1 OpenGL OpenGL OpenGL OpenGL
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- ふみな そや
- 5 years ago
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1 2008 OpenGL
2 1 OpenGL OpenGL OpenGL OpenGL set-opengl.c set-opengl.c draw-graph.c draw-graph.c draw-graph-on-disk.c draw-graph-on-disk.c
3 A ccgl 50 B JPEG 51 B B.2 set-opengl-jpeg.c B.3 ijg-saveimage.c
4 OpenGL OpenGL CG OpenGL GLSC 1 CG OpenGL OpenGL CG OpenGL JPEG 1 Graphic Library for Scientific Computing 3
5 1 OpenGL 1.1 OpenGL OpenGL Open Graphics Library Silicon Graphics 3 UNIX Windows MacOS 3 CG C 1.2 OpenGL
6 1.2.2 OpenGL (1) gltranslatef(t x, t y, t z ); t x, t y, t z (2) glrotatef(θ, x, y, z); θ x, y, z (3) glscalef(s x, s y, s z ); s x, s y, s z [1] 5
7 2 OpenGL 2.1 OpenGL C OpenGL set opengl.c draw-graph.c draw-graph-on-disk.c set-opengl.c draw-graph.c drawgraph-on-disk.c ccgl ( ).c set-opengl.c draw-graph.c ccgl OpenGL GLUT ( A )
8 r w ON/OFF Space / Esc 2.3 main() display(), idle() display() idle() idle() glutpostredisplay() f(x, y, t) = sin( m 2 + n 2 πt) sin(mπx) sin(nπy) ((x, y) [ 1, 1] [ 1, 1], t > 0) /* sample-graph1.c --- * * * ccgl sample-graph.c draw-graph.c set-opengl.c 7
9 * * : *./sample-graph * Nx, Ny: * * : * d --- Dirichlet * n --- Neumann * r --- * w --- ON/OFF * Space --- / * Esc --- * --- * --- * --- */ #include <stdio.h> #include <stdlib.h> #include <GL/glut.h> #include <GL/gl.h> #include <GL/glu.h> #include <math.h> #include "draw-graph.h" #include "set-opengl.h" #define PI M_PI #define M 2 #define N 1 int i, j, n, nmax, skip; /* x, y */ double xmin = -1.0, xmax = 1.0, ymin = -1.0, ymax = 1.0; /* */ int Nx, Ny; /* */ double hx, hy; 8
10 /* */ double tau; /* */ double Tmax; /* */ double dt; double **u; /* z */ double zmin = -1.0, zmax = 1.0; /* */ double distance = 5.0, twist = 0.0, elevation = 60.0, azimuth = 120.0; void display(void) /* t */ if(n % skip == 0) beginopengl(); /* */ drawgraph(nx, Ny, u); endopengl(); void next_step(void) double x, y; /* */ if(n <= nmax) for(i=0;i<=nx;i++) x = xmin + i*hx; for(j=0;j<=ny;j++) y = ymin + j*hy; 9
11 u[i][j] = sin(sqrt(m*m*n*n)*pi*n*tau) * sin(m*pi*x) * sin(n*pi*y); void idle(void) n++; next_step(); glutpostredisplay(); int main(int argc, char **argv) printf("nx, Ny: "); scanf("%d %d", &Nx, &Ny); hx = (xmax - xmin) / Nx; hy = (ymax - ymin) / Ny; printf(" : "); scanf("%lf", &tau); printf("tmax: "); scanf("%lf", &Tmax); printf(" t(>=%g): ", tau); scanf("%lf", &dt); if(dt < tau) dt = tau; skip = rint(dt / tau); n = 0; nmax = rint(tmax / tau); /* u */ u = malloc(sizeof(double *) * (Nx+1)); for(i=0;i<=nx;i++) u[i] = malloc(sizeof(double) * (Ny+1)); 10
12 setview(distance, twist, elevation, azimuth); setframe(xmin, xmax, ymin, ymax, zmin, zmax); OpenGL(&argc, argv); return(0); f(x, y) = x 2 y 2 ((x, y) [ 1, 1] [ 1, 1]) /* sample-graph2.c --- * ( ) * * ccgl sample-graph.c draw-graph.c set-opengl.c * * : *./sample-graph * Nx, Ny: * * : * d --- Dirichlet * n --- Neumann * r --- * w --- ON/OFF * Space --- / * Esc --- * --- * --- * --- */ #include <stdio.h> #include <stdlib.h> 11
13 #include <GL/glut.h> #include <GL/gl.h> #include <GL/glu.h> #include <math.h> #include "draw-graph.h" #include "set-opengl.h" int i, j, n, nmax; /* x, y */ double xmin = -1.0, xmax = 1.0, ymin = -1.0, ymax = 1.0; /* */ int Nx, Ny; /* */ double hx, hy; double **u; /* z */ double zmin = -1.0, zmax = 1.0; /* */ double distance = 5.0, twist = 0.0, elevation = 60.0, azimuth = 120.0; void display(void) double x, y; beginopengl(); for(i=0;i<=nx;i++) x = xmin + i*hx; for(j=0;j<=ny;j++) y = ymin + j*hy; u[i][j] = x*x - y*y; /* */ drawgraph(nx, Ny, u); 12
