Mathematics for Computer Graphics - Ray Tracing III

Transcription

1 Mathematics for Computer Graphics - Ray Tracing III Dr. Philippe B. Laval Kennesaw State University November 12, 2003 Abstract This document is a continuation of the previous documents on ray tracing. It discusses normals to surfaces given by a function, reflection and refraction. The images which appear in this document are taken from [SP1]. 1 Ray Tracing In this document we first discuss another method to find normal vectors. We then discuss how the path of a ray is changed when the ray reflects or refracts on a surface. 1.1 Reflection Vector Calculation Let us assume that an incoming ray with direction vector d (the ray is p (t) = q + td) is hitting a shiny surface with normal n. Let r denote the direction vector of the reflected ray. Given d and n, we want to express r in terms of d and n. We will also assume that n and r has been normalized. To derive the formula, we use the fact that when light is reflected on a shiny surface such as a mirror, the angle of incidence is equal to the angle of reflection. In other words, the angle between d and n will be the same as the angle between n and r. From figure 1, we see that The last line comes from the fact that r = proj n d +(d proj n d) = d 2proj n d r = d 2(d.n) n (1) proj v u = u.v u 2 u 1

2 Figure 1: The reflected ray r and the incoming ray d make the same angle θ with the normal n If the point at which the incoming ray hits the surface is a, then the reflected ray is given by 1.2 Refraction Vector Calculation p (t) = a + tr (2) = a + t (d 2(d.n) n) Certain transparent materials (called dielectric) refract light. As light goes through them, its path is changed slightly. How the path is changed is given by Snell s law. Each transparent material has what is called an index of refraction. If we call n d the index of refraction of the lighter material, and n t the index of refraction of the darker material, then Snell s lawsaysthat n d sin θ = n t sin Φ (3) For more information on index of refraction, visit Thus, if we know the incoming ray and the index of refraction of the two materials the light is going through, we can find how the path of the ray is changed. Figure 2 illustrates how the path of light is changed as it goes through different transparent material. Example 1 Light travels through air (index of refraction equal to 1, and hits water (index of refraction equal to 4 3 )atanangleof60 with the normal to the 2

3 Figure 2: A Ray being refracted as it goes through a different medium surface. What will be the angle of the light with the normal of the surface when it is in the water? Let θ denote the angle between the normal and light in the air. Then, θ =30. Let n a and n w denote the index of refraction of the air and water respectively. Then, n a =1, n w = 4 3. If we call Φ the angle between the normal and light, inside the water, Snell s law gives us Thus, n a sin θ = n w sin Φ sin Φ = n a sin θ n w = = 3 8 Φ = sin The computation to find t is a little bit lengthy, but not too difficult. To help us along the way, we use figure 3. Recall that the vectors d, n and b are unit vectors. This will make the final formula a little less complicated. 3

4 Figure 3: The ray d is refracted ito the ray t 1. We express t in terms of n and b. Let t n be the component of t along n and t b be the component of t along b. Then cos Φ = t n t = t n since t isaunitvector Similarly, Now, and Therefore, sin Φ = t b t n = t n n t b = t b b t = cos Φn +sinφb 2. Express b in terms of known quantities using the fact that b is the unit length vector parallel to the projection of d onto the perpendicular to n. Let d n be the component of d along n and d b be the component of d along b, the direction perpendicular to n. Then, d b + d n = d 4

5 Thus, d b = d d n = d (d.n) n Also, sin θ = d b d = d b since d is a unit vector Therefore d b b = d b = d (d.n) n sin θ 3. Using Snell s law, we express everything in terms of n, d, n d and n t. It follows that t = cos Φn +sinφb = cos Φn +sinφ d (d.n) n sin θ = cos Φn + n d (d (d.n) n) n t The only thing left, is to express cos Φ in terms of known quantities. Using the identity and the fact that we obtain Finally, we have t = n cos Φ = 1 sin 2 Φ n d sin θ = n t sin Φ cos Φ = 1 n2 d n 2 sin 2 θ t 1.3 Normal Vector Calculation 1 n2 d n 2 sin 2 θ n d (d (d.n) n) (4) t n t Until now, we have learned how to find the normal to a flat surface by taking two non-parallel vectors on the surface and taking their cross product. If the surface is given by an implicit function, there is another technique to find the 5

