Implicit and Parametric Surfaces

Transcription

1 Chater 12 Imlicit and Parametric Srfaces In this chater we will look at the two ways of mathematically secifying a geometric object. Yo are robably already familiar with two ways of reresenting a shere of radis r and center at the origin. The first is the imlicit form x 2 + y 2 + z 2 = r 2 or x 2 + y 2 + z 2 r 2 = 0. (12.1) The second is the arametric form exressed in sherical coordinates x = r sin φ cos θ, y = r cos φ, z = r sin φ sin θ, y N x where 0 θ 2π is an angle arameter for longitde measred from the ositive x axis as a ositive rotation abot the y axis, and 0 φ π is an angle arameter for latitde measred from the negative y axis as a ositive rotation abot the z axis. z S x 75

2 76 CHAPTER 12. IMPLICIT AND PARAMETRIC SURFACES 12.1 Imlicit reresentations of srfaces An imlicit reresentation takes the form F (x) = 0 (for examle x 2 + y 2 + z 2 r 2 = 0), where x is a oint on the srface imlicitly described by the fnction F. In other words, oint x is on the srface if and only if the relationshi F (x) = 0 holds for x. With this reresentation, we can easily test a given oint x to see if it is on the srface, simly by checking the vale of F (x). However, if yo examine this reresentation, yo will see that there is no direct way that we can systematically generate consective oints on the srface. This is why the reresentation is called imlicit; it rovides a test for determining whether or not a oint is on the srface, bt does not give any exlicit rles for generating sch oints. Usally, an imlicit reresentation is constrcted so that it has the roerty that F (x) > 0, if x is above the srface, F (x) = 0, if x is exactly on the srface, F (x) < 0, if x is below the srface. For examle, this roerty holds for the shere given by Eqation Rewriting this eqation in vector form we have x 2 r 2 = 0 (where x 2 = x x = x 2 + y 2 + z 2 ). Yo can see that when x > r, then x 2 r 2 > 0, indicating that the oint is otside the shere. Likewise, when x < r, then x 2 r 2 < 0, indicating that the oint is inside the shere. Ths, an imlicit reresentation allows for a qick and easy inside-otside (or above-below) test. Imlicit reresentations are also ideal for raytracing. Given a ray exressed as a starting oint and a direction vector, all oints on the ray can be exressed in arametric form as x(t) = + t, where t is the ray arameter, measring distance along the ray. Inserting this arametric ray eqation into the imlicit reresentation gives F [x(t)] = 0, which can often be conveniently solved for t. We have seen this for ray-lane intersection. The imlicit eqation for a lane is ax + by + cz + d = 0,

3 12.2. PARAMETRIC REPRESENTATIONS OF SURFACES 77 or letting n = a b / a 2 + b 2 + c 2, and e = d/ a 2 + b 2 + c 2, in vector form c we have n x+e = 0. Inserting the ray eqation for x, we have n (+t)+e = 0 and solving for t, we have t = e + n n, which is the distance, along the ray, of the ray-lane intersection. Similarly for the shere, we have the imlicit form Inserting the ray eqation for x gives or x 2 r 2 = 0. ( + t) 2 r 2 = 0, t t + 2 r 2 = 0. Since is a nit vector, 2 = 1, and this redces to t 2 + 2t + 2 r 2 = 0, which by the qadratic formla has soltions t = ± ( ) r 2. d<0 r If the discriminant d = ( ) r 2 < 0 we have no real soltions and the ray misses the shere. If d = 0 we have one soltion t =, so the ray mst be exactly tangent to the shere, and if d > 0 we have two soltions, one where the ray enters the shere and the other where it leaves the shere. The figre to the right shows the cases. d=0 d>0 r r So, an imlicit reresentation can work very well in a raytracer Parametric reresentations of srfaces A arametric srface reresentation has other attribtes that make it sefl in grahics. This reresentation, seaking generally, can be written x = F (, v), where and v are srface arameters, and x is a oint on the srface. The arameters are often chosen so that on the srface being described, 0, v 1. For examle, in the arametric reresentation of the shere introdced at the beginning of this chater, we can normalize the two angle arameters, letting = θ/2π, and v = φ/π.