14 endopengl(); void idle(void) int main(int argc, char **argv) printf("nx, Ny: "); scanf("%d %d", &Nx, &Ny); hx = (xmax - xmin) / Nx; hy = (ymax - ymin) / Ny; /* u */ u = malloc(sizeof(double *) * (Nx+1)); for(i=0;i<=nx;i++) u[i] = malloc(sizeof(double) * (Ny+1)); setview(distance, twist, elevation, azimuth); setframe(xmin, xmax, ymin, ymax, zmin, zmax); OpenGL(&argc, argv); return(0); 2.4 set-opengl.c set-opengl.c set-opengl.c OpenGL polarview() 13
15 : polarview() distance twist elevation azimuth x +5 x 5 resetview() polarview() distance,twist,elevation,azimuth mymotion(x, y) 14
16 x, y mymotion() x, y mymotion xbegin, ybegin xdisp, ydisp xdisp, ydisp polarview() distance, twist, elevation, azimuth /* set-opengl.c */ #include <stdio.h> #include <GL/glut.h> #include <GL/gl.h> #include <GL/glu.h> #include <math.h> #define KEY_ESC 27 #define KEY_SPC 32 #define DBC 0 #define NBC 1 /* / */ static unsigned char moveflag = GL_FALSE; /* */ static int bc; static int xbegin, ybegin; static int mbutton; static double distance, twist, elevation, azimuth; static double D = 10.0, T = 0.0, E = 60.0, A = 120.0; void polarview(void); void resetview(void); void idle(void); void display(void); 15
17 void setview(double d0, double t0, double e0, double a0) D = d0; T = t0; E = e0; A = a0; /* display */ void beginopengl(void) glclear(gl_color_buffer_bit GL_DEPTH_BUFFER_BIT); glenable(gl_depth_test); glpushmatrix(); polarview(); /* display */ void endopengl(void) glpopmatrix(); gldisable(gl_depth_test); glutswapbuffers(); #define MAXCOMMANDS (100) typedef void vvfunc(void); static char command_keys[maxcommands]; vvfunc *command_funcs[maxcommands]; static num_of_reg_commands = 0; /* c f ( ) */ void register_command(char c, vvfunc f) if (num_of_reg_commands < MAXCOMMANDS) command_keys[num_of_reg_commands] = c; command_funcs[num_of_reg_commands++] = f; 16
18 else fprintf(stderr, " \n"); void mykbd(unsigned char key, int x, int y) int i; switch(key) case r : resetview(); break; case w : switchwireflag(); break; case KEY_SPC: moveflag =!moveflag; if(moveflag == GL_TRUE) glutidlefunc(idle); else glutidlefunc(null); break; case KEY_ESC: exit(0); default: for(i=0;i<num_of_reg_commands;i++) if(key == command_keys[i]) command_funcs[i](); return; glutpostredisplay(); 17
19 void mymouse(int button, int state, int x, int y) if(state == GLUT_DOWN) xbegin = x; ybegin = y; mbutton = button; void mymotion(int x, int y) int xdisp, ydisp; xdisp = x - xbegin; ydisp = y - ybegin; switch(mbutton) case GLUT_LEFT_BUTTON: azimuth -= (double)xdisp / 2.0; elevation -= (double)ydisp / 2.0; break; case GLUT_MIDDLE_BUTTON: twist = fmod(twist + xdisp/3.0, 360.0); break; case GLUT_RIGHT_BUTTON: distance += (double)ydisp / 60.0; break; xbegin = x; ybegin = y; glutpostredisplay(); void myinit(char *progname) glutinitwindowposition(0, 0); 18
20 glutinitwindowsize(400, 400); glutinitdisplaymode(glut_double GLUT_RGBA GLUT_DEPTH); glutcreatewindow(progname); glclearcolor(1.0, 1.0, 1.0, 1.0); glutkeyboardfunc(mykbd); glutmousefunc(mymouse); glutmotionfunc(mymotion); resetview(); void myreshape(int width, int height) double aspect = width/(double)height; glviewport(0, 0, width, height); glmatrixmode(gl_projection); glloadidentity(); gluperspective(40.0, aspect, 1.0, 80.0); glmatrixmode(gl_modelview); void polarview(void) gltranslatef(0.0, 0.0, -distance); glrotatef(-twist, 0.0, 0.0, 1.0); glrotatef(-elevation, 1.0, 0.0, 0.0); glrotatef(-azimuth, 0.0, 0.0, 1.0); void resetview(void) distance = D; twist = T; elevation = E; azimuth = A; 19
