Ray Casting II. Courtesy of James Arvo and David Kirk. Used with permission.

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1 y Csting II Courtesy of Jmes Arvo nd Dvid Kirk. Used with permission. MIT EECS Frédo Durnd nd Brb Cutler Some slides courtesy of Leonrd McMilln MIT EECS 6.837, Cutler nd Durnd 1

2 eview of y Csting MIT EECS 6.837, Cutler nd Durnd 2

3 y Csting For every pixel Construct ry from the eye For every object in the scene Find intersection with the ry Keep if closest MIT EECS 6.837, Cutler nd Durnd 3

4 y Trcing Secondry rys (shdows, reflection, refrction) In couple of weeks reflection refrction MIT EECS 6.837, Cutler nd Durnd 4

5 y representtion Two vectors: Origin Direction (normlized is better) Prmetric line P(t) = + t * D P(t) D MIT EECS 6.837, Cutler nd Durnd 5

6 Explicit vs. implicit Implicit Solution of n eqution Does not tell us how to generte point on the plne Tells us how to check tht point is on the plne Explicit Prmetric How to generte points Hrder to verify tht point is on the ry MIT EECS 6.837, Cutler nd Durnd 6

7 A note on shding Norml direction, direction to light Diffuse component: dot product Speculr component for shiny mterils Depends on viewpoint More in two weeks Diffuse sphere shiny spheres MIT EECS 6.837, Cutler nd Durnd 7

8 Textbook ecommended, not required Peter Shirley Fundmentls of Computer Grphics AK Peters MIT EECS 6.837, Cutler nd Durnd 8

9 eferences for ry csting/trcing Shirley Chpter 9 Specilized books: An Introduction to y Trcing, Edited by Andrew S. Glssner elistic y Trcing, Peter Shirley elistic Imge Synthesis Using Photon Mpping, Henrik Wnn Jensen Online resources e.html MIT EECS 6.837, Cutler nd Durnd 9

10 Assignment 1 Write bsic ry cster Orthogrphic cmer Spheres Disply: constnt color nd distnce We provide y Hit Prsing And liner lgebr, imge MIT EECS 6.837, Cutler nd Durnd 10

11 Object-oriented design We wnt to be ble to dd primitives esily Inheritnce nd virtul methods Even the scene is derived from Object3D! Object3D bool intersect(y, Hit, tmin) Plne bool intersect(y, Hit, tmin) Sphere bool intersect(y, Hit, tmin) Tringle bool intersect(y, Hit, tmin) Group bool intersect(y, Hit, tmin) MIT EECS 6.837, Cutler nd Durnd 11

12 y clss y { public: // CONSTUCTO & DESTUCTO y () {} y (const Vec3f &dir, const Vec3f &orig) { direction = dir; origin = orig; } y (const y& r) {*this=r;} // ACCESSOS const Vec3f& getorigin() const { return origin; } const Vec3f& getdirection() const { return direction; } Vec3f pointatprmeter(flot t) const { return origin+direction*t; } privte: }; // EPESENTATION Vec3f direction; Vec3f origin; MIT EECS 6.837, Cutler nd Durnd 12

13 Hit Store intersection point & vrious informtion clss Hit { public: // CONSTUCTO & DESTUCTO Hit(flot _t, Vec3f c) { t = _t; color = c; } ~Hit() {} // ACCESSOS flot gett() const { return t; } Vec3f getcolor() const { return color; } // MODIFIE void set(flot _t, Vec3f c) { t = _t; color = c; } privte: }; // EPESENTATION flot t; Vec3f color; //Mteril *mteril; //Vec3f norml; MIT EECS 6.837, Cutler nd Durnd 13

14 Tsks Abstrct Object3D Sphere nd intersection Group clss Abstrct cmer nd derive Orthogrphic Min function MIT EECS 6.837, Cutler nd Durnd 14

