Paper Summaries • Any takers? Interpolation Projects Factoids • Proposals due next week. • Digital Video Frame Rate – Star Wars was shot at 24 fps. – Keep film look Plan for today Motivation Films • Keyframing and Interpolation • Two early attempts of significant use of CG elements in film • But first… 1
Motivational Film Motivational Film • TRON • Question from SIGGRAPH Bowl – Disney (1982) – Q: Which special effects company was responsible for the effects of TRON? – Clips shown at SIGGRAPH 82 – A: ALL OF THEM • MAGI • Triple I • Robert Abel & Associates • Digital Effects Motivational Film Motivational Film • Interesting TRON Factoid • The Last Starfighter (1985) – No polygons! – Effects by Digital Productions – All objects were defined using mathematical – Universal Pictures formula. • Lesson Learned from TRON – Great effects – Bad movie – Overestimated the “Geek Factor” Motivational Film Motivational Film • Last Starfighter factoids • Life imitating art imitating life… – First movie to do all special effects (except makeup) on – Sort of… a computer. – A prototype for A Last Starfighter video game – All shots of spacecraft, space, etc. were generated on a was produced by Atari (remember them?) but Cray computer. was never distributed. • Lesson Learned from Last Starfighter – Note that there WAS a TRON video game – Great effects (made by Bally/Midway – 1983) – Bad movie – Overestimated the “Geek Factor” 2
Let’s get started 6. Straight ahead action and Pose-to-pose action • Key framing and interpolation • The two main approaches to hand drawn animation. – Straight ahead action • “the animator works straight ahead from his first drawing in the scene.” – Pose-to-pose (keyframing) • the animator plans his actions, figures out just what drawings will be needed to animate the business, makes the drawings concentrating on the poses, ... and then draws the inbetweens.” • Link Key framing Key Framing • The simplest form of animating an object. P 4 • Object has: P 2 P 5 – A beginning state Time = 50 Time = 10 – An end state Time = 55 P 1 – A number of intermediate “key” states P 6 P 3 • It is up to the keyframe system to determine Time = 0 the states for inbetween frames Time = 60 Time = 35 Key Framing Interpolation • 60 sec animation • According to webster.com – Video: 30 frames / sec = 1800 frames – in·ter·po·late (n) - to estimate values of (a function) between two known values – We have 6 key frames – Need to create position of objects for the remaining 1794 frames. – Note: “state” can be defined as a function (in this case of time) • F(t) 3
Interpolation Interpolation • Things to consider: • Things to consider: – Interpolation or approximation? – Continuity – defines “smoothness” of curve. • Interpolation – Control points fall on the curve – Determined by how many derivatives are • Approximation – Control points fall outside the continuous curve. • 0 th order – are the curves attached at the points themselves (positional continuity) • 1 st order – is the first derivative continuous (tangential continuity) • 2 nd order – is the 2 nd derivative continuous (curvature continuity) Interpolation – Continuity Interpolation • Things to consider: – Local vs. Global control • Does changing one point change the entire function? No positional Order 0 2 nd order Order 1 continuity continuity Continuity continuity – Questions? For animation purposes, Tangential (1 st Order) continuity is usually sufficient Linear Interpolation Linear Interpolation • The simplest solution • Pros – Straight-forward = − ⋅ + ⋅ P ( u ) ( 1 u ) P u P 0 1 – Easy to implement P(t) P(1) = P 1 Also can be – Direct relationship between time and value expressed as • Cons = − ⋅ + P ( u ) ( P P ) u P – Lacks tangential continuity 1 0 0 P(0) = P 0 – Why is this important? 4
