Introduction to Computer Graphics. Ray Tracing Review

Transcription

1 Introduction to Computer Graphics Ray Tracing Review

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4 Ray Tracing For each light in scene Emit 1,000,000,000 photons For each photon Find what geometry photon hits Color photon Scatter photon Find what photon hits next... Pray photon hits camera CCD Light pixel that CCD micro-square represents

5 Ray Tracing For each light in scene Emit 1,000,000,000 photons For each photon Find what geometry photon hits Color photon Scatter photon Find what photon hits next... Pray photon hits camera CCD Light pixel that CCD micro-square represents

6 Ray Tracing

7 Ray Tracing

8 Ray Generation

9 Ray Generation

10 Ray Generation

11 Ray Generation

12 Ray Generation

13 Ray Generation

14 Ray Generation

15 Ray Generation screenv eye screenu screencenter

16 Ray Generation screenv eye x: -1 1 y: -1 1 y screenu screencenter x

17 Ray Generation ray origin: eye ray direction: screencenter + x screenu + y screenv screenv eye x: -1 1 y: -1 1 y screenu screencenter x

18 Ray Generation

19 Sphere Intersection Ray

20 Sphere Intersection Circle Ray

21 Sphere Intersection Circle Origin Ray Direction

22 Sphere Intersection Origin Ray Circle Origin Direction

23 Sphere Intersection Origin Ray Circle Origin Radius Direction

24 Sphere Intersection Origin Ray Circle Origin Radius scalar Direction

25 Sphere Intersection Ray: emission point (e) and direction (d) e d

26 Sphere Intersection Ray: emission point (e) and direction (d) Point on Ray: p(t) = e + td e t p d

27 Sphere Intersection e Circle Origin Radius scalar d

28 Sphere Intersection Circle: center point (o) and radius (r) e o r d

29 Sphere Intersection Circle: center point (o) and radius (r) Point on circle: (p - o) (p - o) - r 2 = 0 e o r d

30 Sphere Intersection e Circle: center point (o) and radius (r) Point on circle: (p - o) (p - o) - r 2 = 0 Point on Ray: p(t) = e + td p o e + td r d

31 Sphere Intersection e Circle: center point (o) and radius (r) Point on circle: (p - o) (p - o) - r 2 = 0 Point on Ray: p(t) = e + td p o e + td r d (e + td - o) (e + td - o) - r 2 = 0

32 Sphere Intersection e Circle: center point (o) and radius (r) Point on circle: (p - o) (p - o) - r 2 = 0 Point on Ray: p(t) = e + td p o e + td r d (td + e - o) (td + e - o) - r 2 = 0

33 Sphere Intersection e Circle: center point (o) and radius (r) Point on circle: (p - o) (p - o) - r 2 = 0 Point on Ray: p(t) = e + td p o e + td r d (td + (e - o)) (td + (e - o)) - r 2 = 0

34 Sphere Intersection e Circle: center point (o) and radius (r) Point on circle: (p - o) (p - o) - r 2 = 0 Point on Ray: p(t) = e + td p o e + td r d t 2 (d d) + 2td (e - o) + (e - o) (e - o) - r 2 = 0

35 Sphere Intersection e Circle: center point (o) and radius (r) Point on circle: (p - o) (p - o) - r 2 = 0 Point on Ray: p(t) = e + td p o e + td r d at 2 + bt + c = 0 a = d d b = 2d (e - o) c = (e - o) (e - o) - r 2

36 Sphere Intersection e Circle: center point (o) and radius (r) Point on circle: (p - o) (p - o) - r 2 = 0 Point on Ray: p(t) = e + td p o e + td r d t = -b ± b 2-4ac 2a a = d d b = 2d (e - o) c = (e - o) (e - o) - r 2

37 Sphere Intersection e Circle: center point (o) and radius (r) Point on sphere: (p - o) (p - o) - r 2 = 0 Point on Ray: p(t) = e + td p o e + td r d t = -b ± b 2-4ac 2a a = d d b = 2d (e - o) c = (e - o) (e - o) - r 2

38 Sphere Intersection h = b 2-4ac h < 0 : ray misses sphere h = 0 : ray tangent to sphere h > 0 : ray intersects sphere 2x p o r e d t = -b ± b 2-4ac 2a a = d d b = 2d (e - o) c = (e - o) (e - o) - r 2

39 Sphere Intersection h = b 2-4ac Which intersection h < 0 : ray misses sphere should be used? h = 0 : ray tangent to sphere h > 0 : ray intersects sphere 2x p o r e d t = -b ± b 2-4ac 2a a = d d b = 2d (e - o) c = (e - o) (e - o) - r 2

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42 Lighting

43 Lighting

44 Lighting Idr Idg Idb = l ˆ Ldr kdr nˆ = l ˆ Ldgkdg nˆ = l ˆ nˆ Ldbkdb 0 < Idx< 1 n l ( Ldr, Ldg, Ldb) = light color ( kdr kdg kdb),, = material color Ldx = light channel intensity kdx= material diffuse coefficient for channel

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47 Plane Intersection Plane: (o - p) n = 0 p n o