15 Questions? Imge removed due to copyright considertions. MIT EECS 6.837, Cutler nd Durnd 15

16 Overview of tody y-box intersection y-polygon intersection y-tringle intersection MIT EECS 6.837, Cutler nd Durnd 16

17 y-prllelepiped Intersection Axis-ligned From (X1, Y1, Z1) to (X2, Y2, Z2) y P(t)=+Dt y=y2 y=y1 D x=x1 x=x2 MIT EECS 6.837, Cutler nd Durnd 17

18 Nïve ry-box Intersection Use 6 plne equtions Compute ll 6 intersection Check tht points re inside box Ax+by+Cz+D<0 with proper norml orienttion y=y2 y=y1 D x=x1 x=x2 MIT EECS 6.837, Cutler nd Durnd 18

19 Fctoring out computtion Pirs of plnes hve the sme norml Normls hve only one non-0 component Do computtions one dimension t time Mintin tner nd tfr (closest nd frthest so fr) y=y2 y=y1 D x=x1 x=x2 MIT EECS 6.837, Cutler nd Durnd 19

20 Test if prllel If Dx=0, then ry is prllel If x<x1 or x>x2 return flse y=y2 y=y1 D x=x1 x=x2 MIT EECS 6.837, Cutler nd Durnd 20

21 If not prllel Clculte intersection distnce t1 nd t2 t1=(x1-x)/dx t2=(x2-x)/dx y=y2 y=y1 t1 t2 D x=x1 x=x2 MIT EECS 6.837, Cutler nd Durnd 21

22 Test 1 Mintin tner nd tfr If t1>t2, swp if t1>tner, tner =t1 if t2<tfr, tfr=t2 If tner>tfr, box is missed t1y D t1x tner t2y tfr x=x1 t2x tfr x=x2 y=y2 y=y1 MIT EECS 6.837, Cutler nd Durnd 22

23 Test 2 If tfr<0, box is behind t1x tfr t2x y=y2 y=y1 D x=x1 x=x2 MIT EECS 6.837, Cutler nd Durnd 23

24 Algorithm recp Do for ll 3 xis Clculte intersection distnce t1 nd t2 Mintin tner nd tfr If tner>tfr, box is missed If tfr<0, box is behind If box survived tests, report intersection t tner tner tfr y=y2 t1y D y=y1 x=x1 x=x2 MIT EECS 6.837, Cutler nd Durnd 24

25 Efficiency issues Do for ll 3 xis Clculte intersection distnce t1 nd t2 Mintin tner nd tfr If tner>tfr, box is missed If tfr<0, box is behind If box survived tests, report intersection t tner 1/Dx, 1/Dy nd 1/Dz cn be precomputed nd shred for mny boxes Unroll the loop Loops re costly (becuse of termintion if) Avoids the tner tfr for X dimension MIT EECS 6.837, Cutler nd Durnd 25

26 Questions? Imge removed due to copyright considertions. MIT EECS 6.837, Cutler nd Durnd 26

27 Overview of tody y-box intersection y-polygon intersection y-tringle intersection MIT EECS 6.837, Cutler nd Durnd 27

28 y-polygon intersection y-plne intersection Test if intersection is in the polygon Solve in the 2D plne D MIT EECS 6.837, Cutler nd Durnd 28

29 Point inside/outside polygon y intersection definition: Cst ry in ny direction (xis-ligned is smrter) Count intersection If odd number, point is inside Works for concve nd str-shped MIT EECS 6.837, Cutler nd Durnd 29

30 Precision issue Wht if we intersect vertex? We might wrongly count n intersection for ech djcent edge Decide tht the vertex is lwys bove the ry MIT EECS 6.837, Cutler nd Durnd 30

31 Winding number To solve problem with str pentgon Oriented edges Signed number of intersection Outside if 0 intersection MIT EECS 6.837, Cutler nd Durnd 31

32 Alterntive definitions Sum of the signed ngles from point to vertices 360 if inside, 0 if outside Sum of the signed res of point-edge tringles Are of polygon if inside, 0 if outside MIT EECS 6.837, Cutler nd Durnd 32

33 How do we project into 2D? Along norml Costly Along xis Smrter (just drop 1 coordinte) Bewre of prllel plne D D MIT EECS 6.837, Cutler nd Durnd 33