Cubic Interpolation Remember our principles of animation? • 8. Arcs • x, y, z given as function of t – “The visual path of action from one extreme to • Approximated by cubic polynomials another is always described by an arc.” = + + + 3 2 x ( t ) a t b t c t d x x x x = + + + 3 2 y ( t ) a t b t c t d y y y y = + + + 3 2 z ( t ) a t b t c t d z z z z This is not an arc! Cubic Interpolation Cubic Interpolation • Separate C matrix into basis + geometry ⎡ ⎤ a a a y x z ⎢ ⎥ b [ ] b b ⎢ ⎥ = ⎡ ⎤ ⎡ ⎤ = y Q ( t ) x ( t ) y ( t ) z ( t ) x z G m m m m G G C 1 y ⎢ ⎥ 11 12 13 14 1 x 1 z ⎢ ⎥ ⎢ ⎥ c c c m m m m G G G ⎢ ⎥ ⎢ y ⎥ ⎢ ⎥ [ ] x z = 2 y 21 22 23 24 2 x 2 z C T = 3 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ t t t 1 d ⎣ d d ⎦ G m m m m G G y 3 y x z 31 32 33 34 ⎢ 3 x 3 z ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ m m m m ⎦ ⎣ G G G ⎦ = ⋅ 1 4 4 4 4 2 4 4 4 4 3 4 y 41 42 43 44 1 4 2 4 4 x 4 3 4 4 z Q T C basis matrix control points Cubic Interpolation Cubic Interpolation • Hermite Curves = ⋅ ⋅ – specified by endpoints and tangent vectors from Q ( t ) T M G { { each endpoint basis control • Basis & Control points distinguishes types of curves 5
Cubic Interpolation Cubic Interpolation • Hermite Curves • Problem with Hermite Curves – Animator generally will not enjoy specifying ⎡ − ⎤ ⎡ ⎤ 2 2 1 1 P 1 tangent vectors ⎢ ⎥ ⎢ ⎥ − − − 3 3 2 1 ⎢ ⎥ P ⎢ ⎥ = = 4 M G H ⎢ ⎥ ⎢ ⎥ H 0 0 1 0 R ⎢ ⎥ 1 ⎢ ⎥ – Solution: The Catmull-Rom spline ⎣ 1 0 0 0 ⎦ ⎣ R ⎦ 4 Cubic Interpolation Cubic Interpolation • Catmull-Rom spline • Catmull-Rom – Uses previous and next control point to determine − − ⎡ ⎤ ⎡ ⎤ 1 3 3 1 P tangent vector − i 1 ⎢ ⎥ ⎢ ⎥ − − 2 5 4 1 1 P ⎢ ⎥ ⎢ ⎥ = = i M G ⎢ − ⎥ CR ⎢ ⎥ CR 2 1 0 1 0 P + ⎢ ⎥ ⎢ i 1 ⎥ ⎣ ⎦ 0 2 0 0 ⎣ P ⎦ + i 2 Note: for interpolation between P i and P i+1 you need 4 points – Spline goes through all control points! Cubic Interpolation Cubic Interpolation • Bezier Curves • Bezier Curves – specified by endpoints and intermediate points, − − ⎡ ⎤ ⎡ ⎤ 1 3 3 1 P not on the curve, that approximate the tangent ⎢ ⎥ 0 ⎢ ⎥ − − vectors 3 6 3 0 P ⎢ ⎥ ⎢ ⎥ = = 1 M G H ⎢ ⎥ B − ⎢ ⎥ 3 3 0 0 P ⎢ ⎥ 2 ⎢ ⎥ ⎣ ⎦ 1 0 0 0 ⎣ ⎦ P 3 6
Cubic Interpolation Cubic Interpolation • Bezier Curves suffer from same problem as • B-Splines Hermite Curves – Control points are outside of the curve – Animator generally will not enjoy specifying tangent – Local control vectors…even from control points • Moving one control point only affects a portion of • Exception: Provide animator with an interactive curve editor the entire curve. – DeCastelau Construction – Individual curve segments share control points. • Algorithm to automatically define Bezier control points • Described in Parent Appendix and Shomake Quat. paper – Knots - points connecting individual curve segments. Cubic Interpolation Cubic Interpolation • B-Splines • B-Splines - local control Cubic Interpolation Cubic Interpolation • Segment Q i uses control points P i-3 , P i-2 , P i-1 and P i • B-Splines − − ⎡ ⎤ ⎡ ⎤ 1 3 3 1 P − i 3 ⎢ ⎥ ⎢ ⎥ − − 3 6 3 0 1 P ⎢ ⎥ ⎢ ⎥ = = − i 2 M G ⎢ ⎥ B − ⎢ ⎥ H 6 3 0 3 0 P − ⎢ ⎥ ⎢ i 1 ⎥ ⎣ ⎦ 1 4 1 0 ⎣ ⎦ P i 7
Cubic Interpolation Cubic Interpolation • Rational Splines • Uniform Nonrational B-Splines – expressed using homogeneous coordinates – uniform - knots are equally spaced [ ] = – nonrational - does not use homogeneous Q ( t ) X ( t ) Y ( t ) Z ( t ) W ( t ) coordinates (more on this later) – B - basis -- curves are weighted sums of a basis X ( t ) Y ( t ) Z ( t ) = = = function x ( t ) , y ( t ) , z ( t ) W ( t ) W ( t ) W ( t ) Cubic Interpolation Cubic Interpolation • Rational splines -- why bother? • NURBS – Invariant under rotation, scaling, translation, – Non-uniform Rational B-Splines and perspective transforms – Non-uniform - knots need not be equally – Can define precisely any of the conic sections. spaced. (spheres, ellipsoids, parabolids, etc) – Most useful and used method for representing surface curves in CG. – Probably overkill for cubic interpolation Cubic Interpolation Cubic Splines • Pros – We now have our tangential continuity u=0.0 – More complex than linear but not too complex • Cons u=2/3 – Spline control does not correspond to time. u=1.0 u=1/3 – May require an interactive “curve editor” tool. u does not correspond to t. 8
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