21 void OpenGL(int *argc, char **argv) glutinit(argc, argv); myinit(argv[0]); glutreshapefunc(myreshape); glutidlefunc(null); glutdisplayfunc(display); glutmainloop(); 2.5 draw-graph.c draw-graph.c /* draw-graph.c */ #include <GL/glut.h> #include <GL/gl.h> #include <GL/glu.h> #include <math.h> static double xmin = -2.0, xmax = 2.0, ymin = -2.0, ymax = 2.0, zmin = -1.0, zmax =1.0; static unsigned char wireflag = GL_TRUE; void setframe(double x0, double x1, double y0, double y1, double z0, double z1) xmin = x0; xmax = x1; ymin = y0; ymax = y1; zmin = z0; zmax = z1; 20
22 void setwireflag(unsigned char wireflag0) wireflag = wireflag0; void switchwireflag(void) wireflag =!wireflag; /* */ static void drawframe(void) glcolor3f(0.0, 0.0, 0.0); glbegin(gl_line_loop); glvertex3f(xmin,ymin,zmin); glvertex3f(xmax,ymin,zmin); glvertex3f(xmax,ymax,zmin); glvertex3f(xmin,ymax,zmin); glend(); glbegin(gl_lines); glvertex3f(xmin,ymin,zmin); glvertex3f(xmax,ymin,zmin); glvertex3f(xmax,ymax,zmin); glvertex3f(xmin,ymax,zmin); glend(); glvertex3f(xmin,ymin,zmax); glvertex3f(xmax,ymin,zmax); glvertex3f(xmax,ymax,zmax); glvertex3f(xmin,ymax,zmax); /* */ static void drawwiregraph(int Nx, int Ny, double **u) int i, j; double x0, x1, y0, y1; double hx = (xmax - xmin) / Nx; double hy = (ymax - ymin) / Ny; 21
23 for(i=0;i<nx;i++) x0 = xmin + i * hx; x1 = xmin + (i+1) * hx; for(j=0;j<ny;j++) y0 = ymin + j * hy; y1 = ymin + (j+1) * hy; glcolor3f(0.0, 0.0, 0.0); glbegin(gl_line_loop); glvertex3f(x0, y0, u[i][j]); glvertex3f(x1, y0, u[i+1][j]); glvertex3f(x1, y1, u[i+1][j+1]); glvertex3f(x0, y1, u[i][j+1]); glend(); /* u[i][j] R */ static double red(double v) if(v <= 0.4) return 0.0; else if(v <= 0.5) return (v - 0.4) / 0.1; else return 1.0; /* u[i][j] G */ static double green(double v) if(v <= 0.3) return v / 0.3; else if(v <= 0.6) return 1.0; 22
24 else return -(v - 1) / 0.4; /* u[i][j] B */ static double blue(double v) if(v <= 0.3) return 1.0; else if(v <= 0.4) return -(v - 0.4) / 0.1; else return 0.0; /* u[i][j] */ static void color(double v) double R, G, B; if(v < 0) v = 0; if(v > 1) v = 1; R = red(v); G = green(v); B = blue(v); glcolor3f(r, G, B); /* */ static void drawsolidgraph(int Nx, int Ny, double **u) int i, j; double x0, x1, y0; double hx = (xmax - xmin) / Nx; double hy = (ymax - ymin) / Ny; double v; 23
25 glenable(gl_polygon_offset_fill); /* */ glpolygonoffset(1.0, 1.0); /* */ for(i=0;i<nx;i++) x0 = xmin + i * hx; x1 = xmin + (i+1) * hx; glbegin(gl_quad_strip); for(j=0;j<=ny;j++) y0 = ymin + j * hy; v = (u[i][j] - zmin) / (zmax - zmin); color(v); glvertex3f(x0, y0, u[i][j]); glvertex3f(x1, y0, u[i+1][j]); glend(); gldisable(gl_polygon_offset_fill); /* */ void drawgraph(int Nx, int Ny, double **u) /* */ drawframe(); /* */ drawsolidgraph(nx, Ny, u); if(wireflag == GL_TRUE) /* */ 24
26 drawwiregraph(nx, Ny, u); 2.6 draw-graph-on-disk.c draw-graph-on-disk.c /* draw-graph-on-disk.c */ #include <GL/glut.h> #include <GL/gl.h> #include <GL/glu.h> #include <math.h> #define PI M_PI static double R = 1.0, zmin = -1.0, zmax = 1.0; static unsigned char wireflag = GL_TRUE; void setframe(double r0, double z0, double z1) R = r0; zmin = z0; zmax = z1; void setwireflag(unsigned char wireflag0) wireflag = wireflag0; void switchwireflag(void) wireflag =!wireflag; 25
27 /* */ static void drawframe(void) int i, j; double x, y; glcolor3f(0.0, 0.0, 0.0); glbegin(gl_lines); for(i=0;i<360;i+=90) x = R * cos(pi / * i); y = R * sin(pi / * i); glvertex3f(x, y, zmin); glvertex3f(x, y, zmax); glend(); glbegin(gl_line_loop); for(j=0;j<360;j++) x = R * cos(pi / * j); y = R * sin(pi / * j); glvertex3f(x, y, zmin); glend(); /* */ static void drawwiregraph(int Nr, int Np, double **u) int i, j; double r0, r1, p0, p1; double hr = R / Nr; double hp = 2*PI / Np; for(i=0;i<nr;i++) r0 = i * hr; r1 = (i+1) * hr; 26
