19-surfaces-parametric-implicit

19-surfaces-parametric-implicit - Reading for last week and...

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1 Curves and Surfaces Lecture 19 CPSC 578/478 Spring 2005 Curves: Hermite, Catmull-Rom, Bezier, B-Spline, NURB Surfaces from splines: Revolution, Lofted, Tensor Product Implicit Surfaces Reading for last week and this week: Text, Chapter 13 Supplement with http://graphics.idav.ucdavis.edu/graphics/CAGDNotes/homepage.html And http://www.siggraph.org/education/materials/ HyperGraph/modeling/splines/splines0.htm Class Calendar: April 4: assignment # 3 due April 6: assignment #4 topic/teams due April 11: assignment #4 short proposal due April 13: Quiz #2 April 18: assignment #4 long proposal due April 20: assignment #4 5 min oral proposal/preliminary results May 10: assignment #4 code, executable report due Assignment #3: Check materials section for help sent in by people in class, Jianye. Currently grinberg_modeler_spring2005_ linux.zip Course Syllabus I. Image processing II. Rendering III. Modeling IV. Animation V. Advanced Topics Image Processing Modeling Animation Rendering
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2 Modeling How do we ... Create 3D objects? Store 3D objects? Use 3D objects? Different methods for different object representations In different phases (creation, storage, use) the same object may be represented in different ways. 3D Object Representations Raw data Point cloud Range image Polygon soup Surfaces Mesh Subdivision Parametric Implicit Solids Voxels BSP tree CSG Sweep High-level structures Scene graph Skeleton Application specific Conics: useful shapes inconvenient in 3D Polynomial Functions: hard to control Splines Linear: not smooth parameterization basis functions knots Hermite: global, not always predictable cubic basis functions composite parameterization Catmull-Rom: fast for animation, less smooth, Bezier: predictability, but no local control convex hull property B-Splines: no conics, no perspective transform NURBs!! Last time Correction from last time: Except for the tangent, Frenet Frame undefined for straight line A Physical Spline Instead use a series of segments defined in parametric form. u ) ( ) ( ) ( u z z u y y u x x = = = These functions determined by coordinates of P n
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3 (Exaample for Hermite basis functions) u u 2 u 3 P(u)=(x(u),y(u),z(u)) Matrix representation Basis functions over one segment Bezier Curves and B-Splines: Convex Hull Property 1 ) ( ) ( ) ( 4 .. 1 4 .. 1 = = = = u B u B x u x i i i i i Bezier Curves: Convex Hull Property Passes through end points Continuity not guaranteed Iterative evaluation: de Casteljau algorithm B-Splines: Convex Hull Property Does not pass through points Continuity guaranteed (Iterative evaluation also exists) Bspline One segment Bspline
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