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### Understanding Orthographic Projection and Ray Interpolation Techniques ###

This document explores orthographic (parallel) projection, focusing on determining the starting points of rays and employing various interpolation methods such as nearest neighbor and trilinear interpolation. It details the process of finding projection points, ray casting, and sample point calculations in a 3D image plane environment, specifically at the pixel coordinates (i=4, j=1). The content serves as a guide for understanding the foundation of 3D graphics rendering techniques and the mathematics behind ray tracing in orthographic projection. ###

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### Understanding Orthographic Projection and Ray Interpolation Techniques ###

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  1. Orthographic (parallel) projection: Finding the starting point of the ray P(0,0): origin of image plane P(4,1): image pixel at i=4, j=1 v n P(4,1) P(0,0) u

  2. Orthographic (parallel) projection: Finding the starting point of the ray P(0,0): origin of image plane P(4,1): image pixel at i=4, j=1 v n P(4,1) P(0,0) u

  3. Orthographic (parallel) projection: Finding the starting point of the ray P(0,0): origin of image plane P(4,1): image pixel at i=4, j=1 v n P(4,1) P(0,0) u

  4. Orthographic (parallel) projection: Casting the ray P(0,0): origin of image plane P(4,1): image pixel at i=4, j=1 Q(6): ray sample point at k=6 n n n n n n Q(6) v n P(4,1) P(0,0) u

  5. Orthographic (parallel) projection: Casting the ray P(0,0): origin of image plane P(4,1): image pixel at i=4, j=1 Q(6): ray sample point at k=6 n n n n n n Q(6) v n P(4,1) P(0,0) u

  6. Orthographic (parallel) projection: Casting the ray P(0,0): origin of image plane P(4,1): image pixel at i=4, j=1 Q(6): ray sample point at k=6 n n n n n n Q(6) v n P(4,1) P(0,0) u (assume)

  7. Orthographic (parallel) projection: Interpolation P(0,0): origin of image plane P(4,1): image pixel at i=4, j=1 Q(6): ray sample point at k=6 n n n n n n Q(6) v n P(0,0) u Q(6) (assume)

  8. y Orthographic (parallel) projection: Interpolation x z 32 8 30 13 43 10 Q(6) 20 24

