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Projections  Angel: Interactive Computer Graphics
Road in perspective
Taxonomy of Projections FVFHP Figure 6.10
Taxonomy of Projections
Parallel Projection <ul><li>Center of projection is at infinity </li></ul><ul><ul><li>Direction of projection (DOP) same f...
Orthographic Projections <ul><li>DOP perpendicular to view plane </li></ul>Angel Figure 5.5 Top Side Front
Oblique Projections <ul><li>DOP  not  perpendicular to view plane </li></ul>H&B Cavalier (DOP    = 45 o ) tan(  ) = 1 Ca...
Orthographic Projection <ul><li>Simple Orthographic Transformation </li></ul><ul><li>Original world units are preserved </...
Perspective Transformation <ul><li>First discovered by Donatello, Brunelleschi, and DaVinci during Renaissance </li></ul><...
Perspective Projection <ul><li>How many vanishing points? </li></ul>Angel Figure 5.10 3-Point Perspective 2-Point Perspect...
Perspective Projection <ul><li>In the real world, objects exhibit  perspective foreshortening : distant objects appear sma...
Perspective Projection <ul><li>When we do 3-D graphics, we think of the  screen as a 2-D window onto the 3-D world: </li><...
Perspective Projection <ul><li>The geometry of the situation is that of  similar triangles .  View from above: </li></ul><...
Perspective Projection <ul><li>Desired result for a point  [ x, y, z, 1 ] T  projected onto the view plane: </li></ul><ul>...
A Perspective Projection Matrix <ul><li>Answer: </li></ul>
A Perspective Projection Matrix <ul><li>Example: </li></ul><ul><li>Or, in 3-D coordinates: </li></ul>
Projection Matrices <ul><li>Now that we can express perspective foreshortening as a matrix, we can compose it onto our oth...
Perspective vs. Parallel <ul><li>Perspective projection </li></ul><ul><ul><li>Size varies inversely with distance - looks ...
Classical Projections Angel Figure 5.3
A 3D Scene <ul><li>Notice the presence of the camera, the projection plane, and  the world coordinate axes </li></ul><ul><...
Q1  <ul><li>Using the origin as the centre of projection, derive the perspective transformation onto the plane passing thr...
A1  <ul><li>P(x,y,z) is projected onto P’(x’,y’,z’) </li></ul><ul><li>x’= α x, y’=  α y , z’=  α z </li></ul><ul><li>n 1 x...
Q2  <ul><li>Find the perspective projection onto the view plane z=d where the centre of projection is the origin(0,0,0) </...
Q3  <ul><li>Derive the general perspective transformation onto a plane with reference point R 0 (x 0 ,y 0 ,z 0 ), normal v...
A3  <ul><li>Per N,R0’ =T C . Per N,R0  .T -C </li></ul>
Q4  <ul><li>Derive parallel projection onto xy plane in the direction of projection V=ai+bj+ck </li></ul>
A4  <ul><li>x’-x=ka , y’-y=kb , z’-z=kc </li></ul><ul><li>K=-z/c (z=0 on xy plane) </li></ul><ul><li>1  0  -a/c </li></ul>...
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Lecture 11 Perspective Projection

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perspective projection theory and questions

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Transcript of "Lecture 11 Perspective Projection"

