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10_keyfrm_ik_opt

10_keyfrm_ik_opt - Animation Lecture 10 Slide 1 6.837 Fall...

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Animation Lecture 10 Slide 1 6.837 Fall 2003
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Conventional Animation Draw each frame of the animation great control tedious Reduce burden with cel animation layer keyframe inbetween cel panoramas (Disney’s Pinocchio) ... Lecture 10 Slide 2 6.837 Fall 2003
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Computer-Assisted Animation Lecture 10 Slide 3 6.837 Fall 2003 Keyframing automate the inbetweening good control less tedious creating a good animation still requires considerable skill and talent Procedural animation describes the motion algorithmically express animation as a function of small number of parameteres Example: a clock with second, minute and hour hands hands should rotate together express the clock motions in terms of a “seconds” variable the clock is animated by varying the seconds parameter Example 2: A bouncing ball Abs(sin( ω t+ θ 0 ))*e -kt Image adapted from: Lasseter, John. "Principles of Traditional Animation applied to 3D Computer Animation." ACM SIGGRAPH Computer Graphics 21, no. 4 (July 1987): 35-44.
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Computer-Assisted Animation Physically Based Animation Assign physical properties to objects (masses, forces, inertial properties) Simulate physics by solving equations Realistic but difficult to control Motion Capture Captures style, subtle nuances and realism You must observe someone do something Lecture 10 Slide 4 6.837 Fall 2003 Image removed due to copyright considerations.
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Overview Hermite Splines Keyframing Traditional Principles Articulated Models Forward Kinematics Inverse Kinematics Optimization Differential Constraints Lecture 10 Slide 5 6.837 Fall 2003 Squash & Stretch in Luxo Jr.'s Hop Image adapted from: Lasseter, John. "Principles of Traditional Animation applied to 3D Computer Animation." ACM SIGGRAPH Computer Graphics 21, no. 4 (July 1987): 35-44.
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Keyframing Describe motion of objects as a function of time from a set of key object positions. In short, compute the inbetween frames. Lecture 10 Slide 6 6.837 Fall 2003 Image adapted from: Lasseter, John. "Principles of Traditional Animation applied to 3D Computer Animation." ACM SIGGRAPH Computer Graphics 21, no. 4 (July 1987): 35-44.
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Interpolating Positions Given positions: find curve such that ( , , ), 0, , i i i x y t i n = K = ( ) ( ) ( ) x t t y t C = ( ) i i i x t y C 0 u 0 0 0 ( , , ) x y t 1 1 1 ( , , ) x y t 2 2 2 ( , , ) x y t ( ) t C Lecture 10 Slide 7 6.837 Fall 2003
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Linear Interpolation Simple problem: linear interpolation between first two points assuming : The x-coordinate for the complete curve in the figure: 0 0 0 ( , , ) x y t 1 1 1 ( , , ) x y t 2 2 2 ( , , ) x y t ) 0 1 0 1 0 1 1 0 1 0 2 1 1 2 1 2 2 1 2 1 , , ( ) , , t t t t x x t t t t t t t x t t t t t x x t t t t t t t + = + 0 1 =0 and =1 t t ( ) = + 0 1 ( ) 1 x t x t x t Derivation? Lecture 10 Slide 8 6.837 Fall 2003
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Polynomial Interpolation 0 0 0 ( , , ) x y t 1 1 1 ( , , ) x y t 2 2 2 ( , , ) x y t parabola An n-degree polynomial can interpolate any n+1 points. The Lagrange formula gives the n+1 coefficients of an n-degree polynomial that interpolates n+1 points. The resulting interpolating polynomials are called Lagrange polynomials. On the previous slide, we saw the Lagrange formula for n = 1.
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