b2 – x2 – y2)/2ab • And to find a cosa = (a2 + d2 – b2)/2ad cosa = (a2 + x2 + y2 – b2)/2ad • Use arctangent to find b then f b = atan2(y,x) f = a – b (0,0) (x,y) q f b a (x,y) d (0,0) a q b a (x,y) d b (0,0) f a

  3. Skinning M2R(q 2) M1R(q1) M2R(q2) R(q2) • Elbow joints don’t look realistic because geometry detaches • Transformation hierarchy: • R(q1) rotates upper-arm cylinder about its shoulder at the origin • M1 moves upper-arm cylinder from the origin to its position in world coordinates • R(q2) rotates forearm cylinder about its elbow at the origin • M2 moves forearm elbow from the origin to the end of the upper-arm cylinder when its shoulder is based at the origin • When q2  0 the elbow end of the upper-arm does not align with the elbow end of the forearm M1R(q1) M2 M1R(q1) M2R(q2) M1R(q1)

  4. Skinning M1R(q1) M2R(0) • Solution is to interpolate matrices from the undetached coordinate frame into the correctly oriented coordinate frame per-vertex • Let Mstraight = M1R(q1) M2R(0) Mbent = M1R(q1) M2R(q2) • Distribute (“paint”) weights w on vertices of forearm cylinder • w = 0 at elbow end • w = 1 after elbow • Transform vertices using M(w) = (1 – w) Mstraight + wMbent M1R(q1) M2R(q2) M1R(q1) M2R(2/3q2) w = 1 M1R(q1) M2R(1/3q2) w = 2/3 w= 1/3 M1R(q1)

  5. Build an Elbow glPushMatrix(); glColor3f(0,0,1); glTranslatef(0,-2,0); drawquadstrip(); glPopMatrix(); glPushMatrix(); glColor3f(1,1,0); glRotatef(elbow,0,0,1); glTranslate(0,0,2); drawquadstrip(); glPopMatrix();

  6. Two Coordinate Systems glPushMatrix(); glColor3f(0,0,1); glTranslatef(0,-2,0); drawquadstrip(); glColor3f(1,1,0,.5) glTranslatef(0,4,0); drawquadstrip(); glPopMatrix(); glPushMatrix(); glRotatef(elbow,0,0,1); glColor3f(1,1,0); glTranslatef(0,0,2); drawquadstrip(); glColor3f(0,0,1,.5); glTranslatef(0,0,-4); drawquadstrip(); glPopMatrix(); Yellow limb in blue limb’s coordinate system Blue limb in yellow limb’s coordinate system

  7. Interpolate the Transformations for (i = 0; i < 8; i++) { weight = i/7.0; glPushMatrix(); glRotatef(weight*elbow,0,0,1); glTranslate3f(0,0,-3.5+i); drawquad(); glPopMatrix(); }

  8. Interpolate the Vertices glBegin(GL_QUAD_STRIP); for (i = 0; i <= 8; i++) { weight = i/8.0; glColor3f(weight,weight,1-weight); glPushMatrix(); glRotatef(weight*elbow,0,0,1); glVertex2f(-1,-4.+i); glVertex2f(1,-4.+i); glPopMatrix(); } glEnd(/*GL_QUAD_STRIP*/);

  9. Interpolate the Matrices glLoadIdentity(); glGetMatrixf(A); glRotatef(elbow,0,0,1); glGetMatrixf(B); glBegin(GL_QUAD_STRIP); for (i = 0; i <= 8; i++) { weight = i/8.0; glColor3f(weight,weight,1-weight); C = (1-weight)*A + weight*B; glLoadIdentity(); glMultMatrix(C); glVertex2f(-1,-4.+i); glVertex2f(1,-4.+i); } glEnd(/*GL_QUAD_STRIP*/);

  10. Matrix Palette Skinning • Each vertex has one or more weight attributes associated with it • Each weight determines the effect of each transformation matrix • “Bones” – effect of each transformation is described by motion on bone from canonical position • Weights can be painted on a meshed model to control effect of underlying bone transformations (e.g. chests, faces)

  11. Interpolating Matrices • Skinning interpolates matrices by interpolating their elements • Identical to interpolating vertex positions after transformation • We’ve already seen problems with interpolating rotation matrices • Works well enough for rotations with small angles • Rotations with large angles needs additional processing (e.g. polar decomposition) • Quaternions provide a better way to interpolate rotations… • (aA + bB)p = a(Ap) + b(Bp) • a,b = weights • A,B = matrices • p = vertex position From: J. P. Lewis, Matt Cordner, and Nickson Fong. “Pose space deformation: a unified approach to shape interpolation and skeleton-driven deformation.”Proc. SIGGRAPH 2000

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