  1. 1. Projections Angel: Interactive Computer Graphics
  2. 2. Road in perspective
  3. 3. Taxonomy of Projections FVFHP Figure 6.10
  4. 4. Taxonomy of Projections
  5. 5. Parallel Projection <ul><li>Center of projection is at infinity </li></ul><ul><ul><li>Direction of projection (DOP) same for all points </li></ul></ul>Angel Figure 5.4 DOP View Plane
  6. 6. Orthographic Projections <ul><li>DOP perpendicular to view plane </li></ul>Angel Figure 5.5 Top Side Front
  7. 7. Oblique Projections <ul><li>DOP not perpendicular to view plane </li></ul>H&B Cavalier (DOP  = 45 o ) tan(  ) = 1 Cabinet (DOP  = 63.4 o ) tan(  ) = 2
  8. 8. Orthographic Projection <ul><li>Simple Orthographic Transformation </li></ul><ul><li>Original world units are preserved </li></ul><ul><ul><li>Pixel units are preferred </li></ul></ul>
  9. 9. Perspective Transformation <ul><li>First discovered by Donatello, Brunelleschi, and DaVinci during Renaissance </li></ul><ul><li>Objects closer to viewer look larger </li></ul><ul><li>Parallel lines appear to converge to single point </li></ul>
  10. 10. Perspective Projection <ul><li>How many vanishing points? </li></ul>Angel Figure 5.10 3-Point Perspective 2-Point Perspective 1-Point Perspective
  11. 11. Perspective Projection <ul><li>In the real world, objects exhibit perspective foreshortening : distant objects appear smaller </li></ul><ul><li>The basic situation: </li></ul>
  12. 12. Perspective Projection <ul><li>When we do 3-D graphics, we think of the screen as a 2-D window onto the 3-D world: </li></ul>How tall should this bunny be?
  13. 13. Perspective Projection <ul><li>The geometry of the situation is that of similar triangles . View from above: </li></ul><ul><li>What is x’ ? </li></ul>d P ( x, y, z ) X Z View plane (0,0,0) x’ = ?
  14. 14. Perspective Projection <ul><li>Desired result for a point [ x, y, z, 1 ] T projected onto the view plane: </li></ul><ul><li>What could a matrix look like to do this? </li></ul>
  15. 15. A Perspective Projection Matrix <ul><li>Answer: </li></ul>
  16. 16. A Perspective Projection Matrix <ul><li>Example: </li></ul><ul><li>Or, in 3-D coordinates: </li></ul>
  17. 17. Projection Matrices <ul><li>Now that we can express perspective foreshortening as a matrix, we can compose it onto our other matrices with the usual matrix multiplication </li></ul><ul><li>End result: a single matrix encapsulating modeling, viewing, and projection transforms </li></ul>
  18. 18. Perspective vs. Parallel <ul><li>Perspective projection </li></ul><ul><ul><li>Size varies inversely with distance - looks realistic </li></ul></ul><ul><ul><li>Distance and angles are not (in general) preserved </li></ul></ul><ul><ul><li>Parallel lines do not (in general) remain parallel </li></ul></ul><ul><li>Parallel projection </li></ul><ul><ul><li>Good for exact measurements </li></ul></ul><ul><ul><li>Parallel lines remain parallel </li></ul></ul><ul><ul><li>Angles are not (in general) preserved </li></ul></ul><ul><ul><li>Less realistic looking </li></ul></ul>
  19. 19. Classical Projections Angel Figure 5.3
  20. 20. A 3D Scene <ul><li>Notice the presence of the camera, the projection plane, and the world coordinate axes </li></ul><ul><li>Viewing transformations define how to acquire the image on the projection plane </li></ul>
  21. 21. Q1 <ul><li>Using the origin as the centre of projection, derive the perspective transformation onto the plane passing through the point R 0 (x 0 ,y 0 ,z 0 ) and having normal vector N=n 1 i+n 2 j+n 3 k </li></ul>
  22. 22. A1 <ul><li>P(x,y,z) is projected onto P’(x’,y’,z’) </li></ul><ul><li>x’= α x, y’= α y , z’= α z </li></ul><ul><li>n 1 x’+n 2 y’+n 3 z’=d (where d=n 1 x 0 +n 2 y 0 +n 3 z 0 ) </li></ul><ul><li>α =d/( n 1 x+n 2 y+n 3 z) </li></ul><ul><li>d 0 0 0 </li></ul><ul><li>Per N,R0 = 0 d 0 0 </li></ul><ul><li>0 0 d 0 </li></ul><ul><li>n 1 n 2 n 3 0 </li></ul>
  23. 23. Q2 <ul><li>Find the perspective projection onto the view plane z=d where the centre of projection is the origin(0,0,0) </li></ul>
  24. 24. Q3 <ul><li>Derive the general perspective transformation onto a plane with reference point R 0 (x 0 ,y 0 ,z 0 ), normal vector N=n 1 i+n 2 j+n 3 k, and using C(a,b,c) as the centre of projection </li></ul>
  25. 25. A3 <ul><li>Per N,R0’ =T C . Per N,R0 .T -C </li></ul>
  26. 26. Q4 <ul><li>Derive parallel projection onto xy plane in the direction of projection V=ai+bj+ck </li></ul>
  27. 27. A4 <ul><li>x’-x=ka , y’-y=kb , z’-z=kc </li></ul><ul><li>K=-z/c (z=0 on xy plane) </li></ul><ul><li>1 0 -a/c </li></ul><ul><li>Par V = 0 1 -b/c </li></ul><ul><li>0 0 0 </li></ul>
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