Spring 2017 CSCI 621: Digital Geometry Processing
1.2 Surface Representation & Data Structures
Hao Li http://cs621.hao-li.com 1 Administrative
• No class next Tuesday, due to Siggraph deadline
• Introduction to first programming exercise on Jan 24th
Siggraph Deadline 2013@ILM!
2 Last Time
Geometry Processing
Reconstruction
Rendering
Capture Analysis
Reproduction
Manipulation
3 Geometric Representations
point based quad mesh triangle mesh
implicit surfaces / particles volumetric tetrahedrons4 Geometric Representations
point based quad mesh triangle mesh
Surface Representations
implicit surfaces / particles volumetric tetrahedrons5 High Resolution
6 Large scenes
7 Outline
• Parametric Approximations
• Polygonal Meshes
• Data Structures
8 Parametric Representation
Surface is the range of a function f : IR 2 IR 3, = f( ) ⇥ S
2D example: A Circle f : [0, 2 ] IR 2 r cos(t) r f(t)= r sin(t) ⇥
9 Parametric Representation
Surface is the range of a function f : IR 2 IR 3, = f( ) ⇥ S
2D example: Island coast line f : [0, 2 ] IR 2 r cos(? t) f(t)= r sin(? t) ⇥
10 Piecewise Approximation
Surface is the range of a function f : IR 2 IR 3, = f( ) ⇥ S
2D example: Island coast line f : [0, 2 ] IR 2 r cos(? t) f(t)= r sin(? t) ⇥
11 Polynomial Approximation
Polynomials are computable functions
p p i f(t)= ci t = c˜i i(t) i=0 i=0 Taylor expansion up to degree p p 1 g(h)= g(i)(0) hi + O hp+1 i! i=0 ⇤ ⇥ Error for approximation g by polynomial f
f(t )=g(t ) , 0 t <