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1 Geometry Processing

2 What is Geometry Processing? Understanding the math of 3D shape

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

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

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

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

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

8 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

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12 Simple Geometry: Plane Curves

13 Simple Geometry: Plane Curves

14 Simple Geometry: Plane Curves

15 Curvature What is it? Some formula:

16 Curvature What is it really?

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

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

19 Curvature What is it really?

20 Total Integrated Curvature

21 Total Integrated Curvature

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

23 Inflation Theorem Offset closed curve along normal direction

24 Inflation Theorem Offset closed curve along normal direction

25 Inflation Theorem Offset closed curve along normal direction

26 Inflation Theorem Offset closed curve along normal direction

27 Surfaces in Space

28 Surfaces in Space What is curvature now?

29 Idea #1: Normal Curvature

30 Mean Curvature Average normal curvature at point

31 Idea #2: Look at Normals Again

32 Idea #2: Look at Normals Again Gaussian curvature

33 Mean and Gaussian Curvatue

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

35 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

36 3D Analogues Theorem [Gauss-Bonnet]: Theorem [Steiner]:

37 Discrete Curve

38 Discrete Curve

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

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

41 Internet proof that

42 How do we discretize geometry? Option 2: is not the real curve. It represents an element of a finitedimensional subspace of smooth curves (finite elements)

43 How do we discretize geometry? Option 2: is not the real curve. It represents an element of a finitedimensional subspace of smooth curves (finite elements) What subspace?

44 How do we discretize geometry? Option 3: is the real curve! Construct geometry axiomatically Get the right answer at every level of refinement

45 Discrete Surface

46 Discrete Inflation Theorem

47 Discrete Inflation Theorem

48 Discrete Gauss-Bonnet

49 Chladni Plates Ernst Chladni

50 Isolines of Square Plate

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

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

53 Kirchhoff-Love Theory Plates and shells parameterized by midsurface Volume foliated by offset surfaces

54 Back to the Cone