Last Time• Orthographic projection• Viewing transformations     – Setting up a camera position and orientation10/12/04    ...
Today• Perspective viewing• Homework 3 due10/12/04       © University of Wisconsin, CS559 Spring 2004
Review• View Space is a coordinate system with the viewer looking  down the –z axis, with x to the right and y up• The Wor...
OpenGL Camera• The default OpenGL image plane has u aligned with the x axis, v  aligned with y, and n aligned with z     –...
Typical View Spec. Usage  glMatrixMode(GL_PROJECTION);  glLoadIdentity();  glOrtho(l, r, b, t, n, f);  glMatrixMode(GL_MOD...
Left vs Right Handed View                      Space• You can define u as right, v as up, and n as toward the  viewer: a r...
Perspective Projection• Abstract camera model - box              •   Pinhole cameras work in  with a small hole in it     ...
Distant Objects Are Smaller10/12/04        © University of Wisconsin, CS559 Spring 2004
Parallel lines meetcommon to draw film planein front of the focal point 10/12/04        © University of Wisconsin, CS559 S...
Vanishing points• Each set of parallel lines  (=direction) meets at a different  point: The vanishing point for  this dire...
Basic Perspective Projection• We are going to temporarily ignore canonical view space,  and go straight from view to windo...
Basic Perspective Projection• If you know P(xv,yv,zv) and d, what is P(xs,ys)?     – Where does a point in view space end ...
Basic Case• Similar triangles gives:                                  xs xv                y s yv                         ...
Simple Perspective                     Transformation• Using homogeneous coordinates we can write:     – Our next big adva...
Parallel Lines Meet?• Parallel lines are of the form: x = x 0 + td     – Parametric form: x0 is a point on the line, t is ...
General Perspective• The basic equations we have seen give a flavor of what  happens, but they are insufficient for all ap...
Perspective View Volume• Recall the orthographic view volume, defined by a near, far,  left, right, top and bottom plane• ...
Clipping Planes                                                                    Left Clip                Near Clip     ...
Where is the Image Plane?• Notice that it doesn’t really matter where the image plane is  located, once you define the vie...
Field of View           Assumes a symmetric view volume                                                                   ...
Perspective Parameters• We have seen several different ways to describe a  perspective camera     – Focal distance, Field ...
Focal Distance to FOV• You must have the image size to do this conversion     – Why? Same d, different image size, differe...
OpenGL• gluPerspective(…)     –     Field of view in the y direction, FOV, (vertical field-of-view)     –     Aspect ratio...
gluPerspective to glFrustum• As noted previously, glu functions don’t add basic  functionality, they are just more conveni...
Stereo Viewing  • The aim is to show each eye what it would see if the virtual    scene was physically located in the real...
Stereo Frustum• “Gaze” vector (image plane normal) is not the direction the  viewer is looking – it is perpendicular to th...
Perspective Projection Matrices• We want a matrix that will take points in our perspective  view volume and transform them...
Mapping Lines• We want to map all the lines through the  center of projection to parallel lines     – This converts the pe...
General Perspective     1        0          0   0  n                         0       0    0      0        1      0   ...
Complete Perspective                         Projection   • After applying the perspective matrix, we map the          ort...
OpenGL Perspective Projection• For OpenGL you give the distance to the near and far clipping planes• The total perspective...
Near/Far and Depth Resolution• It may seem sensible to specify a very near clipping plane  and a very far clipping plane  ...
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Cs559 11

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Cs559 11

  1. 1. Last Time• Orthographic projection• Viewing transformations – Setting up a camera position and orientation10/12/04 © University of Wisconsin, CS559 Spring 2004
  2. 2. Today• Perspective viewing• Homework 3 due10/12/04 © University of Wisconsin, CS559 Spring 2004
  3. 3. Review• View Space is a coordinate system with the viewer looking down the –z axis, with x to the right and y up• The World->View transformation takes points in world space and converts them into points in view space• The Projection matrix, or View->Canonical matrix, takes points in view space and converts them into points in Canonical View Space – Canonical View Space is a coordinate system with the viewer looking along –z, x to the right, y up, and everything to be drawn inside the cube [-1,1]x[-1,1]x[-1,1] using parallel projection10/12/04 © University of Wisconsin, CS559 Spring 2004
  4. 4. OpenGL Camera• The default OpenGL image plane has u aligned with the x axis, v aligned with y, and n aligned with z – Means the default camera looks along the negative z axis – Makes it easy to do 2D drawing (no need for any view transformation)• glOrtho(…) sets an orthographic view->canonical matrix – Modifies the GL_PROJECTION matrix• gluLookAt(…) sets the world->view matrix – Takes an image center point, a point along the viewing direction and an up vector – Multiplies a world->view matrix onto the current GL_MODELVIEW matrix – You could do this yourself, using glMultMatrix(…) with the matrix from the previous slides10/12/04 © University of Wisconsin, CS559 Spring 2004
