VIEW VOLUME & CLPPING

Transcription

1 VIEW VOLUME & CLPPING The Stage of View Volume Clipping Plane Equations Clipping a 3-Vertices Facet with an Arbitrary Plane Line-plane Intersection & Coding OpenGL calls for Setting Clipping Planes Clipping in VCS - orthographic view volumes Clipping in VCS - perspective view volume Clipping in NDCS (Normalized Device Coordinate System Viewport Transformation KUAS ME, C. F. Chang The Stage of View Volume Clipping KUAS ME, C. F. Chang 2 國立高雄應用科技大學機械系電腦繪圖學 C. F. Chang

2 View Volume Clipping Sutherland-Hodgman polygon-clipping algorithm can be extended to 3D. We could choose to perform clipping in any one of VCS or NDCS. However, there is a shortcoming to clip in NDCS Both the line-clipping and polygon-clipping algorithms made use of in/out tests for half-spaces. NDCS Y Z (,, (-, -,- x KUAS ME, C. F. Chang 3 Plane Equations N = A i + B j + C k P b P 2 Implicit plane equation: P 0 P or A x + B y + C z + D = 0 N.P + D = 0 O P 0 D P P a where P(x,y,z is the position vector of a point on the plane For any vertex P in space, we have F(P = N.P + D = A x + B y + C z + D The F can be used to determine which side of the plane the vertex P lies on F = 0 point P lies on the plane F > 0 point P lies on the same side as the normal F < 0 point P lies on the opposite side of the normal KUAS ME, C. F. Chang 4 國立高雄應用科技大學機械系電腦繪圖學 C. F. Chang 2

3 Plane Equations (cont Parametric plane equation: N = A i + B j + C k Pb P2 Plane(s, t = P 0 + s(p -P 0 + t(p 2 -P 0 P0 P0 D P P O Pa ( provided P 0, P, and P 2 are non-collinear Explicit plane equation: A x + B y + C z + D = 0 z = - (A/C x - (B/C y - D/C (valid if C is non-zero KUAS ME, C. F. Chang 5 Clipping a 3-Vertices Facet with an Arbitrary Plane There are 8 different cases to consider: Cases -2: all the points are on same side of the clipping plane KUAS ME, C. F. Chang 6 國立高雄應用科技大學機械系電腦繪圖學 C. F. Chang 3

4 Clipping a 3-Vertices Facet with a Plane (cont Cases 3-5: there are only one vertex on the clipping side of the plane Cases 6-8: there are two vertices on the clipping side of the plane KUAS ME, C. F. Chang 7 Line-plane Intersection F(P = N.P + D ( For a line passing through P a and P b, the intersection point of the line and a plane can be written as: N = A i + B j + C k P b P 2 P (t = P a + t ( P b - P a P 0 P P Substituting P into the plane equation (, and solving for t yields O P 0 D P a F(P = N.[P a + t ( P b - P a ] + D = 0 D N P a t N P N P b a F( Pa F( P F ( P b a KUAS ME, C. F. Chang 8 國立高雄應用科技大學機械系電腦繪圖學 C. F. Chang 4

5 Coding- F(P = A x + B y + C z + D int ClipFacet(wcPt3 *p, wcpt3 n, wcpt3 pr { float A,B,C,D; float m; float F[[3]; wcpt3 q; /* Determine the equation of the plane as Ax + By + Cz + D = 0 */ m = (float sqrt(n.x*n.x + n.y*n.y + n.z*n.z; A = n.x / m; B = n.y / m; C = n.z / m; //normalize the normal vector n D = -(n.x*pr.x + n.y*pr.y + n.z*pr.z; /* Evaluate the equation of the plane for each vertex */ /* If F<0 then it is on the side to be retained else it is to be clipped */ F[0] = A*p[0].x + B*p[0].y + C*p[0].z + D; F[] = A*p[].x + B*p[].y + C*p[].z + D; F[2] = A*p[2].x + B*p[2].y + C*p[2].z + D; /* Are all the vertices are on the clipped side */ if (F[0] >= 0 && F[] >= 0 && F[2] >= 0 return(0; /* Are all the vertices on the not-clipped side */ if (F[0] <= 0 && F[] <= 0 && F[2] <= 0 return(3; KUAS ME, C. F. Chang 9 Coding -2 only one vertex on the clipping side of the plane /* Is p0 the only point on the clipped side */ if (F[0] > 0 && F[] < 0 && F[2] < 0 { q.x = p[0].x - F[0] * (p[2].x - p[0].x / (F[2] - F[0]; q.y = p[0].y - F[0] * (p[2].y - p[0].y / (F[2] - F[0]; q.z = p[0].z - F[0] * (p[2].z - p[0].z / (F[2] - F[0]; p[3] = q; q.x = p[0].x - F[0] * (p[].x - p[0].x / (F[] - F[0]; q.y = p[0].y - F[0] * (p[].y - p[0].y / (F[] - F[0]; q.z = p[0].z - F[0] * (p[].z - p[0].z / (F[] - F[0]; p[0] = q; return(4; Q (t = P a + t ( P b - P a F( Pa t F( P F ( P b a KUAS ME, C. F. Chang 0 國立高雄應用科技大學機械系電腦繪圖學 C. F. Chang 5