28 for(j=0;j<np;j++) p0 = j * hp; p1 = (j+1) * hp; glcolor3f(0.0, 0.0, 0.0); glbegin(gl_line_loop); glvertex3f(r0*cos(p0), r0*sin(p0), u[i][j]); glvertex3f(r1*cos(p0), r1*sin(p0), u[i+1][j]); glvertex3f(r1*cos(p1), r1*sin(p1), u[i+1][j+1]); glvertex3f(r0*cos(p1), r0*sin(p1), u[i][j+1]); glend(); /* u[i][j] R */ static double red(double v) if(v <= 0.4) return 0.0; else if(v <= 0.5) return (v - 0.4) / 0.1; else return 1.0; /* u[i][j] G */ static double green(double v) if(v <= 0.3) return v / 0.3; else if(v <= 0.6) return 1.0; else return -(v - 1) / 0.4; 27
29 /* u[i][j] B */ static double blue(double v) if(v <= 0.3) return 1.0; else if(v <= 0.4) return -(v - 0.4) / 0.1; else return 0.0; /* u[i][j] */ static void color(double v) double R, G, B; if(v < 0) v = 0; if(v > 1) v = 1; R = red(v); G = green(v); B = blue(v); glcolor3f(r, G, B); /* */ static void drawsolidgraph(int Nr, int Np, double **u) int i, j; double r0, r1, p0; double hr = R / Nr; double hp = 2*PI / Np; double v; glenable(gl_polygon_offset_fill); /* */ glpolygonoffset(1.0, 1.0); /* 28
30 */ for(i=0;i<nr;i++) r0 = i * hr; r1 = (i+1) * hr; glbegin(gl_quad_strip); for(j=0;j<=np;j++) p0 = j * hp; v = (u[i][j] - zmin) / (zmax - zmin); color(v); glvertex3f(r0*cos(p0), r0*sin(p0), u[i][j]); glvertex3f(r1*cos(p0), r1*sin(p0), u[i+1][j]); glend(); gldisable(gl_polygon_offset_fill); /* */ void drawgraph(int Nr, int Np, double **u) /* */ drawframe(); /* */ drawsolidgraph(nr, Np, u); if(wireflag == GL_TRUE) /* */ drawwiregraph(nr, Np, u); 29
31 u c 2 t 2 = 2 u x u y 2 ((x, y) Ω, t > 0) (3.1) Ω = (A, B) (C, D) u(x, y, 0) = ϕ(x, y) ((x, y) Ω) (3.2) u (x, y, 0) = ψ(x, y) t ((x, y) Ω) (3.3) (Dirichlet ) u(x, y, t) Ω = 0 (t > 0) (3.4) u (Neumann ) (x, y, t) n = 0 (t > 0) (3.5) Ω 3.2 x [A, B] N x y [C, D] N y x i (i = 0,, N x ), y j (j = 0,, N y ) h x = B A N x, h y = D C N y 30
32 x i = A + ih x (i = 0, 1, 2,, N x ), y j = C + jh y (j = 0, 1, 2,, N y ). t τ(> 0) t n = nτ (n = 0, 1, 2, ) (x i, y j, t n ) u u n i,j u(x i, y j, t n ) (x i, y j, t n ) 2 u t, 2 u 2 x, 2 u 2 y 2 2 u t (x i, y 2 j, t n ) = un+1 i,j 2u n i,j + u n 1 i,j + O(τ 2 ) τ 2 (τ +0), 2 u x (x i, y 2 j, t n ) = un i+1,j 2u n i,j + u n i 1,j h 2 x + O(h 2 x) (h x +0), 2 u y (x i, y 2 j, t n ) = un i,j+1 2u n i,j + u n i,j 1 h 2 y + O(h 2 y) (h y +0) u (3.1) h x, h y, τ 1 u n+1 i,j c 2 2u n i,j + u n 1 i,j τ 2 un i+1,j 2u n i,j + u n i 1,j h 2 x + un i,j+1 2u n i,j + u n i,j 1 h 2 y (x i, y j, t n ) u U n i,j 1 U n+1 i,j c 2 2U n i,j + U n 1 i,j τ 2 = U n i+1,j 2U n i,j + U n i 1,j h 2 x + U n i,j+1 2U n i,j + U n i,j 1 h 2 y λ x = cτ/h x, λ y = cτ/h y U n+1 i,j = 2(1 λ 2 x λ 2 y)u n i,j + λ 2 x(u n i 1,j + U n i+1,j) + λ 2 y(u n i,j 1 + U n i,j+1) U n 1 i,j (i = 1, 2,, N x 1; j = 1, 2,, N y 1; n = 1, 2, ) (3.6) (3.2) U 0 i,j = ϕ(x i, y j ) (i = 0, 1, 2,, N x ; j = 0, 1, 2,, N y ), (3.7) 31
33 (3.3) u(x i, y j, τ) t = 0 Taylor u(x i, y j, τ) = u(x i, y j, 0) + τ u t (x i, y j, 0) + τ 2 2 u 2 t (x i, y 2 j, 0) + O(τ 3 ) = u(x i, y j, 0) + τψ(x i, y j ) + τ 2 2 u 2 t (x i, y 2 j, 0) + O(τ 3 ) (3.1) t = 0 ( ) u(x i, y j, τ) = u(x i, y j, 0) + τψ(x i, y j ) + c2 τ 2 2 u 2 x (x i, y 2 j, 0) + 2 u y (x i, y 2 j, 0) + O(τ 3 ). 2 u x 2, 2 u y 2 u(x i, y j, τ) = u(x i, y j, 0) + τψ(x i, y j ) ( + c2 τ 2 u 0 i+1,j 2u 0 i,j + u 0 i 1,j 2 h 2 x + O(h 2 x) + O(h 2 y) + O(τ 3 ). + u0 i,j+1 2u 0 i,j + u 0 ) i,j 1 h 2 y ( Ui,j 1 = Ui,j 0 + τψ(a + ih x, C + jh y ) + c2 τ 2 U 0 i 1,j 2Ui,j 0 + Ui+1,j 0 2 h 2 x + U i,j 1 0 2Ui,j 0 + Ui,j+1 0 ) h 2 y U 1 i,j = (1 λ 2 x λ 2 y)u 0 i,j + λ2 x 2 (U 0 i 1,j + U 0 i+1,j) + λ2 y 2 (U 0 i,j 1 + U 0 i,j+1) + τg(ih r, jh θ ) (i = 1, 2,, N x 1; j = 1, 2,, N y 1) (3.8) Dirichlet (3.4) U n 0,j = U n N x,j = 0 (j = 0, 1, 2,, N y ; n = 1, 2, ), U n i,0 = U n i,n y = 0 (i = 0, 1, 2,, N x ; n = 1, 2, ). Neumann (3.5) u x (x 0, y j, t n ), u y (x i, y 0, t n ) u x (x N x, y j, t n ), u y (x i, y Ny, t n ) U1,j n U0,j n = 0, h x Ui,1 n Ui,0 n = 0, h y UN n x,j U N n x 1,j = 0, h x Ui,N n y Ui,N n y 1 = 0 h y 32
