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Projections: Efficiency and Camera Motion

Learn how to improve efficiency in rendering graphics using display lists and exploit hierarchical structures. Understand camera motion and the transformations involved in world-to-view coordinates. Explore the concepts of pinhole and perspective cameras, as well as projective transformations and the differences between perspective and orthographic projections.

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Projections: Efficiency and Camera Motion

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  1. http://www.ugrad.cs.ubc.ca/~cs314/Vjan2005 ProjectionsWeek 4, Mon Jan 24

  2. News • make sure you’ve signed up for demo slot

  3. Review: Display Lists • reuse block of OpenGL code • efficiency • multiple instances of same object • static objects redrawn often • exploit hierarchical structure when possible • set up list once with glNewList/glEndList • call multiple times

  4. Review: Camera Motion • rotate/translate/scale difficult to control • arbitrary viewing position • eye point, gaze/lookat direction, up vector y lookat x Pref WCS view up z eye Peye

  5. 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

  6. Projections

  7. Pinhole Camera • ingredients • box • film • hole punch • results • pictures! www.kodak.com www.debevec.org/Pinhole www.pinhole.org

  8. Pinhole Camera • theoretical perfect pinhole one ray of projection pinhole film plane

  9. Pinhole Camera • non-zero sized hole multiple rays of projection pinhole film plane

  10. Pinhole Camera • field of view and focal length pinhole field of view focallength film plane

  11. Pinhole Camera • field of view and focal length pinhole focallength field of view film plane

  12. real pinhole camera aperture f camera lens Real Cameras • pinhole camera has small aperture (lens opening) • hard to get enough light to expose the film • lens permits larger apertures • lens permits changing distance to film plane without actually moving the film plane price to pay: limited depth of field

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

  14. General Projection • image plane need not be perpendicular to view plane eye point image plane eye point image plane

  15. Perspective Projection • our camera must model perspective

  16. Perspective Projection • our camera must model perspective

  17. Perspective Projection • our camera must model perspective

  18. one-point perspective Perspective Projections • classified by vanishing points two-point perspective three-point perspective

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

  20. 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)

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

  22. how tall shouldthis bunny be? Perspective Projection projectionplane COP

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

  24. 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?

  25. Perspective Projection Matrix

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

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

  28. 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

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

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

  31. 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

  32. 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

  33. View Volume • 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

  34. 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

  35. 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 Understanding Z • z axis flip changes coord system handedness • RHS before projection (eye/view coords) • LHS after projection (clip, norm device coords)

  36. 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

  37. 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

  38. 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

  39. 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

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

  41. 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

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

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

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

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

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

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