6 Coding -3 two vertices on the clipping side of the plane /* Is p0 the only point on the not-clipped side */ if (F[0] < 0 && F[] > 0 && F[2] > 0 { q.x = p[0].x - F[0] * (p[].x - p[0].x / (F[] - F[0]; q.y = p[0].y - F[0] * (p[].y - p[0].y / (F[] - F[0]; q.z = p[0].z - F[0] * (p[].z - p[0].z / (F[] - F[0]; p[] = q; q.x = p[0].x - F[0] * (p[2].x - p[0].x / (F[2] - F[0]; q.y = p[0].y - F[0] * (p[2].y - p[0].y / (F[2] - F[0]; q.z = p[0].z - F[0] * (p[2].z - p[0].z / (F[2] - F[0]; p[2] = q; return(3; Q (t = P a + t ( P b - P a F( Pa t F( P F ( P b a KUAS ME, C. F. Chang OpenGL calls for Setting Clipping Planes..\Redbook Samples\clip.exe glenable( GL_CLIP_PLANEi glclipplane( GL_CLIP_PLANEi, GLdouble* coeff where i = 0,, 2, Note: At least 6 more clipping planes available Good for cross-sections Modelview matrix moves clipping plane Those vertices (x,y,z satisfying F = A x + B y + C z + D < 0 are clipped Example: void display(void{ GLdouble eqn[4] = {0.0,.0, 0.0, 0.0; // (A, B, C,D /* clip lower half y < 0 */ glclipplane (GL_CLIP_PLANE0, eqn; glenable (GL_CLIP_PLANE0; /* set up other clipping planes /* draw something here */ gldisable (GL_CLIP_PLANE0; KUAS ME, C. F. Chang 2 國立高雄應用科技大學機械系電腦繪圖學 C. F. Chang 6

7 Clipping in VCS - orthographic view volumes For the orthographic view volumes presented previously, the view-volume plane equations can be written in a consistent way, such that all the normals are pointing into the view volume; that is, point P is inside the view volume If F(P > 0. The six clipping planes are: left: x - left = 0 right: -x + right = 0 bottom: y - bottom = 0 top: -y + top = 0 front: -z - near = 0 back: z + far = 0 KUAS ME, C. F. Chang 3 Clipping in VCS - perspective view volume The same can also be done for the perspective view volume, since the six clipping planes can be expressed as: top: -y z (top/near = 0 bottom: y + z (bottom/near = 0 left: x + z (left /near = 0 right: -x z (right/near = 0 front: -z - near = 0 back: -z - far = 0 near slope = top/near top KUAS ME, C. F. Chang 4 國立高雄應用科技大學機械系電腦繪圖學 C. F. Chang 7

8 Clipping in NDCS (Normalized Device Coordinate System NDCS provides a potentially nice coordinate system for clipping operations because the plane equations are simply defined and always remain unchanged. (- <= x/w <= Furthermore, lines in VCS are lines in NDCS and therefore it would seem that correct intersections could be calculated, despite the fact that NDCS-space has a strange warp to it because it is post-perspective division. The potential problem of clipping in NDCS is that the sign of the depth information is lost, as shown in the following example. VCS NDCS P (, 0, -2 (/2, 0, /3 P2 (0, 0, 2 (0, 0, 3 Note: NDCS uses left-hand-side coordinate system KUAS ME, C. F. Chang 5 Viewport Transformation NDCS to DCS simple translate and scale, namely, Y (, (w/2, h/2 (0,0 x X y (-,- (-w/2,-h/2 (w, h NDCS DCS P NDCS = S(,- T(w/2,-h/2 S(w/2, h/2 P NDCS KUAS ME, C. F. Chang 6 國立高雄應用科技大學機械系電腦繪圖學 C. F. Chang 8

9 Assignment & Hints Read the data of an object, then clip it with a plane and show the result. KUAS ME, C. F. Chang 7 How To Define A Cube..\Samples\cube.exe Boundary Representations are used in the following Boundary Representations are also known as b-reps. In this kind of representation, an object is described in terms of its surfaces boundaries GLfloat vertices[][3] = { {-.0,-.0,-.0,{.0,-.0,-.0, //vertex 0, {.0,.0,-.0, {-.0,.0,-.0, {-.0,-.0,.0, //vertex 2, 3, 4 {.0,-.0,.0, {.0,.0,.0, {-.0,.0,.0 ; //vertex 5, 6, 7 x z y GLfloat normals[][3] = { {-.0,-.0,-.0,{.0,-.0,-.0, //vertex 0, {.0,.0,-.0, {-.0,.0,-.0, {-.0,-.0,.0, //vertex 2, 3, 4 {.0,-.0,.0, {.0,.0,.0, {-.0,.0,.0 ; //vertex 5, 6, 7 GLfloat colors[][3] = { {0.0,0.0,0.0,{.0,0.0,0.0, //vertex 0, {.0,.0,0.0, {0.0,.0,0.0, {0.0,0.0,.0, //vertex 2, 3, 4 {.0,0.0,.0, {.0,.0,.0, {0.0,.0,.0 ; //vertex 5, 6, 7 KUAS ME, C. F. Chang 8 國立高雄應用科技大學機械系電腦繪圖學 C. F. Chang 9

