What is Geometry Processing? Understanding the math of 3D shape and applying that math to discrete shape

Transcription

1 Geometry Processing

2 What is Geometry Processing? Understanding the math of 3D shape and applying that math to discrete shape

3 What is Geometry Processing? Understanding the math of 3D shape and applying that math to discrete shape Examples: subdivision and decimation

4 What is Geometry Processing? Understanding the math of 3D shape and applying that math to discrete shape Examples: subdivision and decimation parameterization

5 What is Geometry Processing? Understanding the math of 3D shape and applying that math to discrete shape Examples: subdivision and decimation parameterization remeshing

6 What is Geometry Processing? Understanding the math of 3D shape and applying that math to discrete shape Examples: subdivision and decimation parameterization remeshing smoothing/fairing

7 What is Geometry Processing? Understanding the math of 3D shape and applying that math to discrete shape Examples: subdivision and decimation parameterization remeshing smoothing/fairing Today: a quick taste of surface geometry

8 Simple Geometry: Plane Curves

9 Simple Geometry: Plane Curves

10 Simple Geometry: Plane Curves

11 Curvature What is it? Some formula:

12 Curvature What is it really?

13 Curvature What is it really? small + zeroish small + large - how quickly the normals turn

14 Curvature What is it really? how quickly the normals turn

15 Curvature What is it really?

16 Total Integrated Curvature

17 Total Integrated Curvature

18 Total Integrated Curvature Theorem (Whitney-Graustein): for a closed smooth curve,

19 Inflation Theorem Offset closed curve along normal direction

20 Inflation Theorem Offset closed curve along normal direction

21 Inflation Theorem Offset closed curve along normal direction

22 Inflation Theorem Offset closed curve along normal direction

23 Surfaces in Space

24 Surfaces in Space What is curvature now?

25 Idea #1: Normal Curvature

26 Mean Curvature Average normal curvature at point

27 Idea #2: Look at Normals Again

28 Idea #2: Look at Normals Again Gaussian curvature

29 Mean and Gaussian Curvatue

30 Theorema Egregrium Theorem (Gauss, deep): Gaussian curvature is an isometry invariant all have

31 Informativeness of Curvature Theorem (easy): every curve can be reconstructed (up to rigid motions) from its curvature Theorem (deep): every surface can be reconstructed (up to rigid motions) from its mean and Gaussian curvature

32 3D Analogues Theorem [Gauss-Bonnet]: Theorem [Steiner]:

33 Discrete Curve

34 Discrete Curve

35 How do we Discretize Geometry? Option 1: is not the real curve. It approximates some smooth limit curve.

36 How do we Discretize Geometry? Option 1: is not the real curve. It approximates some smooth limit curve. What is the refinement rule?

37 Internet proof that

38 How do we discretize geometry? Option 2: is the real curve! Construct geometry axiomatically Get the right answer at every level of refinement

39 How do we discretize curvature?

40 How do we discretize curvature?

41 How do we discretize curvature?

42 Discrete Surface

43 Discrete Inflation Theorem

44 Discrete Inflation Theorem

45 Discrete Gauss-Bonnet

46 Chladni Plates Ernst Chladni

47 Isolines of Square Plate

48 Chladni Plates Properties of plate energy: - Stretching negligible - Uniform, local & isotropic - Zero for flat plate - Same in both directions Sophie Germain

49 Chladni Plates Properties of plate energy: - Stretching negligible - Uniform, local & isotropic - Zero for flat plate - Same in both directions Low-order approximation: Sophie Germain