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Review of Selected Surface Graphics Topics (1)

Review of Selected Surface Graphics Topics (1). Jian Huang, CS 594, Spring 2002. Visible Light. 3-Component Color. The de facto representation of color on screen display is RGB. Some printers use CMY(K) Why? The color spectrum can not be represented by 3 basis functions?

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Review of Selected Surface Graphics Topics (1)

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  1. Review of Selected Surface Graphics Topics (1) Jian Huang, CS 594, Spring 2002

  2. Visible Light

  3. 3-Component Color • The de facto representation of color on screen display is RGB. • Some printers use CMY(K) • Why? • The color spectrum can not be represented by 3 basis functions? • But the cone receptors in human eye are of 3 types, roughly sensitive to 430nm, 560nm, and 610nm

  4. RGB • The de facto standard

  5. Raster Displays • Display synchronized with CRT sweep • Special memory for screen update • Pixels are the discrete elements displayed • Generally, updates are visible

  6. Polygon Mesh • Set of surface polygons that enclose an object interior, polygon mesh • De facto: triangles, triangle mesh.

  7. Representing Polygon Mesh • Vertex coordinates list, polygon table and (maybe) edge table • Auxiliary: • Per vertex normal • Neighborhood information, arranged with regard to vertices and edges

  8. Object Space World Space Eye Space Clipping Space Screen Space Viewing Pipeline Canonical view volume • Object space: coordinate space where each component is defined • World space: all components put together into the same 3D scene via affine transformation. (camera, lighting defined in this space) • Eye space: camera at the origin, view direction coincides with the z axis. Hither and Yon planes perpendicular to the z axis • Clipping space: do clipping here. All point is in homogeneous coordinate, i.e., each point is represented by (x,y,z,w) • 3D image space (Canonical view volume): a parallelpipied shape defined by (-1:1,-1:1,0,1). Objects in this space is distorted • Screen space: x and y coordinates are screen pixel coordinates

  9. Spaces: Example Object Space and World Space: Eye-Space:

  10. Spaces: Example • Clip Space: • Image Space:

  11. Homogeneous Coordinates • Matrix/Vector format for translation:

  12. Translation in Homogenous Coordinates • There exists an inverse mapping for each function • There exists an identity mapping

  13. Why these properties are important • when these conditions are shown for any class of functions it can be proven that such a class is closed under composition • i. e. any series of translations can be composed to a single translation.

  14. Rotation in Homogeneous Space The two properties still apply.

  15. Affine Transformation • Property: preserving parallel lines • The coordinates of three corresponding points uniquely determine any Affine Transform!!

  16. Affine Transformations T • Translation • Rotation • Scaling • Shearing

  17. Affine Transformation in 3D • Translation • Rotate • Scale • Shear

  18. Viewing • Object space to World space: affine transformation • World space to Eye space: how? • Eye space to Clipping space involves projection and viewing frustum

  19. F P P Pinhole Model • Visibility Cone with apex at observer • Reduce hole to a point - the cone becomes a ray • Pin hole - focal point, eye point or center of projection.

  20. Image W F I World Perspective Projection and Pin Hole Camera • Projection point sees anything on ray through pinhole F • Point W projects along the ray through F to appear at I (intersection of WF with image plane)

  21. Simple Perspective Camera • camera looks along z-axis • focal point is the origin • image plane is parallel to xy-plane at distance d • d is call focal length for historical reason

  22. Y [Y, Z] [(d/Z)Y, d] Z [0, 0] [0, d] Similar Triangles • Similar situation with x-coordinate • Similar Triangles: point [x,y,z] projects to [(d/z)x, (d/z)y, d]

  23. View Volume • Defines visible region of space, pyramid edges are clipping planes • Frustum :truncated pyramid with near and far clipping planes • Near (Hither) plane ? Don’t care about behind the camera • Far (Yon) plane, define field of interest, allows z to be scaled to a limited fixed-point value for z-buffering.

