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CAP4730: Computational Structures in Computer Graphics

This outline explores the fundamental concepts and operations associated with 3D transformations in computer graphics. It covers essential topics including translation, scaling, and rotation in three dimensions, as well as more advanced subjects like homogeneous coordinates and projection techniques. By conceptualizing transformations as matrix multiplications, we can enhance programming efficiency and grasp the relationship between 3D points and their resulting pixel positions on display screens. This guide also delves into the critical roles of the camera and graphics pipeline, furthering our understanding of rendering in computer graphics.

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CAP4730: Computational Structures in Computer Graphics

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  1. CAP4730: Computational Structures in Computer Graphics 3D Transformations

  2. Outline • 3D World • What we are trying to do • Translate • Scale • Rotate

  3. Transformations in 3D! • Remembering 2D transformations -> 3x3 matrices, take a wild guess what happens to 3D transformations. T=(tx, ty, tz)

  4. Scale, 3D Style S=(sx, sy, sz)

  5. Rotations, in 3D no less! What does a rotation in 3D mean? Q: How do we specify a rotation? R=(rx, ry, rz, ) A: We give a vector to rotate about, and a theta that describes how much we rotate.  Q: Since 2D is sort of like a special case of 3D, what is the vector we’ve been rotating about in 2D?

  6. Rotations about the Z axis What do you think the rotation matrix is for rotations about the z axis? R=(0,0,1,) 

  7. Rotations about the X axis Let’s look at the other axis rotations R=(1,0,0,) 

  8. Rotations about the Y axis R=(0,1,0,) 

  9. Rotations for an arbitrary axis

  10.   Rotations for an arbitrary axis u Steps: 1. Normalize vector u 2. Compute  3. Compute  4. Create rotation matrix

  11. Vector Normalization • Given a vector v, we want to create a unit vector that has a magnitude of 1 and has the same direction as v. Let’s do an example.

  12. Computing the Rotation Matrix • 1. Normalize u • 2. Compute Rx() • 3. Compute Ry() • 4. Generate Rotation Matrix

  13. Rotation Matrix

  14. Applying 3D Transformations P’=TRTP Let’s compute M

  15. Homogenous Coordinates • We need to do something to the vertices • By increasing the dimensionality of the problem we can transform the addition component of Translation into multiplication.

  16. Homogenous Coordinates • Homogenous Coordinates – embed 3D transforms into 4D • All transformations can be expressed as matrix multiplications. • Inverses and combination easier • Equivalence of vectors (4 2 1 1)=(8 4 2 2) • What this means programatically

  17. The Question • Given a 3D point, an eye position, a camera position, and a display plane, what is the resulting pixel position? • Now extend this for a group of three points • Then apply what you know about scan conversion.

  18. Different Phases:Model Definition

  19. Different Phases:Transformations

  20. Different Phases:Projection

  21. Different Phases:Projection

  22. Different Phases:Rasterization

  23. Different Phases:Scan Conversion

  24. What are the steps needed?

  25. Let’s Examine the Camera • If I gave you a world, and said I want to “render” it from another viewpoint, what information do I have to give you? • Position • Which way we are looking • Which way is “up” • Aspect Ratio • Field of View • Near and Far

  26. Camera View Right View Up View Normal View Direction

  27. Camera View Up View Right What are the vectors?

  28. Graphics Pipeline So Far Object Object Coordinates Transformation Object -> World World World Coordinates Projection Xform World -> Projection Normalize Xform & Clipping Projection -> Normalized Camera Projection Coordinates Viewport Normalized Coordinates Viewport Transform Normalized -> Device Screen Device Coordinates

  29. Transformation World->Camera View Right View Up View Normal View Direction

  30. Transformation World->Camera View Right = u View Up = V View Direction = -N

  31. u  Cross Products Given two vectors, the cross product returns a vector that is perpendicular to the plane of the two vectors and with magnitude equal to the area of the parallelogram formed by the two vectors.

  32. Parallel Projections (known aliases): Orthographic or Isometric Projection

  33. Parallel Projection

  34. Parallel Projections (known aliases): Oblique Projection  L 

  35. Projections foreshortening - the farther an object is from the camera , the smaller it appears in the final image

  36. Perspective Projection Side View P=(xp,yp,zp) t=0 P’=(x’,y’,z’) t=? C=(xc,yc,zc) t=1 xp x’ z’ zp

  37. Perspective Projection Side View P=(xp,yp,zp) t=0 P’=(x’,y’,z’) t=? C=(xc,yc,zc) t=1 xp h x’ z’ zp Scale by h

  38. Perspective Divide Foreshortening - look at the x,y, and w values, and how they depend on how far away the object is. Modelview Matrix - describes how to move the world->camera coordinate system Perspective Matrix - describes the camera you are viewing the world with.

  39. Let’s closely examine

  40. What the Perspective Matrix means Note: Normalized Device Coordinates are a LEFT-HANDED Coordinate system

  41. Graphics Pipeline So Far Object Object Coordinates Transformation Object -> World World World Coordinates Projection Xform World -> Projection Normalize Xform & Clipping Projection -> Normalized Camera Projection Coordinates Viewport Normalized Coordinates Viewport Transform Normalized -> Device Screen Device Coordinates

  42. What happens to an object... Transformation Object -> World Object Object Coordinates World World Coordinates

  43. What happens to an object... Transformation - Modelview World -> Eye/Camera World World Coordinates Viewport Viewport Coordinates

  44. What happens to an object... Transformation - Projection (Includes Perspective Divide) Eye/Camera ->View Plane Viewport Viewport Coordinates Rasterization Scan Converting Triangles

  45. Normalized Screen Coordinates

  46. Normalized Screen Coordinates Let’s label all the vectors znsc=0 znsc=1

  47. View Volume (View Frustum) Usually: View Plane = Near Plane Far Plane -1,1,1 Mperspective_Matrix Think about: Clipping 3D Triangles View Frustum Culling 1,-1,-1

  48. Comparison with a camera

  49. Let’s verbalize what’s going on • Review: • Pipeline • Series of steps • What we’ll do next: • Hidden Surface Removal • Depth Buffers • Lighting • Shading • Blending (the elusive alpha) • Textures

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