Ray-tracing Cont. Ray-tracing
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1 Ray-tracing Cont. Ray-tracing light s-ray e-ray s-ray light Intersecting polygons starts with intersecting a plane 2 1
2 Shadow Rays Once an intersection is computed, we send a shadow ray to each light source to see if it is in shadow or not We can treat shadow rays the same as eye rays except: They begin at the point of intersection Ro = {xi,yi,zi} T They are directed toward the light source (normalize vector to find Rd) If no intersection is found, the point is not in shadow from that light source, compute illumination If it is in shadow, only add the ambient term for the light Note, no need to compute exact intersection value or normal 3 Combining with shading Normal is determined by Ray-sphere Angle of incidence = angle of reflection Rn light ray -(s-ray) reflected ray qr qi Sphere 4 2
3 Combining with shading Angle f is between eye-ray and reflected e-ray Rn light ray -(s-ray) reflected ray f qi Sphere 5 Combining with shading For each light source and each color component, the Phong model adds to the illumination as: I += k d L d cosq + k s L s cos a f + k a L a Again, shadow ray lets us determine if the point on the sphere is in shadowed by the particular light If shadowed, do not compute diffuse and specular (only ambient) 6 3
4 Algorithm framework For each pixel j Determine closest intersection with objects in the scene If no hits, color pixel j background color Else Illumination j for = 0; For each light k Compute shadow ray intersection If hit, illumination j += ambient k Else find illum k and add to illum j end light end hit end pixel 7 Intersecting Other Objects Other objects in the scene follow similar procedures We will look at polygons next time 9 4
5 Ray/Plane Intersection Plane is defined by [A,B,C,D] as Ax + By + Cz + D = 0 A 2 + B 2 + C 2 = 1 Normal vector N = [A,B,C] Intersects ray: R(t) = Ro + Rd * t where t > 0 10 Ray/Plane Intersection What can happen? ray // N N N 11 5
6 Ray/Plane Intersection Substitute the ray equation into the plane: A(xo + xd*t) + B(yo + yd*t) + C(zo + zd*t) + D = 0 Solving for t: t = - (Axo + Byo + Czo + D)/(Axd + Byd + Czd) Or t = -(N. Ro + D)/(N. Rd) Note, the normal, N, should face the ray, if not reverse it. If N. Rd > 0 then N = -[A,B,C] 12 Ray/Plane Intersection What happened? N. Rd = 0 aligned or // N N. Rd > 0 reverse & test again possibly hits misses t < 0 t > 0 hits 13 6
7 Ray/Plane Intersection Using t, find exact point of intersection: (xi,yi,zi) = (xo + xd*t,yo + yd*t, zo + zd*t) Use (xi,yi,zi) and N, compute the illumination Summary: - Find N Rd and compare to zero, reverse if appropriate - Calculate t and compare to zero - Compute intersection (xi,yi,zi) 14 Ray/Polygon Intersection Polygon is defined by set of point vertices Planar polygon lies in a plane and we can compute: Ax + By + Cz + D = 0 for any vertex Do ray/plane intersection If intersection, see if ray passes through polygon 15 7
8 Ray/Polygon Intersection If intersection, must still see if ray passes through polygon z (xi,yi,zi) y x 16 Ray/Polygon Intersection Project 3d to 2d based on largest of A,B,C z (xi,yi,zi) y This example: Z (or C) is principal component of N, normal so project on to xy-plane x (xi,yi ) Use 2D test(s) on x-y plane 17 8
9 Ray/Triangle Intersection In the case when we have a triangle... we can be a bit more specialized in our intersection calculation 18 Ray/Triangle Intersection Consider a triangle made up of pts a,b,c c p b For any point in the middle we can define any point p on the plane of the triangle as a weighted average of the combination of vertices: p = w a a + w b b + w c c a 19 9
10 Ray/Triangle Intersection b=-.5 Barycentric coordinates for triangles b=0 c a b=.5 p (b-a) b=1 b Arbitrarily, take the origin to be point a. Define coordinates based on the edges of vertex a as b and g where b = 1 when we hit b g = 1 when we hit c Then, we can define point p in terms of a, b, g: p = a + b (b-a) + g (c-a) 20 Ray/Triangle Intersection Barycentric coordinates for triangles b=-.5 b=0 c b=.5 p b=1 g =.5 b p = a + b (b-a) + g (c-a) Also, we are in the triangle iff : b > 0 g > 0 and b + g < 1 a 21 10
11 Ray/Triangle Intersection Barycentric coords ray/tri intersection? Solve: c Ro + Rd * t = a + b (b-a) + g (c-a) Ro + Rd * t a p b for t, b, g To do this lets rename (b-a) = Eb and (c-a) = Ec 22 Ray/Triangle Intersection Barycentric coords ray/tri intersection? c Then: Ebx Ecx -Rdx b Rox - ax Eby Ecy -Rdy g = Roy - ay Ebz Ecz -Rdz t Roz - az p b Ro + Rd * t a 23 11
12 Ray/Triangle Intersection Barycentric coords ray/tri intersection? t = det(t) det(a) b = det(b) det(a) g = det(c) det(a) T = -Ebx -Ecx (ax - Rox) -Eby -Ecy (ay - Roy) -Ebz -Ecz (az - Roz) C = -Ebx (ax - Rox) Rdx -Eby (ay - Roy) Rdy -Ebz (az - Roz) Rdz B = (ax - Rox) -Ecx Rdx (ay - Roy) -Ecy Rdy (az - Roz) -Ecz Rdz A= -Ebx -Ecx Rdx -Eby -Ecy Rdy -Ebz -Ecz Rdz 24 Ray/Triangle Intersection Barycentric coords ray/tri intersection? From t, b, g: if ( t > 0 ) //tri is in front of us if (b > 0, g > 0) if (b + g < 1){ we intersect (return t) } } }//else we didn't hit the tri 25 12
13 Ray/Triangle Intersection Barycentric coords ray/tri intersection Can you implement this? 26 Ray-tracing surfaces? Given what we already know... Break the surface into polygons (or triangles!) Raytrace the polygons 27 13
14 Ray-tracing surfaces? N To determine the normal: Find the tangent in u and v N = Tu X Tv 28 Recursive Ray Tracing The reflected intensity (or color) at a surface point is computed by: - Local (reflection) model: no interaction with other objects, ambient, diffuse, and specular. - Global model: perfect reflection and refraction. What if we spawn many reflected rays? 29 14
15 Recursive Ray Tracing 30 Recursive Ray Tracing 31 15
16 Recursive Ray Tracing 32 Recursive Ray Tracing 33 16
17 Recursive Ray Tracing Recursively resend rays until some limit is reached 34 Distributed Ray Tracing Distribute rays - rays add noise or jitter to account for: Soft shadows, motion blur, specular on rough surface
18 36 Phong Smoothing - AKA: normal-vector interpolation shading Can you implement this? v 3 v 2 v
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