  24. Why do clipping • Clipping is a visibility preprocess. In real-world scenes clipping can remove a substantial percentage of the environment from consideration. • Clipping offers an important optimization

  25. Difficulty • It is difficult to do clipping directly in the viewing frustum

  26. Canonical View Volume • Normalize the viewing frustum to a cube, canonical view volume • Converts perspective frustum to orthographic frustum – perspective transformation

  27. Perspective Transform • The equations alpha = yon/(yon-hither) beta = yon*hither/(hither - yon) s: size of window on the image plane z’ alpha 1 z yon hither

  28. About Perspective Transform • Some properties

  29. Perspective + Projection Matrix • AR: aspect ratio correction, ResX/ResY • s= ResX, • Theta: half view angle, tan(theta) = s/d

  30. Camera Control and Viewing Focal length (d), image size/shape and clipping planes included in perspective transformation • r                Angle or Field of view (FOV) • AR  Aspect Ratio of view-port • Hither, YonNearest and farthest vision limits (WS). Lookat - coi Lookfrom - eye View angle - FOV

  31. Complete Perspective • Specify near and far clipping planes - transform z between znear and zfar on to a fixed range • Specify field-of-view (fov) angle • OpenGL’s glFrustum and gluPerspective do these

  32. More Viewing Parameters • Camera, Eye or Observer: • lookfrom:location of focal point or camera • lookat: point to be centered in image • Camera orientation about the lookat-lookfromaxis • vup: a vector that is pointing straight up in the image. This is like an orientation.

  33. Point Clipping (x, y) is inside iff AND

  34. One Plane At a Time Clipping(a.k.a. Sutherland-Hodgeman Clipping) • The Sutherland-Hodgeman triangle clipping algorithm uses a divide-and-conquer strategy. • Clip a triangle against a single plane. Each of the clipping planes are applied in succession to every triangle. • There is minimal storage requirements for this algorithm, and it is well suited to pipelining. • It is often used in hardware implementations.

  35. Sutherland-HodgmanPolygon Clipping Algorithm • Subproblem: • clip a polygon (input: vertex list) against a single clip edges • output the vertex list(s) for the resulting clipped polygon(s) • Clip against all four planes • generalizes to 3D (6 planes) • generalizes to any convex clip polygon/polyhedron • Used in viewing transforms

  36. Polygon Clipping At Work

  37. With Pictures

  38. 4DPolygonClip • Use Sutherland Hodgman algorithm • Use arrays for input and output lists • There are six planes of course !

  39. 4D Clipping • Point A is inside, Point B is outside. Clip edge AB x = Ax + t(Bx – Ax) y = Ay + t(By – Ay) z = Az + t(Bz – Az) w = Aw + t(Bw – Aw) • Clip boundary: x/w = 1 i.e. (x–w=0); w-x = Aw – Ax + t(Bw – Aw – Bx + Ax) = 0 Solve for t.

  40. Why Homogeneous Clipping • Efficiency/Uniformity: A single clip procedure is typically provided in hardware, optimized for canonical view volume. • The perspective projection canonical view volume can be transformed into a parallel-projection view volume, so the same clipping procedure can be used. • But for this, clipping must be done in homogenous coordinates (and not in 3D). Some transformations can result in negative W : 3D clipping would not work.

  41. Hidden Lines and Surfaces

  42. Hidden Surfaces Removed

  43. y clipped line x 1 y 1 near far clipped line x z 0 1 z image plane near far Where Are We ? • Canonical view volume (3D image space) • Clipping done • division by w • z > 0

  44. Painters Algorithm • Sort objects in depth order • Draw all from Back-to-Front (far-to-near) • Is it so simple?

  45. 3D Cycles • How do we deal with cycles? • Deal with intersections • How do we sort objects that overlap in Z?

  46. Form of the Input Object types: what kind of objects does it handle? • convex vs. non-convex • polygons vs. everything else - smooth curves, non-continuous surfaces, volumetric data

  47. Form of the output Precision: image/object space? • Object Space • Geometry in, geometry out • Independent of image resolution • Followed by scan conversion • Image Space • Geometry in, image out • Visibility only at pixels

  48. Object Space Algorithms • Volume testing – Weiler-Atherton, etc. • input: convex polygons + infinite eye pt • output: visible portions of wireframe edges

  49. Image-space algorithms • Traditional Scan Conversion and Z-buffering • Hierarchical Scan Conversion and Z-buffering • input: any plane-sweepable/plane-boundable objects • preprocessing: none • output: a discrete image of the exact visible set

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