48 Plane Intersection Point on Ray: p(t) = e + td Plane: (o - p) n = 0 p e d n o

49 Plane Intersection (o - (e + td)) n = 0 Point on Ray: p(t) = e + td Plane: (o - p) n = 0 p e d n o

50 Plane Intersection (o - e - td) n = 0 Point on Ray: p(t) = e + td Plane: (o - p) n = 0 p e d n o

51 Plane Intersection (-td + (o - e)) n = 0 Point on Ray: p(t) = e + td Plane: (o - p) n = 0 p e d n o

52 Plane Intersection -td n + (o - e) n = 0 Point on Ray: p(t) = e + td Plane: (o - p) n = 0 p e d n o

53 Plane Intersection -td n = -(o - e) n Point on Ray: p(t) = e + td Plane: (o - p) n = 0 p e d n o

54 Plane Intersection td n = (o - e) n Point on Ray: p(t) = e + td Plane: (o - p) n = 0 p e d n o

55 Plane Intersection t = (o - e) n d n Point on Ray: p(t) = e + td Plane: (o - p) n = 0 p e d n o

56 Plane Intersection (o - e) n t = Point on Ray: p(t) = e + td d n Plane: (o - p) n = 0 if d n = 0 : ray parallel to plane p e d n o

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59 Shadow

60 Shadow

61 Shadow

62 Shadow

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65 Circle Intersection t = (o - e) n d n Point on Ray: p(t) = e + td Plane: (o - p) n = 0 if d n = 0 : ray parallel to plane p e d n o

66 Circle Intersection t = (o - e) n d n Point on Ray: p(t) = e + td Plane: (o - p) n = 0 if d n = 0 : ray parallel to plane p e d n o r

67 Circle Intersection t = (o - e) n d n Point on Ray: p(t) = e + td Plane: (o - p) n = 0 if d n = 0 : ray parallel to plane p e d n o r if (p - o) (p - o) > r 2 Outside Circle

68 Triangle Intersection t = (o - e) n d n Point on Ray: p(t) = e + td Plane: (o - p) n = 0 if d n = 0 : ray parallel to plane p e d n o n = (b - a) (c - a)

69 Triangle Intersection 2D c p a b

70 Triangle Intersection 2D c p a b

71 Triangle Intersection 3D c acp p bcp a abp b

72 Triangle Intersection 3D c acp p bcp a b = a b sinθ a abp b

73 Triangle Intersection 3D c acp p bcp a b = area(parallelogram) a abp b

74 Triangle Intersection 3D c acp λ2 p λ1 bcp a b = area(parallelogram) a λ3 abp b area( abc) = (0.5) a b

75 Triangle Intersection λ1 = bcp abc λ2 = acp abc λ3 = abp abc area( abc) = (0.5) a b acp a c λ2 p λ3 λ1 abp 3D bcp b

76 Triangle Intersection λ1 = 0.5 (b-p) (c-b) 0.5 (b-a) (c-b) λ2 = 0.5 (c-p) (a-c) 0.5 (b-a) (c-b) λ3 = 0.5 (a-p) (b-a) 0.5 (b-a) (c-b) area( abc) = (0.5) a b acp a c λ2 p λ3 λ1 abp 3D bcp b

77 Triangle Intersection λ1 = λ2 = λ3 = (b-p) (c-b) (b-a) (c-b) (c-p) (a-c) (b-a) (c-b) (a-p) (b-a) (b-a) (c-b) acp a c λ2 p λ3 λ1 abp 3D bcp b

78 Triangle Intersection λ1 = λ2 = λ3 = (b-p) (c-b) (b-a) (c-b) (c-p) (a-c) (b-a) (c-b) (a-p) (b-a) (b-a) (c-b) sign? acp a c λ2 p λ3 λ1 abp 3D bcp b

79 Triangle Intersection λ1 = λ2 = λ3 = (b-p) (c-b) (b-a) (c-b) (c-p) (a-c) (b-a) (c-b) (a-p) (b-a) (b-a) (c-b) acp a c λ2 p λ3 λ1 abp bcp,sign(((b-p) (c-b)) ((b-a) (c-b))), sign( ((c-p) (a-c)), sign( ((a-p) (b-a)) b ((b-a) (c-b)) ((b-a) (c-b))

80 Triangle Intersection λ1 = λ2 = λ3 = (b-p) (c-b) (b-a) (c-b) (c-p) (a-c) (b-a) (c-b) (a-p) (b-a) (b-a) (c-b) acp a c λ2 p λ3 λ1 abp bcp,sign(((b-p) (c-b)) ((b-a) (c-b))), sign( ((c-p) (a-c)), sign( ((a-p) (b-a)) b ((b-a) (c-b)) ((b-a) (c-b)) p(t) = e + td t = (o - e) n d n λ1 > 0, λ2 > 0, λ3 > 0

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83 Reflection

84 Reflection n l r

85 Reflection n l θ θ r

86 Reflection l n = l n cosθ n l θ θ r

87 Reflection l n = l n cosθ n l (l n)n r

88 Reflection 2(l n) = 2 l n cosθ n l 2(l n)n r

89 Reflection 2(l n) = 2 l n cosθ n l r -l 2(l n)n r = 2(l n)n - l r

90 Reflection r

91 Reflection What now gets hit? r

92 Reflection What now gets hit? Recursive! r

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98 Texture Coordinates

99 Texture Coordinates 3D 2D

100 Texture Coordinates?

101 Texture Coordinates

102 Texture Coordinates θ = atan2(z,x) ɸ = asin(y/r)

103 Texture Coordinates θ = atan2(z,x) ɸ = asin(y/r) -π < θ < π -π/2 < ɸ < π/2

104 Texture Coordinates θ = atan2(z,x) ɸ = asin(y/r) -π < θ < π 0<u<1 -π/2 < ɸ < π/2 0<v<1

105 Texture Coordinates θ = atan2(z,x) ɸ = asin(y/r) u = θ / 2π + 1/2 v = ɸ / π + 1/2 -π < θ < π 0<u<1 -π/2 < ɸ < π/2 0<v<1

106 Texture Coordinates (0.6,0.2) (0,0) (1,0) (0.15,0.1) (0.6,0.9) (0,1)

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