  5. 5. Typical View Spec. Usage glMatrixMode(GL_PROJECTION); glLoadIdentity(); glOrtho(l, r, b, t, n, f); glMatrixMode(GL_MODELVIEW); glLoadIdentity(); gluLookAt(ex,ey,ez,cx,cy,cx,ux,uy,uz);• GLU functions, such as gluLookAt(…), are not part of the core OpenGL library – They can be implemented with other core OpenGL commands – For example, gluLookAt(…) uses glMultMatrix(…) with the matrix from the previous slides – They are not dependent on a particular graphics card10/12/04 © University of Wisconsin, CS559 Spring 2004
  6. 6. Left vs Right Handed View Space• You can define u as right, v as up, and n as toward the viewer: a right handed system u×v=w – Advantage: Standard mathematical way of doing things• You can also define u as right, v as up and n as into the scene: a left handed system v×u=w – Advantage: Bigger n values mean points are further away• OpenGL is right handed• Many older systems, notably the Renderman standard developed by Pixar, are left handed10/12/04 © University of Wisconsin, CS559 Spring 2004
  7. 7. Perspective Projection• Abstract camera model - box • Pinhole cameras work in with a small hole in it practice10/12/04 © University of Wisconsin, CS559 Spring 2004
  8. 8. Distant Objects Are Smaller10/12/04 © University of Wisconsin, CS559 Spring 2004
  9. 9. Parallel lines meetcommon to draw film planein front of the focal point 10/12/04 © University of Wisconsin, CS559 Spring 2004
  10. 10. Vanishing points• Each set of parallel lines (=direction) meets at a different point: The vanishing point for this direction – Classic artistic perspective is 3- point perspective• Sets of parallel lines on the same plane lead to collinear vanishing points: the horizon for that plane• Good way to spot faked images10/12/04 © University of Wisconsin, CS559 Spring 2004
  11. 11. Basic Perspective Projection• We are going to temporarily ignore canonical view space, and go straight from view to window – Easier to understand and use for Project 2• Assume you have transformed to view space, with x to the right, y up, and z back toward the viewer• Assume the origin of view space is at the center of projection (the eye)• Define a focal distance, d, and put the image plane there (note d is negative) – You can define d to control the size of the image10/12/04 © University of Wisconsin, CS559 Spring 2004
  12. 12. Basic Perspective Projection• If you know P(xv,yv,zv) and d, what is P(xs,ys)? – Where does a point in view space end up on the screen? P(xv,yv,zv) yv P(xs,ys) d -zv xv10/12/04 © University of Wisconsin, CS559 Spring 2004
  13. 13. Basic Case• Similar triangles gives: xs xv y s yv = = d zv d zv yv P(xs,ys) P(xv,yv,zv) d -zv View Plane10/12/04 © University of Wisconsin, CS559 Spring 2004
  14. 14. Simple Perspective Transformation• Using homogeneous coordinates we can write: – Our next big advantage to homogeneous coordinates  xv  1 0 0 0  xs   y  0 y  ≡  v  1 0 0  s   zv  Ps =  P 0 0 1 0 v  d  z     1   vd   0  0 d 0 10/12/04 © University of Wisconsin, CS559 Spring 2004
  15. 15. Parallel Lines Meet?• Parallel lines are of the form: x = x 0 + td – Parametric form: x0 is a point on the line, t is a scalar (distance along the line from x0) and d is the direction of the line (unit vector) – Different x0 give different parallel lines• Transform and go from homogeneous to regular:  x0 + txd   x  1 0 0 0  x0  1 0 0 0   xd   z + tz   y  0 1 0 0   y  0 1 0 0  y   0 d   =  0  + t   d  = y0 + tyd  f  z  0 0 1 0   z  0 0 1 0  z   z0 + tz d    0 0 1  0   0   1  0 0 1  d  0  0   1   w  f   f       fxd fyd f• Limit as t→∞ is  zd zd   10/12/04 © University of Wisconsin, CS559 Spring 2004
  16. 16. General Perspective• The basic equations we have seen give a flavor of what happens, but they are insufficient for all applications• They do not get us to a Canonical View Volume• They make assumptions about the viewing conditions• To get to a Canonical Volume, we need a Perspective Volume …10/12/04 © University of Wisconsin, CS559 Spring 2004
  17. 17. Perspective View Volume• Recall the orthographic view volume, defined by a near, far, left, right, top and bottom plane• The perspective view volume is also defined by near, far, left, right, top and bottom planes – the clip planes – Near and far planes are parallel to the image plane: zv=n, zv=f – Other planes all pass through the center of projection (the origin of view space) – The left and right planes intersect the image plane in vertical lines – The top and bottom planes intersect in horizontal lines10/12/04 © University of Wisconsin, CS559 Spring 2004
  18. 18. Clipping Planes Left Clip Near Clip Planexv Plane Far Clip View Plane n l Volume r f -zv Right Clip Plane10/12/04 © University of Wisconsin, CS559 Spring 2004
  19. 19. Where is the Image Plane?• Notice that it doesn’t really matter where the image plane is located, once you define the view volume – You can move it forward and backward along the z axis and still get the same image, only scaled• The left/right/top/bottom planes are defined according to where they cut the near clip plane• Or, define the left/right and top/bottom clip planes by the field of view10/12/04 © University of Wisconsin, CS559 Spring 2004
