Ray Tracing. Yogitha Golla Indiana State University Terre Haute IN, USA. December 13, Abstract This project is about Ray Tracing Mechanism.

Transcription

1 Ray Tracing Yogitha Golla Indiana State University Terre Haute IN, USA December 13, 2011 Abstract This project is about Ray Tracing Mechanism. The purpose of this project is to show how to generate images by tracing the paths of light rays from an object to an observer s eye. This procedure is used to generate each pixel in an image. An actual ray tracing algorithm and the history of the development of such algorithms are discussed in detail. Ray tracing is method for calculating the path of waves or particles through a system with regions of varying propagation velocity, absorption characteristics, and reflecting surfaces.ray tracing has become a very popular method for image synthesis due to its unparalleled flexibility and its ability to generate images of high quality and realism.simulation of effects such as reflection and refraction have been the hallmarks of ray tracing.with the advent of stochastic ray tracing [cook84], the range of effects expanded to include motion blur, soft shadows dept of field, and both blurry reflections and translucency.many phenomena that are difficult or impossible with other techniques are simple with ray tracing. This project is to perform a Ray-Sphere intersection test using Algebric and Geometric solutions. A ray is tested against the sphere( object of bounded volume) and return a simple hit of no hit. The idea behind Ray-Sphere intersections is to put the mathematical equation of the ray into the mathematical equation of the sphere and determining if there is a real solution. If there is a real solution then there is a intersection, that is the ray passes through the sphere. For the obtained real solution the closest point of intersection and the Normal at the intersection point must be returned. 1 Introduction Ray Tracing was first introduced by Appel(1968) and by Goldstein and Nagel(1971), and the innovative paper by Whitted(1980) made this method popular for generating high ly realistic images. 1

2 We mainly use this concept in Entertainment, Games Pre -Production, Simulation. As we know, it is very hard to stimulate effects with Rasterizatio techniques, it also requires many passes.but Ray Tracing is easier to implement. 2 Statement of Problem For a ray spawned from the eye,an object intersector must return (at least) the closest intersection point and the surface s normal at this point.for a ray sent towards a light, all that is needed is whether the intersection point is closer than the light,if so,it blocks or filters the light.when the particle A is projected towards the target T, there might be an obstracle O in the path of A preventing it to reach T. So a test can be performed to find whether there is a obstracle O in the path of A. Move A in a line towards T and check for collision of A with O before reaching T. The occurance of collision confirms the presence of obstracle in the path from A to T. But this process requires lots of calculation. Inorder to avoid the complex process and calculation Ray-Sphere Intersection is used to find the presence of an obstracle. A ray is passed from the particle A orgin towards the target T. Sphere is considered as an obstracle. As sphere is ease of testing for intersection with a ray and makes it useful as a bounding volume. If the ray intersects the sphere before reaching Target T, then it is concluded that obstracle is present inbetween the particle A and the target T. 3 Basic Concepts in Ray Tracing 3.1 Reflection Reflection is the direction of a ray at an interface between two different media so that the ray returns into the medium from which it orginated. a - incident ray Figure 1: Reflection 2

3 r - reflected ray n - normal to the surface(medium) θ i - angle of incidence θ r - angle of reflection angle of incidence = angle of reflection θ i = θ r 3.2 Refraction Refraction is the bending of light rays as it crosses interface between media having different refractive indices. m 1 - medium 1 m 2 - medium 2 c 1 - refractive index of medium 1 c 2 - refractive index of medium 2 a - incident ray r - refracted ray n - normal θ 1 - angle of incidence θ 2 - angle of refraction c 1 sinθ 1 = c 2 sinθ Forward Ray Tracing Figure 2: Refraction Forward Ray Tracing follows the light particles from the light source to the objects. Forward Ray Tracing is also known as light ray tracing and photon tracing. Forward Ray Tracing is inefficient because we need to trace all the light rays from the source. While we track every light ray from the light source many rays will go waste because they never contribute to the final image. Many rays from the light source never come through the viewplane and into the eye. 3

