14-curve-surface

# 14-curve-surface - Lecture 15 Curve Surface and Subdivision...

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Lecture 15 Curve, Surface, and Subdivision

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Part I: Curve
Reading (for Curve & Surface) Required • Hearn & Baker, 10.6 -10.9 • Foley, 11.2 Optional •B artels, B eatty, and B arsky. An Introduction to Splines for use in Computer Graphics and Geometric Modeling. 1987. • Farin. Curves and Surfaces CAGD: A Practical Guide. 4th ed. 1997.

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Curves Before Computers • The loftsman’s or carpenter’s spline: – long, narrow strip of wood or metal – shaped by lead weights called “ducks” – gives curves with second-order continuity, usually • Used for designing cars, ships, airplanes etc. • But curves based on physical artifacts cannot be replicated well, since there is no exact definition of what the curve is. • Around 1960, a lot of industrial designers were working on this problem.
Motivation for curves • What do we use curves for in Computer Graphics? – Building models – Movement paths –A n im a t i o n –T r u e t y p ef o n t s – Vector Graphics, e.g. pdf or Flash [Matthew]

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Motivation for curves Advantages of using curves –E x a c t C u r v a t u r e – A few coefficients (just a few numbers) can represent a curve e.g., small data storage and data transmission – No Aliasing problem in the representation level •D i s a d v a n t a g e s o f u s i n g c u r v e s – Computation is needed to convert the curves to polygons or line segments before rendering because the graphics hardware works on triangles and lines – This could slow down the rendering performance Note: we may pre-compute the polygons/lines for curves when the program starts.
1. Choosing Mathematical Representation 1. Explicit y = f ( x ) – What if the curve is not a function? 2. Implicit f(x,y,z)=0 x 2 + y 2 -R=0 – Hard to work with 3. Parametric (x(u),y(u)) – Easier to work with x(u) = cos 2 π u y(u) = sin 2 π u Candidates:

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Parametric Polynomial Curves • We’ll use parametric curves where the functions are all polynomials in the parameter. • Advantages – Easy (and efficient) to compute – Infinitely differentiable
2. Choosing Linear, Quadratic, and Cubic C0=G0 continuity C1/G1 continuity C2/G2 continuity Note: at joining points Candidates: Need 2 control points Need 3 control points Need 4 control points Linear: Quadratic: -Cont inuous - But JAGGED - Continuous -Smoothder ivat ive - But JAGGED 2nd derivative - 2nd derivatives of pieces can be chosen to match (between segments) - Good enough for most graphics . Cubic: { same t for x, y, z … }

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Parametric continuity • C0: matching endpoints • C1: matching endpoints, first derivatives equal • C2: matching endpoints, first and second derivatives equal • Cn: matching endpoints, first n derivatives equal Geometric continuity • G0: matching endpoints (G0 = C0) • G1: matching endpoints, first derivatives proportional • G2: matching endpoints, first and second derivatives proportional • Gn: matching endpoints, first n derivatives proportional
Cubic Curves •T h u s , U s e n=3 • For simplicity, we define each cubic function with the range x ( t ) = y ( t ) = z ( t )= Turn this into matrix form ……

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Compact Representation • Place all coefficients into a matrix: *This is tangent to the curve
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14-curve-surface - Lecture 15 Curve Surface and Subdivision...

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