34 U1,j n = U0,j, n UN n x,j = U N n x 1,j (j = 0, 1, 2,, N y ; n = 1, 2, ) Ui,1 n = Ui,0, n Ui,N n y = Ui,N n y 1 (i = 0, 1, 2,, N x ; n = 1, 2, ). 3.3 /* wave2-opengl.c --- ( ) * * ccgl wave2-opengl.c set-opengl.c draw-graph.c * * : *./wave2-opengl * Nx, Ny: * : 0.01 * Tmax: 100( ) * t: 0.02 * * : * d --- Dirichlet * n --- Neumann * r --- * w --- ON/OFF * Space --- / * Esc --- * --- * --- * --- */ #include <stdio.h> #include <stdlib.h> #include <GL/glut.h> #include <GL/gl.h> #include <GL/glu.h> #include <math.h> 33
35 #include "draw-graph.h" #include "set-opengl.h" #define PI M_PI #define DBC 0 #define NBC 1 int i, j, n, nmax, skip; /* x, y */ double xmin = -2.0, xmax = 2.0, ymin = -2.0, ymax = 2.0; /* */ int Nx, Ny; /* */ double hx, hy; /* */ double tau; /* x, y, x^2, y^2 */ double lambdax, lambday, lambdax2, lambday2; /* */ double Tmax; /* */ double dt; /* u1[i][j]=u_i,j^n-1 u2[i][j]=u_i,j^n u3[i][j]=u_i,j^n+1 */ double **u1, **u2, **u3; /* */ static int bc = DBC; /* z */ double zmin = -1.0, zmax = 1.0; /* */ double distance = 10.0, twist = 0.0, elevation = 60.0, azimuth = 120.0; void setbc(int bc0) bc = bc0; 34
36 void setdbc() setbc(dbc); void setnbc() setbc(nbc); void display(void) /* t */ if(n % skip == 0) beginopengl(); /* t=0 */ if(n == 0) /* */ drawgraph(nx, Ny, u1); /* t=n (n>=1) */ else /* */ drawgraph(nx, Ny, u2); endopengl(); double phi(double x, double y) 35
37 return sin(pi*x)/2.0 + sin(pi*y)/2.0; double psi(double x, double y) return 0.0; void first_step(void) /* */ /* u_i,j^0 = (xi,yj) */ for(i=0;i<=nx;i++) for(j=0;j<=ny;j++) u1[i][j] = phi(xmin + i*hx, ymin + j*hy); /* u_i,j^1 = (1- x^2- y^2)... */ for(i=1;i<nx;i++) for(j=1;j<ny;j++) u2[i][j] = (1.0 - lambdax2 - lambday2) * u1[i][j] * lambdax2 * (u1[i-1][j] + u1[i+1][j]) * lambday2 * (u1[i][j-1] + u1[i][j+1]) + tau * psi(xmin + i*hx, ymin + j*hy); void next_step(void) /* */ if(n <= nmax) for(i=1;i<nx;i++) for(j=1;j<ny;j++) 36
38 u3[i][j] = 2.0 * (1.0 - lambdax2 - lambday2) * u2[i][j] + lambdax2 * (u2[i-1][j] + u2[i+1][j]) + lambday2 * (u2[i][j-1] + u2[i][j+1]) - u1[i][j]; /* Dirichlet */ if(bc == DBC) for(i=0;i<=nx;i++) u3[i][0] = u3[i][ny] = 0.0; for(j=0;j<=ny;j++) u3[0][j] = u3[nx][j] = 0.0; /* Neumann */ if(bc == NBC) for(i=0;i<=nx;i++) u3[i][0] = u3[i][1]; u3[i][ny] = u3[i][ny-1]; for(j=0;j<=ny;j++) u3[0][j] = u3[1][j]; u3[nx][j] = u3[nx-1][j]; /* u1 <- u2, u2 <- u3 */ for(i=0;i<=nx;i++) for(j=0;j<=ny;j++) u1[i][j] = u2[i][j]; u2[i][j] = u3[i][j]; 37
39 void idle(void) n++; next_step(); glutpostredisplay(); int main(int argc, char **argv) printf("nx, Ny: "); scanf("%d %d", &Nx, &Ny); hx = (xmax - xmin) / Nx; hy = (ymax - ymin) / Ny; printf(" : "); scanf("%lf", &tau); lambdax = tau / hx; lambday = tau / hy; lambdax2 = lambdax * lambdax; lambday2 = lambday * lambday; printf("tmax: "); scanf("%lf", &Tmax); printf(" t(>=%g): ", tau); scanf("%lf", &dt); if(dt < tau) dt = tau; skip = rint(dt / tau); n = 0; nmax = rint(tmax / tau); /* u1, u2, u3 */ u1 = malloc(sizeof(double *) * (Nx+1)); u2 = malloc(sizeof(double *) * (Nx+1)); 38
40 u3 = malloc(sizeof(double *) * (Nx+1)); for(i=0;i<=nx;i++) u1[i] = malloc(sizeof(double) * (Ny+1)); u2[i] = malloc(sizeof(double) * (Ny+1)); u3[i] = malloc(sizeof(double) * (Ny+1)); register_command( d, setdbc); register_command( n, setnbc); setview(distance, twist, elevation, azimuth); setframe(xmin, xmax, ymin, ymax, zmin, zmax); first_step(); OpenGL(&argc, argv); return(0); 39
