Computer Graphics

1. Computer Graphics Modeling

2. Modeling • Modeling is simply the process of creating 3D objects • Many different processes to create models • Many different representations of model data • Once the models are obtained, one can transform them to the correct locations in space, place a virtual camera in space, and render a 2D image of the scene

3. Model Representations • Polygonal • CSG (constructive solid geometry) • Spatial subdivision techniques • Implicit representation • Parametric patches

4. Polygonal • Complex objects arebroken down intomany simple polygons • Polygons form the “skin” of the object • Objects are hollow • Polygons have a front face and a back face

5. Triangles • Triangles are the #1 choice of polygons • They are always planar • Graphics hardware is often optimized for triangles • Nvidia TNT2  8M triangles per second • Nvidia Geforce2  31M triangles per second • Nvidia Geforce4  136M vertices per second • Note: Don’t read too much into these numbers. They are similar to processor speed numbers in that they don’t tell the whole story of how fast the computer actually is.

6. Triangulation • How are triangles obtained? • Triangulating a set of surface points • Several different triangulation approaches • Delaunay triangulations attempt to equalize triangle angles (reduces long skinny triangles)

7. How Many Triangles? • More triangles are needed in surface areas that require more geometric detail • Higher curvature  more triangles necessary

8. Level Of Detail (LOD) • It can be very costly to always display an entire object at the greatest level of detail • Less detail is necessary the farther away you are from on object • Also depends on the viewing angle and screen size • Solution: create several models of the object at different levels of detail • Display the correct one for the viewing distance

9. LOD 50 Vertices 500 Vertices 2000 Vertices

10. LOD

11. LOD • How to create the same model in multiple levels of detail? • Ex: start with most detailed, resample with less vertices, and then re-triangulate • When to switch models when rendering? • Based on distance and screen size • Ex: 640x480 screen  307,200 pixels Object takes up half of screen  150K pixels  Any more than 300K triangles (half are facing away from the camera) is overkill at this distance and screen size • Visual artifacts can occur at switching point

12. Creating the Surface Points • Triangulation works on a set of points • One needs to create this surface point set • Many different approaches: • Manual placement • Mathematical (geometrical) generation • Scanning real objects

13. Mathematical Generation • Solids of Revolution • Rotation of a cross-section around an axis • Spheres • Cones • Cylinders • Bundt cakes?

14. Mathematical Generation • Extrusion • Extrude a cross-section along a profile curve • Scale may vary along the profile • Many metal and plastic parts • Cones • Cylinders • Bottles

15. Scanning Real Objects • Laser scanning • Tomographic methods • Medical scanning (Xray, CT, MRI) • Radar Recovered 3D model Hand-held laser scanner Slice of brain from CT scan Recovered 3D model of lungs

16. Scanning Real Objects • Computer vision Actual photograph of Hoover Tower Recovered 3D model Model rendered from novel view and texture mapped

17. Triangle Representations • Graphics cards do most of the triangle work • Need an efficient way to send triangles to the card • Some typical primitive triangle representations: • Lists • Fans • Strips 1.6 vertices per triangle 1.5 vertices per triangle 3.0 vertices per triangle

18. Triangle Representations • Adding a level of indirection with indexed triangle representations • Store the vertices (in a vertex buffer) independently of the rendering order (which is stored in a index buffer) • Enables one to reuse a vertex multiple times without sending 3 coordinates each time • Conventional wisdom is that indexed triangle strips are the most efficient triangle representation

19. Surface Normal • Each triangle has a single surface normal • Easy way to defines the orientation of the surface • Again, the normal is just a vector (no position) C N A B

20. Computing the Surface Normal • Let V1 be the vector from point A to point B • Let V2 be the vector from point A to point C • N = V1x V2 • N is often normalized • Note that order of vertices becomes important • Triangle ABC has an outward facing normal • Triangle ACB has an inward facing normal C N A B

21. Modeling Approaches • Polygonal • CSG (constructive solid geometry) • Spatial subdivision techniques • Implicit representation • Parametric patches

22. Constructive Solid Geometry • Unlike polygonal approaches, CSG models are “solids” • Polygonal models are “skins” • Solid approaches are often better suited for medical applications because • Cutting slices through objects • Representing internal functional data • Temperature • Blood flow

23. CSG Trees • CSG models are stored in trees • Leaves are primitive shapes • Spheres, Cubes, Cones, etc. • Nodes are Boolean operations • Union, Difference, Intersection

24. CSG Operations • Primitives: • Union: • Intersection: • Difference:

25. CSG Rendering • CSG models can be rendered by: • Computing surface points and triangulating • Rendering can then be performed by standard hardware • Use of special CSG rendering routines • Not hardware optimized

26. Modeling Approaches • Polygonal • CSG (constructive solid geometry) • Spatial subdivision techniques • Implicit representation • Parametric patches

27. Spatial subdivision techniques • The idea is to divide up space into pieces that are the object are pieces that are not the object • A “solid” approach to modeling • Usually pieces are cubes • Amount of detail that can be represented is controlled by how small the cubes are

28. 2D Images • B/W images • Image is evenly divided up into pixels • Pixels size controls the level of detail • Pixels are on/off to represent presence/absence of object • Representation designed for ease of blitting onto screen, not for efficient data representation

