Rendering Algebraic Surfaces CS348B Final Project

Transcription

1 Rendering Algebraic Surfaces CS348B Final Project Gennadiy Chuyeshov Overview This project is devoted to the visualization of a class consisting of implicit algebraic surfaces in three-dimensional space, i.e. the surfaces given by an equation of the form,, 0 where is polynomial with real coefficients. Due to the idea of ray tracing algorithm instead of this equation we can consider the equation of intersection between the ray and the surface which can be transformed to the algebraic equation 0, where is the polynomial of a single variable constructed from. Implicit Surfaces Let f be a continuous scalar function defined on a domain. The implicit surface generated by the function is the locus of points at which the function has zero value, i.e.,, :,, 0 As mentioned in the previous subsection to nd a ray-surface intersection we must put the parametric representation for the ray into the equation of the surface, namely,, 0. This yields the following equation,, 0 with respect to a single variable. Thus on the semi-axis we have the equation 0; where,,. The next step is to determine whether there exists a root of this equation, because if this equation has no real roots then the ray does not intersect the surface. It is natural to restrict the visualization of a surface to some volume which represents a part of the real space. Since we use convex extents, a ray may intersect it through an interval only and it is quite easy to calculate the end points of this interval, and where. Surely, the interval, depends on the parameters of the ray,,,,,. The task is now to check whether this ray intersects the surface. If the ray intersects the surface, then the parameter of the intersection point lies between and. Assume that,,, are roots of equation. It is clear that to visualize the surface we should find the smallest root from the interval,. Having found this root we can calculate the coordinates,, of the point on the surface and a normal vector at this point by the following formula,,,,,,,,,, Algebraic Surfaces As it was mentioned in the overview an algebraic surface is the surface given by an equation of the form,, 0 1

2 where is a polynomial with real coefficients. It can be written as,,, where and deg is the degree of the polynomial. Plugging in the parametric representation of the ray into this equation we obtain the polynomial,, of a single variable t with real coefficients. Thus we get the equation 0,, where, is the ray-extent intersection interval. According to previous section the problem is now to find, i.e. to find the smallest root of the equation on the specified interval. We solve this problem in two steps: 1. Root Isolation. This step allows us to find an interval (if it exists), where is isolated, i.e. an interval, where there is the only root. 2. Root-Finding calculates a prescribed approximation of in an isolating interval (provided this interval exists and we have found it). Root Isolation To isolate the root we involve Descartes' Sign Rule [1] a method of determining the maximum number of positive real roots of a polynomial. Let,, be a sequence of real numbers and let,, be the subsequence of non-zero elements of. Then the number var of variations in is the number of integers such that 0 and 0. Let be a real polynomial. It is uniquely determined by a string of its coefficients,,. We define var as var. Descartes' Sign Rule asserts that the number of positive real roots (taking into account their multiplicity) of a real polynomial is equal to var 2 where is a non-negative integer. In other words, a number of allowable roots can be var, var 2, var 4 and so on. We emphasize that this rule is applicable on the positive semi-axis. In the implementation of Descartes' Sign Rule in our problem we should first transform the polynomial to another one, which is defined on but not on,. Below for the sake of simplicity we denote,. We first consider the polynomial If α,, α are the roots of such that α a, b for 1 i k, then,, are the roots of from the interval 0,1. In order to avoid the consideration of the roots which are greater than one we further transform into the polynomial

3 It is easy to see that,, are the roots of from the semi-interval, if and only if 1,, 1 are the positive real roots of. We consider the following cases. 1. The case when var 0. In this case according Descartes' Sign Rule there is no positive roots of. Hence there are no roots of on,. It means that the ray does not intersect the surface. 2. The case when var 1. We obviously have a single positive root of, which implies that has only one root on,. 3. The case when 2. We cannot determine the exact number of roots, but we can bisect, and consider two subintervals separately. Thus after a number of steps we can determine whether there exist any roots of in, and if they exist, find an interval containing the smallest one. Root Finding To find a prescribed approximation of we use the Dekker-Brent method as it was described in [2]. The Dekker-Brent method combines the bisection and some more advanced root-finding algorithm (either quadratic interpolation or secant). On each step this method operates with three abscissas, and, where is the latest and the closest approximation of the root, is the previous approximation, and is either previous or even an older approximation (it is possible that ). Using the values of the polynomial in,, and and involving either the inverse quadratic interpolation method (if ) or the secant method (when ), we obtain an approximation of the root. The key of the Dekker-Brent method is that we take as the next approximation of the root only if the following criteria hold 3 4 and 1 2 Otherwise instead of we take following the bisection method. Then we take new, and we keep the same if 0 otherwise we take. We stop our process when or 4 max, 1 where is the machine precision and is the prescribed tolerance. Ideally, to find the best approximation for the root we should take, but in our case we empirically obtain the tolerance basing on the quality of image. As it is mentioned in [2] numerous computer experiments have shown that this method has a faster convergence in comparison with the customary methods. The point is that Dekker-Brent method combines the sureness and the universality of the standard bisection method with relatively fast convergence of the secant or the quadratic interpolation method. Implementation The algorithm was implemented within pbrt physically based ray tracer [4]. I created the new type of object algebraicsurface which makes it possible to specify the equation of an algebraic surface directly in the pbrt-script file as follows: Shape "algebraicsurface" "integer elements" [ ] "float coefs" [ ] "float xmin" [-1.5] "float xmax" [1.5] 3

4 "float ymin" [-1.5] "float ymax" [1.5] "float zmin" [-1.5] "float zmax" [1.5] In this example I specified a unit sphere centered at the origin and placed into a bounding cube with the side of length 3. The elements array of triples of integers describes which particular components are present in the equation (in this example we have,,, and ); coefs array of floats contains the coefficients for every component we specified in the elements array. Thus, the equation we specified here is or 1. 4

5 Images Kiss Surface [4] 1 Tooth Surface [5] 5

6 Tanglecube [6] Chmutov Surface of 6 th order [7]

7 Chmutov Surface of 6 th order, close-up render References [1] Collins G. E., Akritas A. G., Polynomial Real Root Isolation Using Descarte's Rule of Signs. Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computations, pp , [2] Forsythe G. E., Malcolm M. A., Moler C. B., Computer Methods for Mathematical Computations. Prentice-Hall, Englewood Cliffs, NJ, [3] Hanrahan P., Ray Tracing Algebraic Surfaces. Computer Graphics, 17(3), pp 83-90, 1983 (Proceedings of the SIGGRAPH 83). [4] Pharr M., Humphreys G., Physically Based Rendering, Morgan Kaufmann, San Francisco, CA, [5] Kiss Surface from Wolfram MathWorld. [6] Tooth Surface from Wolfram MathWorld. [7] Tanglecube from Wolfram MathWorld. [8] Chmutov Surface from Wolfram MathWorld. 7