41 u c 2 t 2 = 2 u x u y 2 (t > 0, (x, y) Ω) (4.1) Ω = (x, y) R 2 x 2 + y 2 < 1 x = r cos θ y = r sin θ 1 2 u c 2 t 2 = 2 u r + 1 u 2 r r u ((r, θ) (0, 1) [0, 2π), t > 0) (4.2) r 2 θ 2 u(r, θ, 0) = f(r, θ) ((r, θ) [0, 1] [0, 2π]) (4.3) u (r, θ, 0) = g(r, θ) t ((r, θ) [0, 1] [0, 2π]) (4.4) (Dirichlet ) u(1, θ, t) = 0 (θ [0, 2π), t > 0) (4.5) (Neumann ) u (1, θ, t) = 0 r (θ [0, 2π), t > 0) (4.6) Bessel Fourier [5] 40
42 4.2 r [0, R] N r θ [0, 2π] N θ r i (i = 0,, N r ), θ j (j = 0,, N θ ) h r = R N r, h θ = 2π N θ r i = ih r (i = 0, 1, 2,, N r ), θ j = jh θ (j = 0, 1, 2,, N θ ). t τ(> 0) t n = nτ (n = 0, 1, 2, ) (r i, θ j, t n ) u u n i,j u(r i, θ j, t n ) (r i, θ j, t n ) u u r 2 t, 2 u 2 r, 2 u 2 θ 2 u r (r i, θ j, t n ) = un i+1,j u n i 1,j + O(h 2 2h r) r (h r +0), 2 u t (r i, θ 2 j, t n ) = un+1 i,j 2u n i,j + u n 1 i,j + O(τ 2 ) τ 2 (τ +0), 2 u r (r i, θ 2 j, t n ) = un i+1,j 2u n i,j + u n i 1,j h 2 r + O(h 2 r) (h r +0), 2 u θ (r i, θ 2 j, t n ) = un i,j+1 2u n i,j + u n i,j 1 h 2 θ + O(h 2 θ) (h θ +0) u (4.1) h r, h θ, τ 1 u n+1 i,j c 2 2u n i,j + u n 1 i,j τ 2 un i+1,j 2u n i,j + u n i 1,j h 2 r + 1 r i u n i+1,j u n i 1,j 2h r + 1 r 2 i u n i,j+1 2u n i,j + u n i,j 1 h 2 θ (r i, θ j, t n ) u U n i,j 1 U n+1 i,j c 2 2U n i,j + U n 1 i,j τ 2 U n i+1,j 2U n i,j + U n i 1,j h 2 r + 1 r i U n i+1,j U n i 1,j 2h r + 1 r 2 i U n i,j+1 2U n i,j + U n i,j 1 h 2 θ 41
43 λ r = cτ/h r, λ θ = cτ/h θ ( ) U n+1 i,j = 2 1 λ 2 r λ2 θ U n ri 2 i,j + λ 2 r(ui 1,j n + Ui+1,j) n + 1 c 2 τ 2 (Ui+1,j n U r i 2h i 1,j) n + λ2 θ (Ui,j 1 n + Ui,j+1) n U n 1 i,j r (4.3) r 2 i (i = 1, 2,, N r 1; j = 1, 2,, N θ 1; n = 1, 2, ) (4.7) U 0 i,j = f(r i, θ j ) (i = 0, 1, 2,, N r ; j = 0, 1, 2,, N θ ), (4.8) (4.4) u(r, θ, τ) t = 0 Taylor u(r, θ, τ) = u(r, θ, 0) + τ u 2 τ (r, θ, 0) + t 2 = u(r, θ, 0) + τg(r, θ) + τ 2 2 (4.1) t = 0 u(r, θ, τ) = u(r, θ, 0) + τg(r, θ) + c2 τ 2 u u r 2 r, 2 u 2 θ 2 u(r, θ, τ) = u(r, θ, 0) + τg(r, θ) ( + c2 τ 2 u 0 i+1,j 2u 0 i,j + u 0 i 1,j r h 2 r + 2O(h 2 r) + O(h 2 θ) + O(τ 3 ). Ui,j 1 = Ui,j 0 + τg(ih r, jh θ ) ( + c2 τ 2 U 0 i+1,j 2Ui,j 0 + Ui 1,j 0 2 h 2 r 2 2 u t 2 (r, θ, 0) + O(τ 3 ) 2 u t 2 (r, θ, 0) + O(τ 3 ) ( 2 u r (r, θ, 0) + 1 u 2 r r (r, θ, 0) u r 2 θ u 0 i+1,j u 0 i 1,j 2h r + 1 r i U 0 i+1,j U 0 i 1,j 2h r ( ) Ui,j 1 = 1 λ 2 r λ2 θ Ui,j 0 + λ2 r 2 (U i 1,j 0 + Ui+1,j) r 2 i τ 2 r 2 i + 1 r 2 i ) (r, θ, 0) u 0 i,j+1 2u 0 i,j + u 0 ) i,j 1 r 2 h 2 θ Ui,j+1 0 2Ui,j 0 + Ui,j 1 0 ) h 2 θ (Ui+1,j 0 U 2h i 1,j) 0 + λ2 θ (Ui,j Ui,j+1) 0 + τg(ih r, jh θ ) r r 2 i (i = 1, 2,, N r 1; j = 1, 2,, N θ 1) (4.9) + O(τ 3 ). 42
44 Dirichlet (4.5) U n N r,j = 0 (j = 0, 1, 2,, N θ ; n = 1, 2, ) Neumann (4.6) u r (r N r, θ j, t n ) U n N r,j U n N r 1,j h r = 0 U n N r,j = U n N r 1,j (j = 0, 1, 2,, N θ ; n = 1, 2, ). 4.3 [5] 4.4 /* wave2d-disk-e-opengl.c --- ( ) * * ccgl wave2d-disk-e-opengl.c set-opengl.c draw-graph-on-disk.c * * : *./wave2d-disk-e-opengl * Nx, Ny: * : 0.01 * Tmax: 100( ) * t: 0.02 * * : * d --- Dirichlet * n --- Neumann * r --- * w --- ON/OFF 43
45 * Space --- / * Esc --- * --- * --- * --- */ #include <stdio.h> #include <stdlib.h> #include <GL/glut.h> #include <GL/gl.h> #include <GL/glu.h> #include <math.h> #include "draw-graph-on-disk.h" #include "set-opengl.h" #define PI M_PI #define DBC 0 #define NBC 1 int i, j, n, nmax, skip; /* r */ double R = 2.0; /* */ int Nr, Np; /* */ double hr, hp; /* */ double tau; /* r,, r^2, ^2 */ double lambdar, lambdap, lambdar2, lambdap2; /* */ double Tmax; /* */ double dt; /* u1[i][j]=u_i,j^n-1 u2[i][j]=u_i,j^n 44
46 u3[i][j]=u_i,j^n+1 */ double **u1, **u2, **u3; /* */ static int bc = DBC; /* z */ double zmin = -0.8, zmax = 0.8; /* */ double distance = 5.0, twist = 0.0, elevation = 60.0, azimuth = 120.0; void setbc(int bc0) bc = bc0; void display(void) /* t */ if(n % skip == 0) beginopengl(); /* t=0 */ if(n == 0) /* */ drawgraph(nr, Np, u1); /* t=n (n>=1) */ else /* */ drawgraph(nr, Np, u2); endopengl(); 45
