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Viewing III Week 4, Wed Jan 27

http://www.ugrad.cs.ubc.ca/~cs314/Vjan2010. Viewing III Week 4, Wed Jan 27. News. extra TA office hours in lab 005 Tue 2-5 (Kai) Wed 2-5 (Garrett) Thu 1-3 (Garrett), Thu 3-5 (Kai) Fri 2-4 (Garrett) Tamara's usual office hours in lab Fri 4-5. Review: Convenient Camera Motion.

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Viewing III Week 4, Wed Jan 27

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  1. http://www.ugrad.cs.ubc.ca/~cs314/Vjan2010 Viewing IIIWeek 4, Wed Jan 27

  2. News • extra TA office hours in lab 005 • Tue 2-5 (Kai) • Wed 2-5 (Garrett) • Thu 1-3 (Garrett), Thu 3-5 (Kai) • Fri 2-4 (Garrett) • Tamara's usual office hours in lab • Fri 4-5

  3. Review: Convenient Camera Motion • rotate/translate/scale versus • eye point, gaze/lookat direction, up vector y lookat x Pref WCS view up z eye Peye

  4. y lookat x Pref WCS view v VCS up z eye Peye u w Review: World to View Coordinates • translate eye to origin • rotate view vector (lookat – eye) to w axis • rotate around w to bring up into vw-plane

  5. Review: W2V vs. V2W • MW2V=TR • we derived position of camera in world • invert for world with respect to camera • MV2W=(MW2V)-1=R-1T-1

  6. Review: Graphics Cameras • real pinhole camera: image inverted eye point image plane • computer graphics camera: convenient equivalent eye point center of projection image plane

  7. Review: Projective Transformations • planar geometric projections • planar: onto a plane • geometric: using straight lines • projections: 3D -> 2D • aka projective mappings • counterexamples?

  8. Projective Transformations • properties • lines mapped to lines and triangles to triangles • parallel lines do NOT remain parallel • e.g. rails vanishing at infinity • affine combinations are NOT preserved • e.g. center of a line does not map to center of projected line (perspective foreshortening)

  9. -z Perspective Projection • project all geometry • through common center of projection (eye point) • onto an image plane x y z z x x

  10. Perspective Projection projectionplane center of projection (eye point) how tall shouldthis bunny be?

  11. Basic Perspective Projection • nonuniform foreshortening • not affine similar triangles P(x,y,z) y P(x’,y’,z’) z z’=d but

  12. Perspective Projection • desired result for a point [x, y, z, 1]T projected onto the view plane: • what could a matrix look like to do this?

  13. Simple Perspective Projection Matrix

  14. Simple Perspective Projection Matrix is homogenized version of where w = z/d

  15. Simple Perspective Projection Matrix is homogenized version of where w = z/d

  16. Perspective Projection • expressible with 4x4 homogeneous matrix • use previously untouched bottom row • perspective projection is irreversible • many 3D points can be mapped to same (x, y, d) on the projection plane • no way to retrieve the unique z values

  17. Moving COP to Infinity • as COP moves away, lines approach parallel • when COP at infinity, orthographic view

  18. Orthographic Camera Projection • camera’s back plane parallel to lens • infinite focal length • no perspective convergence • just throw away z values

  19. Perspective to Orthographic • transformation of space • center of projection moves to infinity • view volume transformed • from frustum (truncated pyramid) to parallelepiped (box) x x Frustum Parallelepiped -z -z

  20. orthographic view volume perspective view volume y=top y=top x=left x=left y y z x=right VCS z=-near x VCS y=bottom z=-far z=-far x y=bottom x=right z=-near View Volumes • specifies field-of-view, used for clipping • restricts domain of z stored for visibility test z

  21. Canonical View Volumes • standardized viewing volume representation perspective orthographic orthogonal parallel x or y x or y = +/- z backplane x or y backplane 1 frontplane frontplane -z -z -1 -1

  22. Why Canonical View Volumes? • permits standardization • clipping • easier to determine if an arbitrary point is enclosed in volume with canonical view volume vs. clipping to six arbitrary planes • rendering • projection and rasterization algorithms can be reused

  23. Normalized Device Coordinates • convention • viewing frustum mapped to specific parallelepiped • Normalized Device Coordinates (NDC) • same as clipping coords • only objects inside the parallelepiped get rendered • which parallelepiped? • depends on rendering system

  24. Normalized Device Coordinates left/right x =+/- 1, top/bottom y =+/- 1, near/far z =+/- 1 NDC Camera coordinates x x x=1 right Frustum -z z left x= -1 z=1 z= -1 z=-n z=-f