34 Questions? Imge removed due to copyright considertions. MIT EECS 6.837, Cutler nd Durnd 34

35 Overview of tody y-box intersection y-polygon intersection y-tringle intersection MIT EECS 6.837, Cutler nd Durnd 35

36 y tringle intersection Use ry-polygon Or try to be smrter Use brycentric coordintes c P b D MIT EECS 6.837, Cutler nd Durnd 36

37 Brycentric definition of plne P( α, β, γ)=α+βb+γc with α + β + γ =1 Is it explicit or implicit? [Möbius, 1827] c P b MIT EECS 6.837, Cutler nd Durnd 37

38 Brycentric definition of tringle P( α, β, γ)=α+βb+γc with α + β + γ =1 0< α <1 0< β <1 0< γ <1 c P b MIT EECS 6.837, Cutler nd Durnd 38

39 Given P, how cn we compute α, β, γ? Compute the res of the opposite subtringle tio with complete re α=a /A, β=a b /A γ=a c /A Use signed res for points outside the tringle c P T T b MIT EECS 6.837, Cutler nd Durnd 39

40 Intuition behind re formul P is brycenter of nd Q A is the interpoltion coefficient on Q All points on line prllel to bc hve the sme α All such T tringles hve sme height/re c P Q b MIT EECS 6.837, Cutler nd Durnd 40

41 Simplify Since α + β + γ =1 we cn write α =1 β γ P(β, γ)=(1βγ) + βb +γc c P b MIT EECS 6.837, Cutler nd Durnd 41

42 Simplify P(β, γ)=(1βγ) + βb +γc P(β, γ)= + β(b-) +γ(c-) Non-orthogonl coordinte system of the plne c P b MIT EECS 6.837, Cutler nd Durnd 42

43 How do we use it for intersection? Insert ry eqution into brycentric expression of tringle P(t)= +β (b-)+γ (c-) Intersection if β+γ<1; 0<β nd 0<γ c P b D MIT EECS 6.837, Cutler nd Durnd 43

44 Intersection x +td x = x +β (b x - x )+γ (c x - x ) y +td y = y +β (b y - y )+γ (c y - y ) z +td z = z +β (b z - z )+γ (c z - z ) c P b D MIT EECS 6.837, Cutler nd Durnd 44

45 MIT EECS 6.837, Cutler nd Durnd 45 Mtrix form x +td x = x +β (b x - x )+γ (c x - x ) y +td y = y +β (b y - y )+γ (c y - y ) z +td z = z +β (b z - z )+γ (c z - z ) b P D = z z y y x x z z z z z y y y y y x x x x x t D c b D c b D c b γ β c

46 MIT EECS 6.837, Cutler nd Durnd 46 Crmer s rule denotes the determinnt Cn be copied mechniclly in the code b P D A D c D c D c z z z z z y y y y y x x x x x = β A D b D b D b z z z z z y y y y y x x x x x = γ A c b c b c b t z z z z z z y y y y y y x x x x x x = c

47 Advntge Efficient Store no plne eqution Get the brycentric coordintes for free Useful for interpoltion, texture mpping c P b D MIT EECS 6.837, Cutler nd Durnd 47

48 Plucker computtion Plucker spce: 6 or 5 dimensionl spce describing 3D lines A line is point in Plucker spce MIT EECS 6.837, Cutler nd Durnd 48

49 Plucker computtion The rys intersecting line re hyperplne A tringle defines 3 hyperplnes The polytope defined by the hyperplnes is the set of rys tht intersect the tringle MIT EECS 6.837, Cutler nd Durnd 49

50 Plucker computtion The rys intersecting line re hyperplne A tringle defines 3 hyperplnes The polytope defined by the hyperplnes is the set of rys tht intersect the tringle y-tringle intersection becomes polytope inclusion Couple of dditionl issues MIT EECS 6.837, Cutler nd Durnd 50

51 Next week: Trnsformtions Permits 3D IFS ;-) Imge removed due to copyright considertions. MIT EECS 6.837, Cutler nd Durnd 51

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