47 double phi(double r, double p) return r*sin(p)*cos(p); double psi(double r, double p) return 0.0; void first_step(void) /* */ /* u_i,j^0 = (xi,yj) */ for(i=0;i<=nr;i++) for(j=0;j<=np;j++) u1[i][j] = phi(i*hr, j*hp); /* u_i,j^1 = (1- r^2- ^2/ri^2)... */ for(i=1;i<nr;i++) for(j=1;j<np;j++) u2[i][j] = (1.0 - lambdar2 - (lambdap2 / (i*hr*i*hr))) * u1[i][j] * lambdar2 * (u1[i-1][j] + u1[i+1][j]) + ((tau*tau) / (2.0*i*hr*hr)) * (u1[i+1][j] - u1[i-1][j]) + (lambdap2 / (i*hr*i*hr)) * (u1[i][j-1] + u1[i][j+1]) + tau * psi(i*hr,j*hp); void next_step(void) 46
48 /* */ if(n <= nmax) for(i=1;i<nr;i++) for(j=1;j<np;j++) u3[i][j] = 2.0 * (1.0 - lambdar2 - (lambdap2 / (i*hr*i*hr))) * u2[i][j] + lambdar2 * (u2[i-1][j] + u2[i+1][j]) + ((tau*tau) / (2.0*i*hr*hr)) * (u2[i+1][j] - u2[i-1][j]) + (lambdap2 / (i*hr*i*hr)) * (u2[i][j-1]+u2[i][j+1]) - u1[i][j]; /* Dirichlet */ if(bc == DBC) u3[nr][j] = 0.0; /* Neumann */ if(bc == NBC) for(j=0;j<=np;j++) u3[nr][j] = u3[nr-1][j]; /* =2 =0 */ for(i=0;i<=nr;i++) u3[i][np] = u3[i][0]; /* u1 <- u2, u2 <- u3 */ for(i=0;i<=nr;i++) for(j=0;j<=np;j++) u1[i][j] = u2[i][j]; u2[i][j] = u3[i][j]; 47
49 void idle(void) n++; next_step(); glutpostredisplay(); int main(int argc, char **argv) printf("nr, Ntheta: "); scanf("%d %d", &Nr, &Np); hr = R / Nr; hp = 2.0 * PI / Np; printf(" : "); scanf("%lf", &tau); lambdar = tau / hr; lambdap = tau / hp; lambdar2 = lambdar * lambdar; lambdap2 = lambdap * lambdap; printf("tmax: "); scanf("%lf", &Tmax); printf(" t(>=%g): ", tau); scanf("%lf", &dt); if(dt < tau) dt = tau; skip = rint(dt / tau); n = 0; nmax = rint(tmax / tau); /* u1,u2,u3 */ u1 = malloc(sizeof(double *) * (Nr+1)); u2 = malloc(sizeof(double *) * (Nr+1)); 48
50 u3 = malloc(sizeof(double *) * (Nr+1)); for(i=0;i<=nr;i++) u1[i] = malloc(sizeof(double) * (Np+1)); u2[i] = malloc(sizeof(double) * (Np+1)); u3[i] = malloc(sizeof(double) * (Np+1)); setview(distance, twist, elevation, azimuth); setframe(r, zmin, zmax); first_step(); OpenGL(&argc, argv); return(0); 49
51 A ccgl OpenGL GLUT Cygwin ccgl #!/bin/sh name= basename $1.c gcc -finput-charset=cp932 -fexec-charset=cp932 \ -Wl,--enable-auto-import \ -I/usr/X11R6/include -o $name "$@" \ -L/usr/X11R6/lib -lglut -lgl -lglu -lx11 50
52 B JPEG B.1 Independent JPEG Group 1 JPEG set-opengl-jpeg.c draw-graph.c draw-graph-on-disk.c ijg-saveimage.c ccgl ( ).c set-opengl-jpeg.c draw-graph.c ijg-saveimage.c -ljpeg B.2 set-opengl-jpeg.c /* set-opengl-jpeg.c */ #include <stdio.h> #include <GL/glut.h> #include <GL/gl.h> #include <GL/glu.h> #include <math.h> #define KEY_ESC 27 #define KEY_SPC 32 #define DBC 0 #define NBC
53 /* / */ static unsigned char moveflag = GL_FALSE; /* */ static int bc; static int xbegin, ybegin; static int mbutton; static double distance, twist, elevation, azimuth; static double D = 10.0, T = 0.0, E = 60.0, A = 120.0; void polarview(void); void resetview(void); void idle(void); void display(void); /* */ int id = 0; char fname[100]; #define DIRNAME "wave2-images" void setview(double d0, double t0, double e0, double a0) D = d0; T = t0; E = e0; A = a0; /* display */ void beginopengl(void) glclear(gl_color_buffer_bit GL_DEPTH_BUFFER_BIT); glenable(gl_depth_test); glpushmatrix(); polarview(); /* display */ 52
54 void endopengl(void) glpopmatrix(); gldisable(gl_depth_test); glutswapbuffers(); /* 1 10 ( 1000 ) */ if(id < 10000) if(id % 10 == 0) /* 10 */ snap_ijg_image(); sprintf(fname, "%s/image%03d.jpg", DIRNAME, id/10); save_ijg_image(fname); id++; #define MAXCOMMANDS (100) typedef void vvfunc(void); static char command_keys[maxcommands]; vvfunc *command_funcs[maxcommands]; static num_of_reg_commands = 0; /* c f ( ) */ void register_command(char c, vvfunc f) if (num_of_reg_commands < MAXCOMMANDS) command_keys[num_of_reg_commands] = c; command_funcs[num_of_reg_commands++] = f; else fprintf(stderr, " \n"); void mykbd(unsigned char key, int x, int y) 53