29. 2D Quad Trees • Quad trees adaptively subdivide the image • Subdivision can continue until the required level of detail is reached • Significantly more efficient than using a 2D block of uniform small squares

30. 3D Space Subdivision • Voxels are the 3D extension of pixels • Again, the brute-force approach is to fill 3D space with uniform small voxels and mark the space as occupied or empty

31. Octrees • Octrees are the 3D extension of quad trees and use adaptive subdivision

32. Modeling Approaches • Polygonal • CSG (constructive solid geometry) • Spatial subdivision techniques • Implicit representation • Parametric patches

33. Implicit Representation • An implicit definition of a sphere: x2 + y2 + z2 = r2 • Defines a type of “isosurface” • A set of 3D points that satisfy the equation • However, they are difficult to work with • Quickly tell me all the points that satisfy the above equation given a particular value of r

34. Metaballs • Metaballs (a.k.a. blobby objects) are an implicit modeling technique

35. Metaballs in Action • Implicit representations find most of the use in shape changing animations

36. Modeling Approaches • Polygonal • CSG (constructive solid geometry) • Spatial subdivision techniques • Implicit representation • Parametric patches

37. Parametric Patches • Parametric patches are used to model smooth curved surfaces and to allow dynamic control over the shape of the surface • Parametric patches are uses heavily in CAD application and animation • Currently real-time applications are dominated by triangle meshes • Patches hold many advantages over triangle meshes and if hardware support for them becomes widespread there could be a major shift to them

38. Parametric Patches • One of the first uses of parametric patches is the “Utah Teapot” • The actual teapot is on the left

39. Parametric Patches • We will start be examining curves • Bézier curves • B-Spline curves • Then we will expand to surface patches • Bézier patches • B-Spline patches

40. Bézier Curves • Pierre Bézier, a French designer, first used them in the 1970’s in the design of Renault car bodies • Since, they have been heavily used by Adobe in their fonts as well as in graphic applications • A Bézier curve is a parametric curve • The function C(u) defines the curve points (x, y, z) as the parameter u varies through [0..1]

41. Bézier Curves • The space curve C(u) is defined by: C(u) = i0..N Pi Bi (u) where the Pi are the control points and the Bi(u) are the blending functions

42. Bézier Curves • The control points are simply 3D points • The blending functions are defined as: Bi(u) = N ui (1 – u)N-i i • Most of the time, the cubic version is used (n=3) • This implies 4 control points are needed and 4 blending functions: B0(u) = (1 – u)3 B1(u) = 3u (1 – u)2 B2(u) = 3u2 (1 – u) B3(u) = u3

43. Bézier Curves • The blending functions make much more sense graphically • Note that t in the graph below is our u

44. Bézier Curves • Due to the setup of the blending functions: • The curve goes through the end points • control points P0 and P3 • The curve doesn’t go through the interior control points • The interior control influence the curve • The curve is pulled in the direction of the control point • Moving a control point changes the shape of the entire curve (called global control)

45. Bézier Curves • A Bézier Curve can only “curve” as much as its degree allows • Degree 1 (linear) is a line • Degree 2 (quadratic) can curve once • Degree 3 (cubic) can curve twice • Recall that the number of control points is always 1 larger than the degree

46. Bézier Curves • So what if you want a really “curvy” curve? • You could increase the degree of the curve • The downfall of this approach is that each control point has global control over the curve shape • That is, changing a single control point will modify the shape of the entire curve (not great for modeling) • High degree curves are “unstable” • It is computationally expensive • You could join multiple cubic curves together • Each control point only has influence over its piece of the curve (local control w.r.t. the entire joined curve) • Hardware can be optimized for cubic curves • But, we must maintain continuity across the joins!

47. Bézier Curves • So how are we going to maintain continuity? • Notice that the line between the first 2 control points specify the tangent of the curve at u=0 • And the line between the last 2 control points specify the tangent of the curve at u=1

48. Bézier Curves • So if we line up P2P3 with Q0Q1, then we have the same tangent at end u=1 point of curve P and u=0 point of curve Q

49. Bézier Curves • Continuity put into more formal terms: • C0 continuity means the curve points vary smoothly • That is, the curve is a set of connected points • For our joined example, it means the two joined segments share the same endpoints • P3 is the same point as Q0 • C1 continuity means the curve’s first derivatives vary smoothly (plus C0 continuity) • For our joined example, it means the tangents at the end endpoints must be the same (P2, P3, Q0, Q1 must form a line) • And P0, P1 must form a line with the curve before it and Q2, Q3 must form a line with the curve after it  all 4 control points are constrained to some extent

50. Bézier Curves • C2 continuity means the curve’s second derivatives vary smoothly (plus C1 continuity) • Second derivative corresponds to curvature • For our joined example, it implies constrains on P1, P2, P3, Q0, Q1, and Q2 • Plus P0, P1, and P2 are constrained from the curve before it and Q1, Q2, and Q3 are constrained from the curve after it • Thus, although it can be achieved, there are so many constrains on the positions of the control points that joined cubic Bézier curves are almost never C2 continuous