  9. Orthographic (parallel) projection: Nearest Neighbor Interpolation y x z 32 8 30 13 43 10 Q(6) 20 24

  10. Orthographic (parallel) projection: Nearest Neighbor Interpolation y x z 32 8 30 13 43 10 Q(6) 20 24

  11. Orthographic (parallel) projection: Nearest Neighbor Interpolation y x z 32 8 30 13 43 10 Q(6) 20 24

  12. Orthographic (parallel) projection: Nearest Neighbor Interpolation y x z 32 8 30 13 43 10 Q(6) 20 24

  13. Orthographic (parallel) projection: Trilinear Interpolation y x z 32 8 30 13 ww 43 10 Q(6) vv uu 20 24

  14. Orthographic (parallel) projection: Trilinear Interpolation y x z 32 8 30 13 ww 43 10 Q(6) vv uu 20 24

  15. Orthographic (parallel) projection: Trilinear Interpolation y x z 32 8 30 13 ww 43 10 Q(6) vv uu 20 24

  16. Orthographic (parallel) projection: Trilinear Interpolation y x z 32 8 30 13 ww 43 10 Q(6) vv uu 20 24

  17. Orthographic (parallel) projection: Trilinear Interpolation y x z 32 8 30 13 ww 43 10 Q(6) vv uu 20 24

  18. Orthographic (parallel) projection: Trilinear Interpolation y x z 32 8 30 13 ww 43 10 Q(6) vv uu 20 24

  19. Orthographic (parallel) projection: Trilinear Interpolation y x z 32 8 30 13 ww 43 10 Q(6) vv uu 20 24

  20. Orthographic (parallel) projection: Trilinear Interpolation y x z 32 8 30 13 ww 43 10 Q(6) vv uu 20 24

  21. Orthographic (parallel) projection: Trilinear Interpolation y x z 32 8 30 13 ww 43 10 Q(6) vv uu 20 24

  22. Orthographic (parallel) projection: Trilinear Interpolation y x z 32 8 30 13 ww 43 10 Q(6) vv uu 20 24

  23. Orthographic (parallel) projection: Trilinear Interpolation y x z 32 8 30 13 ww 43 10 Q(6) vv uu 20 24

  24. Orthographic (parallel) projection: 1st linear interpolation: along x y x z 32 8 30 13 43 10 Q(6) uu 20 24

  25. Orthographic (parallel) projection: 1st linear interpolation: along x y x z 32 8 30 13 43 10 Q(6) uu 20 24

  26. Orthographic (parallel) projection: 1st linear interpolation: along x y x z 32 8 30 13 43 10 Q(6) uu 20 24

  27. Orthographic (parallel) projection: 1st linear interpolation: along x y x z 32 8 30 13 43 10 Q(6) uu 20 24

  28. Orthographic (parallel) projection: 1st linear interpolation: along x y x z 32 8 30 13 43 10 Q(6) uu 20 24

  29. Orthographic (parallel) projection: 1st linear interpolation: along x y x z 32 8 30 13 43 10 Q(6) uu 20 24

  30. Orthographic (parallel) projection: 1st linear interpolation: along x y x z 32 8 30 13 43 10 Q(6) uu 20 24

  31. Orthographic (parallel) projection: 1st linear interpolation: along x y x z 32 8 30 13 43 10 Q(6) uu 20 24

  32. Orthographic (parallel) projection: 1st linear interpolation: along x y x z 32 8 30 13 43 10 Q(6) uu 20 24

  33. Orthographic (parallel) projection: 1st linear interpolation: along x y x z 32 8 30 13 43 10 Q(6) uu 20 24

  34. Orthographic (parallel) projection: 1st linear interpolation: along x y x z 32 8 30 13 43 10 Q(6) uu 20 24

  35. Orthographic (parallel) projection: 1st linear interpolation: along x y x z 32 8 30 13 43 10 Q(6) uu 20 24

  36. Orthographic (parallel) projection: 1st linear interpolation: along x y x z 32 8 30 13 43 10 Q(6) uu 20 24

  37. Orthographic (parallel) projection: 2nd linear interpolation: along y y x z 32 8 30 13 43 10 Q(6) vv 20 24

  38. Orthographic (parallel) projection: 2nd linear interpolation: along y y x z 32 8 30 13 43 10 Q(6) vv 20 24

  39. Orthographic (parallel) projection: 2nd linear interpolation: along y y x z 32 8 30 13 43 10 Q(6) vv 20 24

  40. Orthographic (parallel) projection: 2nd linear interpolation: along y y x z 32 8 30 13 43 10 Q(6) vv 20 24

  41. Orthographic (parallel) projection: 2nd linear interpolation: along y y x z 32 8 30 13 43 10 Q(6) vv 20 24

  42. Orthographic (parallel) projection: 2nd linear interpolation: along y y x z 32 8 30 13 43 10 Q(6) vv 20 24

  43. Orthographic (parallel) projection: 2nd linear interpolation: along y y x z 32 8 30 13 43 10 Q(6) vv 20 24

  44. Orthographic (parallel) projection: 3rd linear interpolation: along z y x z 32 Val(Q(6)) 8 30 13 ww 43 10 Q(6) 20 24

  45. Orthographic (parallel) projection: 3rd linear interpolation: along z y x z 32 Val(Q(6)) 8 30 13 ww 43 10 Q(6) 20 24

  46. Orthographic (parallel) projection: 3rd linear interpolation: along z y x z 32 Val(Q(6)) 8 30 13 ww 43 10 Q(6) 20 24

  47. Orthographic (parallel) projection: 3rd linear interpolation: along z y x z 32 Val(Q(6)) 8 30 13 ww 43 10 Q(6) 20 24

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