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12_animation_ii - Computer Animation III Quaternions...

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Computer Animation III Quaternions Dynamics Some slides courtesy of Leonard McMillan and Jovan Popovic
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Recap: Euler angles 3 angles along 3 axis Poor interpolation, lock But used in flight simulation, etc. because natural http://www.fho-emden.de/~hoffmann/gimbal09082002.pdf
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Assignment 5: OpenGL Interactive previsualization ± OpenGL API ± Graphics hardware ± Jusr send rendering commands ± State machine Solid textures ± New Material subclass ± Owns two Material* ± Chooses between them ± “Shader tree”
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Final project First brainstorming session on Thursday Groups of three Proposal due Monday 10/27 ± A couple of pages ± Goals ± Progression Appointment with staff
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Final project Goal-based ± Simulate a visual effect ± Natural phenomena ± Small animation ± Game ± Reconstruct an existing scene Technique-based ± Monte-Carlo Rendering ± Radiosity ± Fluid dynamics
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Overview Interpolation of rotations, quaternions ± Euler angles ± Quaternions Dynamics ± Particles ± Rigid body ± Deformable objects
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Quaternion principle A quaternion = point on unit 3-sphere in 4D = orientation. We can apply it to a point, to a vector, to a ray We can convert it to a matrix We can interpolate in 4D and project back onto sphere ± How do we interpolate? ± How do we project?
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Quaternion recap 1 (wake up) 4D representation of orientation q = {cos( θ/2 ); v sin( θ/2 )} Inverse is q -1 =(s, - v ) Multiplication rule ± Consistent with rotation composition How do we apply rotations? How do we interpolate? v ( ) () 12 1 2 21 1 2 , ss vv sv v qq =− ++ × r rrr r r
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Quaternion Algebra Two general quaternions are multiplied by a special rule: Sanity check : {cos( α/2 ); v sin( α/2 )} {cos( β/2 ); v sin( β/2 )} {cos( α/2 )cos( β/2 ) - sin( α/2 ) v . sin( β/2 )} v , cos( β/2 ) sin( α/2 ) v + cos( α/2 )sin( β/2 ) v + v × v } {cos( α/2 )cos( β/2 ) - sin( α/2 )s in( β/2 ), v (cos( β/2 ) sin( α/2 ) + cos( α/2 ) sin( β/2 ))} {cos( (α+β)/2 ), v sin( (α+β)/2 ) } ( ) () 12 1 2 21 1 2 , ss vv sv v qq =− ++ × r rrr r r
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Quaternion Algebra Two general quaternions are multiplied by a special rule: To rotate 3D point/vector p by q , compute ± q {0; p } q -1 p= (x,y,z) q ={ cos( θ /2), 0,0,sin( θ /2) } = {c, 0,0,s} q {0, p }= {c, 0, 0, s} {0, x, y, z} ( ) () 12 1 2 21 1 2 , ss vv sv v qq =− ++ × r rrr r r = {c.0- zs, c p +0(0,0,s)+ (0,0,s) × p } = {-zs, c p + (-sy,sx,0) } q {0, p } q -1 = {-zs, c p + (-sy,sx,0) } {c, 0,0,-s} = {-zsc-(c p +(-sy,sx,0)).(0,0,-s), -zs(0,0,-s)+c(c p +(-sy, sx,0))+ (c p + (-sy,sx,0) ) x (0,0,-s) } = {0, (0,0,zs 2 )+c 2 p +(-csy, csx,0)+(-csy, csx, 0)+(s 2 x, s 2 y, 0)} = {0, (c 2 x-2csy-s 2 x, c 2 y+2csx-s 2 y, zs 2+ sc 2 )} = {0, x cos( θ )-y sin( θ ), x sin( θ )+y cos( θ ), z }
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Quaternion Interpolation (velocity) The only problem with l inear int erp olation (lerp) of quaternions is that it interpolates the straight line (the secant) between the two quaternions and not their spherical distance.
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This note was uploaded on 12/14/2011 for the course EECS 6.837 taught by Professor Durand during the Fall '03 term at MIT.

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12_animation_ii - Computer Animation III Quaternions...

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