6 normal vector. It is a technique you more than likely learned in multivariable calculus. We review it here for convenience. We derive the formula for the 3D case. The 2D case is similar. Suppose that f (x, y, z) represents a surface S. In other words, a point (x, y, z) is on S if and only if f (x, y, z) =0. We say that this is an implicit representation of S. Let C beacurveons defined by the differentiable parametric function (x (t),y(t),z(t)). The tangent vector T to the curve C at the point (x (t),y(t),z(t)) is given by ( d dt x (t), d dt y (t), d dt z (t)). Since the curve lies on S, T is also tangent to S. Furthermore, f (x (t),y(t),z(t)) = 0 for any value of t since C is on S. It follows that df dt =0for every t. Using the chain rule, we get 0 = df dt = f x = dx dt + f dy y dt + f ).T ( f y, f ) ( This means that the vector f y, f is perpendicular to any tangent to S. ( ) Thus, f y, f is normal to S. ( ) The vector f y, f is called the gradient of f and is denoted (del). Thus, ( f f = y, f ) Example 2 Find the normal to the ellipsoid x2 a + y2 2 b + 2 z2 c =1. 2 The implicit function which gives this surface is f (x, y, z) = x2 dz dt a + y2 2 b 2 + z2 c 2 1, and the surface is given by the set of points (x, y, z) which satisfy f (x, y, z) =0. Thus, if we call n the normal, we have 1.4 Examples n = f (x, y, z) ( f = y, f ) ( 2x = a 2, 2y b 2, 2z ) c 2 1. Consider the ray from the origin, with direction vector (1, 1, 1). (a) Find the point at which this ray intersect with the plane x =2. (b) What is the parametric equation of the ray. 6

7 (c) Assuming we have a reflecting surface on the plane x =2, find the equation of the reflected ray. 2. Consider the ellipsoid x2 4 + y2 9 + z2 16 =1. (a) Find the normal to the surface at the point ( ) 1, 1, (b) Find the parametric equation of the ray from (10, 10, 0) to the point above. (c) Assuming the surface of the ellipsoid reflects light, find the equation of the reflected ray. 2 Assignment 1. Consider the parabola given by f (x, y) =0, where f (x, y) =x y 2. Assume the inside of the parabola is made of a reflective surface. Consider a ray originating at the point (c, 0) for some real number c [0, ) and hitting the inside of the parabola. Also assume that the ray is moving in the positive x direction. Answer the following questions: (a) Find the equation of the ray between (c, 0) and a point on the parabola. (b) Find the equation of the reflected ray. (c) Is there a value of c for which all the rays are parallel to the x-axis? What does this suggest? (d) Can you think of applications of the result you found in the previous question? 3 Resources This is a list of books and other resources I used to compile these notes. References [BF1] Burden, Richard L., and Faires, Douglas J., Numerical Analysis,Brooks/Cole, sixth edition, [BG1] Burger, Peter, and Gillies, Duncan, Interactive Computer Graphics, Addison-Wesley, [DD1] Deitel, H.M., and Deitel, P. J., Java, How to Program, Prentice Hall, [DP1] Dunn, Fletcher and Parberry, Ian, 3D Math Primer for Graphics and Game Development, Wordware Publishing, Inc.,

8 [FD1] Foley, J.D., Van Dam, A., Feiner, S.K., and Hughes, J.F., Computer Graphics, Principles and Practices, Addison-Wesley, [FD2] Foley, J.D., Van Dam, A., Feiner, S.K., Hughes, J.F., and Philipps, R.L., Introduction to Computer Graphics, Addison-Wesley, [H1] Hill, F.S. JR., Computer Graphics Using Open GL, Prentice Hall, [LE1] Lengyel, Eric, Mathematics for 3D Game Programming & Computer Graphics, Charles River Media, Inc., [SH1] Schildt, Herbert, Java2, The Complete Reference, McGraw-Hill, 2001 [SE1] Schneider, Philip J., and Eberly, David H., Geometric Tools for Computer Graphics, Morgan Kaufman, [SP1] Shirley, Peter, Fundamentals of Computer Graphics, AKPeters, [SJ1] Stewart, James, Calculus, Concepts and Contexts, second edition, Brooks/Cole, [WG1] Wall, David, and Griffith, Arthur, Graphics Programming with JFC, Wiley, [AW1] Watt, Alan, 3D Computer Graphics, Addison-Wesley, [AW2] Watt, Alan, The Computer Image, Addison-Wesley,