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• 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 CCSA 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: RayCasting 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.
 Spring '08
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