  20. 20. Field of View Assumes a symmetric view volume Left Clip Near Clip Planexv Plane Far Clip View Plane FOV Volume f -zv Right Clip Plane10/12/04 © University of Wisconsin, CS559 Spring 2004
  21. 21. Perspective Parameters• We have seen several different ways to describe a perspective camera – Focal distance, Field of View, Clipping planes• The most general is clipping planes – they directly describe the region of space you are viewing• For most graphics applications, field of view is the most convenient – It is image size invariant – having specified the field of view, what you see does not depend on the image size – But, not sufficient for stereo viewing (more later)• You can convert one thing to another10/12/04 © University of Wisconsin, CS559 Spring 2004
  22. 22. Focal Distance to FOV• You must have the image size to do this conversion – Why? Same d, different image size, different FOV height/2 d FOV/2  height  FOV d = tan   2d 2 −1  height  FOV = 2 tan    2d • Project 2 does the inverse conversion to find the focal length10/12/04 © University of Wisconsin, CS559 Spring 2004
  23. 23. OpenGL• gluPerspective(…) – Field of view in the y direction, FOV, (vertical field-of-view) – Aspect ratio, a, should match window aspect ratio – Near and far clipping planes, n and f – Defines a symmetric view volume• glFrustum(…) – Give the near and far clip plane, and places where the other clip planes cross the near plane – Defines the general case – Used for stereo viewing, mostly10/12/04 © University of Wisconsin, CS559 Spring 2004
  24. 24. gluPerspective to glFrustum• As noted previously, glu functions don’t add basic functionality, they are just more convenient – So how does gluPerspective convert to glFrustum? – Symmetric, so only need t and ly t ? FOV / 2 n z10/12/04 © University of Wisconsin, CS559 Spring 2004
  25. 25. Stereo Viewing • The aim is to show each eye what it would see if the virtual scene was physically located in the real world • “Fish tank” Virtual RealityLeft eye Virtual Scene (fixed)Head movesRight eye Screen (fixed) 10/12/04 © University of Wisconsin, CS559 Spring 2004
  26. 26. Stereo Frustum• “Gaze” vector (image plane normal) is not the direction the viewer is looking – it is perpendicular to the screen• View volume is not symmetric Left eye g l n r Screen (fixed)10/12/04 © University of Wisconsin, CS559 Spring 2004
  27. 27. Perspective Projection Matrices• We want a matrix that will take points in our perspective view volume and transform them into the orthographic view volume – This matrix will go in our pipeline before an orthographic projection matrix (r,t,f) (r,t,n) (r,t,f) (r,t,n) (l,b,n) (l,b,n)10/12/04 © University of Wisconsin, CS559 Spring 2004
  28. 28. Mapping Lines• We want to map all the lines through the center of projection to parallel lines – This converts the perspective case to the orthographic case, we can use all our existing methods• The relative intersection points of lines with the near clip plane should not change• The matrix that does this looks like the matrix for our simple perspective case10/12/04 © University of Wisconsin, CS559 Spring 2004
  29. 29. General Perspective 1 0 0 0  n 0 0 0  0 1 0 0  0 n 0 0 MP =  ≡  0 0 ( n + f ) n − f  0 0 n+ f − nf      0 0 1n 0  0 0 1 0 • This matrix leaves points with z=n unchanged• It is just like the simple projection matrix, but it does some extra things to z to map the depth properly• We can multiply a homogenous matrix by any number without changing the final point, so the two matrices above have the same effect 10/12/04 © University of Wisconsin, CS559 Spring 2004
  30. 30. Complete Perspective Projection • After applying the perspective matrix, we map the orthographic view volume to the canonical view volume:  2 − (r + l)  ( r − l) 0 0  ( r − l )  n 0  0 0  − ( t + b)   0   0 2 0 0 n 0M view − >canonical = M O M P =  ( t − b) ( t − b )  0 0 ( n + f ) − nf     2 − ( n + f )    0 0  0 0  (n − f ) (n − f )  1 0   0  0 0 1   M world − >canonical = M view − >canonical M world − >view x canonical = M world − >canonical x world 10/12/04 © University of Wisconsin, CS559 Spring 2004
  31. 31. OpenGL Perspective Projection• For OpenGL you give the distance to the near and far clipping planes• The total perspective projection matrix resulting from a glFrustum call is:  2n 0 (r + l) 0    ( r − l) (r − l)   0 2n ( t + b) 0   M OpenGL = ( t − b) ( t − b)    0 0 (n + f ) 2f n    (n − f ) ( n − f )   0 0 −1 0 10/12/04 © University of Wisconsin, CS559 Spring 2004
  32. 32. Near/Far and Depth Resolution• It may seem sensible to specify a very near clipping plane and a very far clipping plane – Sure to contain entire scene• But, a bad idea: – OpenGL only has a finite number of bits to store screen depth – Too large a range reduces resolution in depth - wrong thing may be considered “in front” – See Shirley for a more complete explanation• Always place the near plane as far from the viewer as possible, and the far plane as close as possible10/12/04 © University of Wisconsin, CS559 Spring 2004

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