  25. Understanding Z • z axis flip changes coord system handedness • RHS before projection (eye/view coords) • LHS after projection (clip, norm device coords) VCS NDCS y=top y (1,1,1) x=left y z (-1,-1,-1) z x x=right x z=-far y=bottom z=-near

  26. perspective view volume y=top x=left y z=-near VCS y=bottom z=-far x x=right Understanding Z near, far always positive in OpenGL calls glOrtho(left,right,bot,top,near,far); glFrustum(left,right,bot,top,near,far); glPerspective(fovy,aspect,near,far); orthographic view volume y=top x=left y z x=right VCS x z=-far y=bottom z=-near

  27. Understanding Z • why near and far plane? • near plane: • avoid singularity (division by zero, or very small numbers) • far plane: • store depth in fixed-point representation (integer), thus have to have fixed range of values (0…1) • avoid/reduce numerical precision artifacts for distant objects

  28. NDCS y (1,1,1) z (-1,-1,-1) x Orthographic Derivation • scale, translate, reflect for new coord sys VCS y=top x=left y z x=right x z=-far y=bottom z=-near

  29. NDCS y (1,1,1) z (-1,-1,-1) x Orthographic Derivation • scale, translate, reflect for new coord sys VCS y=top x=left y z x=right x z=-far y=bottom z=-near

  30. Orthographic Derivation • scale, translate, reflect for new coord sys

  31. Orthographic Derivation • scale, translate, reflect for new coord sys VCS y=top x=left y z x=right x z=-far y=bottom z=-near same idea for right/left, far/near

  32. Orthographic Derivation • scale, translate, reflect for new coord sys

  33. Orthographic Derivation • scale, translate, reflect for new coord sys

  34. Orthographic Derivation • scale, translate, reflect for new coord sys

  35. Orthographic Derivation • scale, translate, reflect for new coord sys

  36. Orthographic OpenGL glMatrixMode(GL_PROJECTION); glLoadIdentity(); glOrtho(left,right,bot,top,near,far);

  37. Demo • Brown applets: viewing techniques • parallel/orthographic cameras • projection cameras • http://www.cs.brown.edu/exploratories/freeSoftware/catalogs/viewing_techniques.html

  38. Projections II

  39. Asymmetric Frusta • our formulation allows asymmetry • why bother? x x right right Frustum Frustum -z -z left left z=-n z=-f

  40. Asymmetric Frusta • our formulation allows asymmetry • why bother? binocular stereo • view vector not perpendicular to view plane Left Eye Right Eye

  41. Simpler Formulation • left, right, bottom, top, near, far • nonintuitive • often overkill • look through window center • symmetric frustum • constraints • left = -right, bottom = -top

  42. Field-of-View Formulation • FOV in one direction + aspect ratio (w/h) • determines FOV in other direction • also set near, far (reasonably intuitive) x w h fovx/2 Frustum -z  fovy/2 z=-n z=-f

  43. Perspective OpenGL glMatrixMode(GL_PROJECTION); glLoadIdentity(); glFrustum(left,right,bot,top,near,far); or glPerspective(fovy,aspect,near,far);

  44. Demo: Frustum vs. FOV • Nate Robins tutorial (take 2): • http://www.xmission.com/~nate/tutors.html

  45. viewing transformation projection transformation modeling transformation viewport transformation Projective Rendering Pipeline OCS - object/model coordinate system WCS - world coordinate system VCS - viewing/camera/eye coordinate system CCS - clipping coordinate system NDCS - normalized device coordinate system DCS - device/display/screen coordinate system object world viewing O2W W2V V2C VCS WCS OCS clipping C2N CCS perspectivedivide normalized device N2D NDCS device DCS

  46. Projection Warp • warp perspective view volume to orthogonal view volume • render all scenes with orthographic projection! • aka perspective warp x x z=d z=d z=0 z=

  47. Perspective Warp • perspective viewing frustum transformed to cube • orthographic rendering of cube produces same image as perspective rendering of original frustum

  48. Predistortion

  49. viewing transformation projection transformation modeling transformation viewport transformation Projective Rendering Pipeline OCS - object/model coordinate system WCS - world coordinate system VCS - viewing/camera/eye coordinate system CCS - clipping coordinate system NDCS - normalized device coordinate system DCS - device/display/screen coordinate system object world viewing O2W W2V V2C VCS WCS OCS clipping C2N CCS perspectivedivide normalized device N2D NDCS device DCS

  50. Separate Warp From Homogenization normalized device clipping • warp requires only standard matrix multiply • distort such that orthographic projection of distorted objects is desired persp projection • w is changed • clip after warp, before divide • division by w: homogenization viewing V2C C2N CCS VCS NDCS projection transformation perspective division alter w / w

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