Lecture_Chapter3_c

Representation 72 rendering parametric curves

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Unformatted text preview: e is special case of B-Spline •  Affine invariance ECS 175 Chapter 3: Object Representation 71 Parametric Surfaces •  Extend notion to 3D – two parameters instead of one f (u, v ) = nm ￿￿ i bi (u)bj (v )pij i original image CC-SA Wojciech Mola ECS 175 Chapter 3: Object Representation 72 Rendering Parametric Curves •  Triangles and lines are suitable for rendering •  Flexible triangle mesh generation a)  Create vertices by sampling the curve/surface along its parameter u (or u,v) b)  Use subdivision to approximate curve by its convex hull •  Alternative: Ray-Casting ECS 175 Chapter 3: Object Representation 73 Parametric Curves - Summary •  Multiple established definitions •  Curves defined by control points and parameter values •  Suitable for modeling/editing of complex shapes •  Control over continuity, smoothness •  Rendering requires sampling or subdivision ECS 175 Chapter 3: Object Representation 74 Implicit Representations •  Object surface is the (zero) level set of a function f (x, y ) = 0 Contour defined by zero level set: “All (x,y) locations, where function evaluates to 0” Divides space into regions (positive sign and negative sign) ECS 175 Chapter 3: Object Representation 75 Implicit Representations •  Easy curve/surface/volume operations •  Examples: g (x, y, z ) = 0 f (x, y, z ) = 0 Surface 1 f (x, y, z ) = 0 AND g (x, y, z ) = 0 Intersection curve f (x, y, z ) > 0 AND g (x, y, z ) < 0 Volume subtraction f (x, y, z ) > 0 OR g (x, y, z ) < 0 Volume union ECS 175 Chapter 3: Object Representation Surface 2 76 Implicit Representations •  Subdivision and rasterization useful for parametric representations •  Triangulation of implicit representations is harder ECS 175 Chapter 3: Object Representation 77 Triangulating Implicit Representations •  Implicit function given in discretized form •  Extract curve that corresponds to f(x,y) = 0 ECS 175 Chapter 3: Object Representation 78 Triangulating Implicit Representations •  Extract curve that corresponds to f(x,y) = 0 •  Thresholding? ECS 175 Chapter 3: Object Representation 79 Triangulating Implicit Representations •  Extract curve that corresponds to f(x,y) = 0 •  Better: Marching Quads •  Fast computation of curve intersection with grid edges •  Piecewise linear representation of curve •  Output suitable for rendering •  Generalizes to 3D (Marching Cubes) ECS 175...
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This document was uploaded on 03/12/2014 for the course ECS 175 at UC Davis.

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