4 78 CHAPTER 12. IMPLICIT AND PARAMETRIC SURFACES Unlike an imlicit reresentation of a srface, a arametric reresentation allows s to directly generate oints on the srface. All that is reqired is to choose vales of the arameters and v and then x(, v) = F (, v). If this is done in a systematic way over the range of ossible and v vales, it is ossible to generate a set of oints samling the entire srface. However, a arametric srface is difficlt to raytrace, since there is no direct way to take an arbitrary oint in sace and test to see if it is on the srface. Becase a arametric reresentation allows s to systematically generate oints on a srface, it is the ideal form if we want to be able to generate a olygonal srface aroximating the mathematical srface. This is exactly what we want in alications tilizing grahics hardware and grahics APIs like OenGL and DirectX to render a scene. These reqire a olygonal reresentation of the geometry, since their architectre is designed to efficiently rocess triangles. Prodcing a olygonal srface simly reqires iterating over the range of the two arameters and v, as in the sdocode below, which shows how to generate all of a model s vertices for a chosen resoltion: m = nmber of stes in v; n = nmber of stes in ; v = 1.0 / m; = 1.0 / n; for(i = 0; i < m; i++){ v = i * v; for(j = 0; j < n; j++){ = j * ; x = F(, v); insertvertex(x); In the sedocode, the call to insertvertex(x) adds vertex x to a table of vertices for the model. A similar loo cold then be sed to generate all of the olygonal faces from the vertices. For examle, let s consider the arametric reresentation of the shere. The figre to the right shows how we might generate olygons to tile a shere. The tiling has trianglar faces at the oles, bt qadrilateral faces away from the oles. All of the triangles at one ole share a common vertex. Ths, the code to generate a shere, inclding srface normals and textre coordinates at a vertex might be as follows: shere: m = 4, n = 6, = 1/6, v = 1/4

5 12.2. PARAMETRIC REPRESENTATIONS OF SURFACES 79 v = 1.0 / m; = 1.0 / n; // create all of the vertices insertvertex((0, r, 0)); // soth ole, vertex 0 insertvertex((0, r, 0)); // north ole, vertex 1 for(i = 1; i < m - 1; i++){ // iterate over latitdes v = i * v; φ = πv; for(j = 0; j < n; j++){ // iterate over longitdes = j * ; θ = 2π; r sin φ cos θ x = r cos φ ; r sin φ sin θ insertvertex(x); // vertex i n * j insertnormal(normalize(x)); inserttexcoords(, v); // create trianglar faces at the two oles for(j = 0; j < n - 1; j++){ insertface(3, 0, j+3, j+2); insertface(3, 1, (m-2)*n + j+2, (m-2)*n+j+3); // create qadrilateral faces for(i = 1; i < m - 2; i++) for(j = 0; j < n - 1; j++) insertface(4, (i-1)*n+j+2, (i-1)*n+j+3, i*n+j+3, i*n+j+2); Note, that in the above we are sing the convention that the call insertface( nmber of vertices in face, list of vertex indices) adds a list of vertex indices to a table of faces describing the shere. Ths, we see that a arametric reresentation is ideal if one wants to create an aroximating olygonal model from a srface. Another advantage of a arametric srface, is that since the srface is exlicitly arameterized, for every oint on the srface, we have arametric coordinates (, v) = F 1 (x), so if the inverse of F is easily comtable, we have a air of arameters for each srface oint that can be sed to index a textre ma for coloring the srface. For examle for or arametric shere examle, and so that z x r sin φ sin θ = = tan θ, r sin φ cos θ y = cos φ, r θ = tan 1 z x, φ = cos 1 y r,

6 80 CHAPTER 12. IMPLICIT AND PARAMETRIC SURFACES and = 1 z tan 1 2π x, v = 1 y π cos 1 r. In ractice, we often have and v available at the time when we generate a olygonal srface, so we do not actally need to know the inverse fnction, since we can jst store the (, v) coordinates of a oint in a data strctre along with the osition of the oint. So, a arametric reresentation can work very well when textre maing is being sed Imlicit and arametric lane reresentations Or imlicit definition of a lane, in vector form, is given by n x n 0 = 0, where n is the nit srface normal of the lane and 0 is any oint known to be on the lane. In this case, the ray intersection with the lane is given by n t = ( 0 ) n n for ray x = + t. 0 We can also create a arametric reresentation of the lane and se this to tile the lane with olygons. What we need to do this is to create a coordinate frame on the lane. Ths, we need two non-arallel vectors we will call a and a v and a oint 0, on the lane, that will serve as the origin. If a and a v are orthogonal nit vectors, we have an orthogonal coordinate system on the lane. If we then aly some distance measre, for examle w =width in the a direction and h = height in the a v direction, any oint x on the lane is given by 0 a av x = 0 + wa + hva v. Now, if and v are made to vary from 0 to 1 in a nested loo, we can generate a set of qadrilaterals forming a rectangle of width w and height h on the lane. These qadrilaterals can be easily sbdivided to create triangles if we need them.