4 Figure 3: Forward Ray Tracing 3.4 Backward Ray Tracing Backward Ray Tracing trace rays from the eye point through the viewplane and on to the objects. Backward ray tracing is also known as eye ray tracing. Figure 4: Backward Ray Tracing 4 Algorithm Intersection of the Ray-Sphere : Algebraic Solution. Define a ray as R origin = R 0 = [X 0 Y 0 Z 0 ] R direction = R d = [X d Y d Z d ] where Xd 2 + Y d 2 + Z2 d = 1(Normalized) which defines a ray as set of points on line R(t) = R 0 + R d t, wheret > o (A1) equation A1 is parametric form of the ray equation.so all the points on the ray can be generated directly by varying it.points on the line where t 0 are behind the ray s origin. Sphere is defined by Spheres Center = S c = [X c Y c Z c ] 4

5 Figure 5: Intersection of the Ray-Sphere : Algebraic Solution Spheres radius = S r Spheres Surface is the set of points [X s Y s Z s ] where (X s X c ) 2 +(Y s Y c ) 2 +(Z s Z c ) 2 = Sr 2 (A2) Spheres Surface is expressed as an implicit equation.each point [X s Y s Z s ] can be tested by the implicit equation, if it fulfills the equations conditions,point is on the surface. To solve intersection problem eq (A1) is substituted into the sphere eq. and the result is solved for t.express eq (A1) as a set of equations for the set of equations for the set of points [x y z] in terms of it: X = X 0 + X d t Y = Y 0 + Y d t Z = Z 0 +Z d t (A3) Substitute eq. (A3) into the spheres eq.[x s Y s Z s ] (X 0 +X d t X c ) 2 +(Y 0 +Y d t Y c ) 2 +(Z 0 +Z d t Z c ) 2 = Sr 2 (A4) In terms of t this simplifies to A t 2 + B t + c = 0 (A5) STEP I : Calculation of A, B, C of the quadratic. where A = X 2 d + Y 2 d + Z2 d = 1 B = 2 (X d (X 0 X c ) + Y d (Y 0 Y c ) + Z d (Z 0 Z c ) C = (X 0 X c ) 2 + (Y 0 Y c ) 2 + (Z 0 Z c ) 2 S 2 r Coefficient A is always equal to 1, as ray direction is normalized. Step II Calculation of discriminant, t 0, t 1 : This equation is quadratic, and the solution for t is (with A=1) t 0 = B (B 2 4 c)/2 t 1 = B+ (B 2 4 c)/2 (A6) if the part in Sqrt() function is negative, the line misses the sphere.since t > 0, t 0 and t 1 are examined. The smaller,positive real root is the closest intersection point on the ray. 5

6 Step III Calculation of intersection point: (A7) r intersect = r i = [x i y i z i ] = [X 0 + X d t Y 0 + Y d t Z 0 + Z d t] Step IV: Calculation of Normal at point: The unit vector normal at the surface is then simply: r normal = r n = [(x i X c )/S r (y i Y c )/S r (z i Z c )/S r ] (A8) If the ray originates inside the sphere,r n should be negated so that it points back towards the ray. 5 Example Given a ray with an origin at [1 2 1] and a direction vector of [1 2 4], find the nearest intersection point with a sphere of radiuss r = 3 centered at [3 0 5]. First normalize the direction vector,which yields: direction vector magnitude = ( ) = 21 R d = [1/ 21 2/ 21 3/ 21] = [ ]. Now find A,B, and C using equation (A5): A = 1 (because the ray direction is normalized) B = 2 (0.218 (1 3) ( 2 0) ( 1 5)) = C = (1 3) 2 + ( 2 0) 2 + ( 1 5) = 35. We now check if the discriminant is positive(a6): IsB 2 4 c > 0? Substituting : is > 0? Y es, > 0. This means the ray intersects the sphere. From this we can calculate t 0 from (A6): 6

7 t o = B (B 2 4 C)/2 = (31.400) = Since t 0 is positive,we dont have to calculate t 1. The actual intersection point is, by (A7): r i = [X 0 + X d t Y 0 + Y d t Z 0 + Z d t] = [ ] = [ ]. The unit vector normal is, by (A8); r n = [(x i X c )/S r (y i y c )/S r (z i Z c )/S r ] = [( )/3 ( )/3 ( )/3] = [ ] 6 Time Complexity Ray sphere intersection by Algebraic solution takes O(n) time to find whether the ray intersects the sphere before reaching the target point. References [1] Andrew S. Glassner An Introduction to ray tracing 1989 [2] Timothy J. Purcell, Ian Buck, William R. Mark, Pat Hanraha Ray Tracing on Programmable Graphics Hardware 7