Lecture_Chapter3_c

# Representation 72 rendering parametric curves

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 ) &gt; 0 AND g (x, y, z ) &lt; 0 Volume subtraction f (x, y, z ) &gt; 0 OR g (x, y, z ) &lt; 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...
View Full Document

## This document was uploaded on 03/12/2014 for the course ECS 175 at UC Davis.

Ask a homework question - tutors are online