Barycentric Coordinates and Parameterization

Transcription

1 Barycentric Coordinates and Parameterization

2 Center of Mass Geometric center of object

3 Center of Mass Geometric center of object Object can be balanced on CoM How to calculate?

4 Finding the Center of Mass Plumb line method

5 Special Case: Points CoM is average

6 Special Case: Points CoM is average Center of mass is inside convex hull

7 Special Case: Points CoM is average Center of mass is inside convex hull What if points have different mass?

8 Special Case: Points

9 Special Case: Points Weighted average Still in convex hull Scaling the masses doesn t affect CoM

10 Special Case: Points Weighted average Still in convex hull Scaling the masses doesn t affect CoM can assume masses sum to one

11 Inverse Problem Given three points p i and a target point c: For what masses is c the CoM?

12 Inverse Problem Given three points p i and a target point c: For what masses is c the CoM? Special case 1:

13 Inverse Problem Given three points p i and a target point c: For what masses is c the CoM? Special case 1:

14 Inverse Problem Given three points p i and a target point c: For what masses is c the CoM? Special case 1: Special case 2: c outside triangle

15 Inverse Problem Given three points p i and a target point c: For what masses is c the CoM? Special case 1: Special case 2: c outside triangle not possible (needs antigravity )

16 Inverse Problem Given three points p i and a target point c: For what masses is c the CoM? Observation:

17 Inverse Problem Given three points p i and a target point c: For what masses is c the CoM?

18 Inverse Problem Given three points p i and a target point c: For what masses is c the CoM?

19 Inverse Problem Given three points p i and a target point c: For what masses is c the CoM?

20 Inverse Problem Given three points p i and a target point c: For what masses is c the CoM? These are barycentric coordinates of c

21 Barycentric Coordinates Can be interpreted as weighted point sum point in edge coordinates

22 Barycentric Coordinates Properties:

23 Barycentric Coordinates Properties: corners are (0,0), (1,0), (0,1) unique for any inside point

24 Barycentric Coordinates Properties: corners are (0,0), (1,0), (0,1) unique for any inside point

25 Barycentric Coordinates Properties: corners are (0,0), (1,0), (0,1) unique for any inside point Why do we care?

26 Barycentric Interpolation Extends any function from corners to triangle

27 Barycentric Interpolation Extends any function from corners to triangle colors normals whatever

28 Negative Barycentric Coordinates Points outside triangle also have coords

29 Negative Barycentric Coordinates Points outside triangle also have coords Alternate inside-triangle check: compute barycentric coords check they re valid

30 Barycentric Coords in 3D Given c in plane of tri: find coords with

31 Barycentric Coords in 3D Given c in plane of tri: find coords with Problem: too many equations!!

32 Barycentric Coords in 3D Given c in plane of tri: find coords with Problem: too many equations!! Can we eliminate one of the variables?

33 Barycentric Coords in 3D Given c in plane of tri: find coords with Both sides vectors in normal direction

34 Barycentric Coords in 3D Given c in plane of tri: find coords with Both sides vectors in normal direction

35 Barycentric Coords in 3D Given c in plane of tri: find coords with Both sides vectors in normal direction

36 Barycentric Coords in 3D Given c in plane of tri: find coords with

37 Barycentric Coords in 3D Given c in plane of tri: find coords with What if c is not in the plane of triangle?

38 Ray Tracing Triangles 1. Find point where ray hits triangle plane 2. Calculate barycentric coordinates 3. Check coords valid 4. Linearly interpolate normals etc. 5. Shade pixel

39 Beyond Triangles Much carries over Are coords still unique?

40 Beyond Triangles Much carries over Are coords still unique? No!

41 Beyond Triangles Much carries over Are coords still unique? No! Many generalized barycentric coords schemes exist

42 Barycentric Coords as a Map Maps from a triangle in 2D to 3D triangle Called parameterization of triangle

43 Barycentric Coords as a Map Maps from a triangle in 2D to 3D triangle Called parameterization of triangle from now on, 2D coords are u and v

44 Parameterization Map between region of plane and arbitrary surface why do we want to do this?

45 Parameterization Map between region of plane and arbitrary surface Can then use parameterization to paint image on 3D surface: texture map

46 Texture Map Parameterization == texture map == UV coordinates == UV unwrapping

47 Texture Map Parameterization == texture map == UV coordinates == UV unwrapping Usually means assigning U and V coordinates to every pixel

48 Texture Map Parameterization == texture map == UV coordinates == UV unwrapping Usually means assigning U and V coordinates to every pixel Or U and V for every vertex, then interpolate

49 Parameterization History How to parameterize the earth (sphere)? Very practical, important problem in Middle Ages

50 Latitude & Longitude Distorts areas and angles

51 Planar Projection Covers only half of the earth Distorts areas and angles

52 Stereographic Projection Distorts areas

53 Albers Projection Preserves areas, distorts aspect ratio

54 Fuller Parameterization

55 No Free Lunch Every parameterization of the earth either: distorts areas distorts distances distorts angles

56 Good Parameterizations low area distortion low angle distortion no obvious seams one piece

57 Soup Parameterization

58 Planar Parameterization Project surface onto plane

59 Planar Parameterization Project surface onto plane quite useful in practice

60 Planar Parameterization Project surface onto plane quite useful in practice only partial coverage bad distortion when normals perpendicular

61 Planar Parameterization In practice: combine multiple views

62 Cube Map

63 Cylindrical Parameterization

64 Conformal Parameterization Conformal = angle-preserving

65 Conformal Parameterization Conformal = angle-preserving Riemann mapping theorem can map any surface conformally

66 Conformal Parameterization Conformal = angle-preserving Riemann mapping theorem can map any surface conformally Area distortion can be bad

67 Texture Atlas Break up surface into easy pieces, parameterize separately

68 Texture Atlas Some automatic methods exist but often artists hand-paint UV coords

69 Projection Mapping

70 Projection Mapping Scan 3D geometry, compute texture map Then, project anything you want on object

71