10 Data Structure of a Cube void colorcube(void { drawtriangle(0, 3, 2; drawtriangle(2,, 0; drawtriangle(3, 7, 6; drawtriangle(6, 2, 3; drawtriangle(0, 4, 7; drawtriangle(7, 3, 0; drawtriangle(, 2, 6; drawtriangle(6, 5, ; drawtriangle(4, 5, 6; drawtriangle(6, 7, 4; drawtriangle(0,, 5; drawtriangle(5, 4, 0; KUAS ME, C. F. Chang 9 Draw 3-Vertex3 Facets void drawtriangle(int a, int b, int c{ glbegin(gl_polygon; glcolor3fv(colors[a]; glnormal3fv(normals[a]; glvertex3fv(vertices[a]; glcolor3fv(colors[b]; glnormal3fv(normals[b]; glvertex3fv(vertices[b]; glcolor3fv(colors[c]; glnormal3fv(normals[c]; glvertex3fv(vertices[c]; glend(; drawtriangle( 0, 3, 2 ; a b c KUAS ME, C. F. Chang 20 國立高雄應用科技大學機械系電腦繪圖學 C. F. Chang 0

11 Construct a data file A A Simple Format 8 //number of vertices //vertex // // // // // // // 7 2 //number of facets //vertex indices for facet x z y KUAS ME, C. F. Chang 2 Structures for Polygonal Objects struct wcpt3 { float x, y, z; ; struct Plane { ; float a, b, c, d; //for a plane ax + by + cz + d = 0 /* structure for a 3-vertex facet */ struct Facet { int vertexindices[3]; // Index of the vertices that make up a 3-vertices facet wcpt3 normals[3]; // Normals to the three vertices Plane coeff; // Coefficients of the plane that contains the facet ; struct PolyObject { int wcpt3 * int Facet* ; nvertices; pvertices; nfaces; pfaces; KUAS ME, C. F. Chang 22 國立高雄應用科技大學機械系電腦繪圖學 C. F. Chang

12 Read Data From a File ReadRenderObj.dsw bool readobject(const char* filename, PolyObject& object { int i; FILE* iobuffer = fopen(filename, "r"; if (iobuffer == NULL { cerr << "Cannot open file: " << filename << endl; return false; /* Read the coordinates of vertices */ fscanf(iobuffer, "%d", &object.nvertices; object.pvertices = new wcpt3[object.nvertices]; for (i = 0; i < object.nvertices; i++ { fscanf(iobuffer, "%f", &object.pvertices[i].x; fscanf(iobuffer, "%f", &object.pvertices[i].y; fscanf(iobuffer, "%f", &object.pvertices[i].z; /* Read facets */ fscanf(iobuffer, "%d", &object.nfaces; object.pfaces = new Facet[object.nFaces]; for (i = 0; i < object.nfaces; i++ { Facet *pface = &object.pfaces[i]; for (int j = 0; j < 3; j++ fscanf(iobuffer, "%d", &pface->vertexindices[j]; return true; KUAS ME, C. F. Chang 23 Intersection of Objects f 0, f 2, 2 以物 2 之每一面來切割 f 0 面 2 因為前面程序所得之結果, 只有原屬於物 之面, 無法得到屬於物 2 之面 From Fast Algorithms for 3D-Graphics KUAS ME, C. F. Chang 24 國立高雄應用科技大學機械系電腦繪圖學 C. F. Chang 2

13 Clipping a Polygon with a Plane (Another Expression n, normal of the face T V 2 切割面 R R V 2 I n (I-V n = (R-V n (V 2 - V n V ( R V n V ( V2 V, where ( V V n I V 2 I V (I V n (R V n V V (V V n (V V n 2 一個頂點 V 若滿足 :V n-r n>0, 則該頂點在切割面之外側 ( 因為法線恒指向外側 若一線段之兩頂點皆在切割面之外側, 則去除該線段若一線段之兩頂點皆在切割面之內側, 則保留該線段若一線段之兩頂點分別在切割面之內 外側, 則以下式求交點 I, 並去除外側之頂點 2 KUAS ME, C. F. Chang 25 2 國立高雄應用科技大學機械系電腦繪圖學 C. F. Chang 3