1. CS 395: Adv. Computer Graphics Overview Parametric Surfaces Watt: Chapter 3 + readings Jack Tumblin jet@cs.northwestern.edu

2. Curves and Surfaces Basic Problem: Polygons are easy, fast, renderable, BUTPolygons are easy, fast, renderable, BUT Polygons meshes are not smooth; no derivatives: poor silhouettes, reflections...Polygons meshes are not smooth; no derivatives: poor silhouettes, reflections... Polygons can only approximate curves,Polygons can only approximate curves, Polygons are less compact:Polygons are less compact: Previous methods: metal/wood 'splines'... (see Farin book)

3. What's a Parametric Curve? Vary one or more 'parameter' to explore a curve or surface Example: parametric circle, in z=1 plane: x(u) = R*cos(u) y(u) = R*sin(u) z(u) = 1 x y z

4. Background Many Historical Parametric Curve Makers:Many Historical Parametric Curve Makers: –Lissajous Curves, http://kosmoi.com/Science/Mathematics/Graphs/Encyclo/ –Spirographs, http://math.dartmouth.edu/~dlittle/java/SpiroGraph/ –Harmonographs, http://astronomy.swin.edu.au/~pbourke/curves/harmonograph/ –Epicycles, etc. http://www.astronomynotes.com/history/epicycle.htm

5. Background Few found use in design until computers:Few found use in design until computers: –Paul DeCastlejau (1950s, Citroen) –Pierre Bezier (1960s, Renault) –70's, 80's explosion of Comp. Geometry; –GREAT results: now faded as research area

6. OUTLINE Historical Parametrics: transcendentalsHistorical Parametrics: transcendentals in CG: mostly polynomialin CG: mostly polynomial Key Idea 1: blending points...Key Idea 1: blending points...

7. OUTLINE Key Idea 2: Linear Interpolation, NestingKey Idea 2: Linear Interpolation, Nesting –Paul DeCastlejau (1950s, Citroen) –Pierre Bezier (1960s, Renault) http://www.ibiblio.org/e-notes/Splines/Bezier.htmhttp://www.ibiblio.org/e-notes/Splines/Bezier.htm How can we connect multiple Bezier curves? How can we make a Bezier surface?

8. Efficient! 9 unique Bezier Patches (some were mirrored around z axis: total is ?17?)

9. ‘Digital’ Image: a 2D Grid of Numbers NO intrinsic meaning—use it for anything:NO intrinsic meaning—use it for anything: reflectance, transparency, illumination, normal direction, material, velocity... reflectance, transparency, illumination, normal direction, material, velocity... u v u v

10. OUTLINE Key Idea 3: Generalize: Blending Fcns., in Matrix formKey Idea 3: Generalize: Blending Fcns., in Matrix form –Uniform B-splines –Other Basis Functions –Non-uniform? 'Duplicate Control Pts' http://www.ibiblio.org/e-notes/Splines/Bezier.htm

11. Useful Goals Continuity: are all derivatives smooth?Continuity: are all derivatives smooth? w.r.t. parameters; w.r.t. space; Global / Local Control: move 1 control pt: does entire curve change?Global / Local Control: move 1 control pt: does entire curve change? Convex Hull: is curve within its control pts?Convex Hull: is curve within its control pts? Interpolating:does curve touch desired pts?Interpolating:does curve touch desired pts? Affine Invariant; Projective Invariant:Affine Invariant; Projective Invariant: transform control pts, then draw curve, OR draw curve, then transform, SAME result!

12. Useful Goals Invertible; find ray-surface intersection in 3D (for rendering, shading) in u,v parameters (for texture, etc.) find surface-surface intersection in 3D (for 'trimming', fairing, etc.) in u,v parametersInvertible; find ray-surface intersection in 3D (for rendering, shading) in u,v parameters (for texture, etc.) find surface-surface intersection in 3D (for 'trimming', fairing, etc.) in u,v parameters

13. Further Sources Endless books on curves and surfaces: G. Farin, "Curves and Surfaces for CAGD" (recommended; most rigorous & complete)Endless books on curves and surfaces: G. Farin, "Curves and Surfaces for CAGD" (recommended; most rigorous & complete) On-line tutorials, Java AppletsOn-line tutorials, Java Applets