55 int i; switch(key) case r : resetview(); break; case w : switchwireflag(); break; case KEY_SPC: moveflag =!moveflag; if(moveflag == GL_TRUE) glutidlefunc(idle); else glutidlefunc(null); break; case KEY_ESC: exit(0); default: for(i=0;i<num_of_reg_commands;i++) if(key == command_keys[i]) command_funcs[i](); return; glutpostredisplay(); void mymouse(int button, int state, int x, int y) if(state == GLUT_DOWN) xbegin = x; ybegin = y; mbutton = button; 54
56 void mymotion(int x, int y) int xdisp, ydisp; xdisp = x - xbegin; ydisp = y - ybegin; switch(mbutton) case GLUT_LEFT_BUTTON: azimuth -= (double)xdisp / 2.0; elevation -= (double)ydisp / 2.0; break; case GLUT_MIDDLE_BUTTON: twist = fmod(twist + xdisp/3.0, 360.0); break; case GLUT_RIGHT_BUTTON: distance += (double)ydisp / 60.0; break; xbegin = x; ybegin = y; glutpostredisplay(); void myinit(char *progname) glutinitwindowposition(0, 0); glutinitwindowsize(400, 400); prepare_ijg_buffer(400, 400); /* */ glutinitdisplaymode(glut_double GLUT_RGBA GLUT_DEPTH); glutcreatewindow(progname); 55
57 glclearcolor(1.0, 1.0, 1.0, 1.0); glutkeyboardfunc(mykbd); glutmousefunc(mymouse); glutmotionfunc(mymotion); resetview(); void myreshape(int width, int height) double aspect = width/(double)height; glviewport(0, 0, width, height); glmatrixmode(gl_projection); glloadidentity(); gluperspective(40.0, aspect, 1.0, 80.0); glmatrixmode(gl_modelview); void polarview(void) gltranslatef(0.0, 0.0, -distance); glrotatef(-twist, 0.0, 1.0, 0.0); glrotatef(-elevation, 1.0, 0.0, 0.0); glrotatef(-azimuth, 0.0, 0.0, 1.0); void resetview(void) distance = D; twist = T; elevation = E; azimuth = A; void OpenGL(int *argc, char **argv) /* DIRNAME */ 56
58 char cmd[1024]; sprintf(cmd, "mkdir %s", DIRNAME); system(cmd); glutinit(argc, argv); myinit(argv[0]); glutreshapefunc(myreshape); glutidlefunc(null); glutdisplayfunc(display); glutmainloop(); B.3 ijg-saveimage.c /* * ijg-saveimage.c --- Independent JPEG Group * OpenGL (2008/8/16, by mk) * (2008/8/30) * img ( ) * ijg_buffer (2008/11/2) * * : * prepare_ijg_buffer(, ); * snape_ijg_image(); * save_ijg_image( ); * * : */ #include <stdio.h> #include <stdlib.h> #include <GL/glut.h> #include <GL/gl.h> #include <jpeglib.h> 57
59 #define MAX_HEIGHT (2048) static int ijg_width = 0, ijg_height = 0; static JSAMPLE *ijg_buffer = NULL; void prepare_ijg_buffer(int w, int h) /* */ if (w == ijg_width && h == ijg_height) return; /* */ if (h > MAX_HEIGHT) fprintf(stderr, "h=%d is too large (must be <= %d)\n", h, MAX_HEIGHT); return; ijg_width = w; ijg_height = h; if (ijg_buffer!= NULL) free(ijg_buffer); ijg_buffer = malloc(w * h * 3); void snap_ijg_image() /* */ glreadbuffer(gl_front); /* buffer */ glreadpixels(0, 0, ijg_width, ijg_height, GL_RGB, GL_UNSIGNED_BYTE, ijg_buffer); void save_ijg_image(char *fname) struct jpeg_compress_struct cinfo; struct jpeg_error_mgr jerr; FILE *outfile; JSAMPROW img[2048]; // JSAMPARRAY *img; 58
60 int i,j; // JPEG cinfo.err = jpeg_std_error(&jerr); jpeg_create_compress(&cinfo); // outfile = fopen(fname, "wb"); jpeg_stdio_dest(&cinfo, outfile); // cinfo.image_width = ijg_width; cinfo.image_height = ijg_height; cinfo.input_components = 3; cinfo.in_color_space = JCS_RGB; // jpeg_set_defaults(&cinfo); jpeg_set_quality(&cinfo, 100, TRUE); // jpeg_start_compress(&cinfo, TRUE); // for (i = 0; i < ijg_height; i++) img[i] = ijg_buffer + (ijg_height - i) * 3 * ijg_width; jpeg_write_scanlines(&cinfo, img, ijg_height); // jpeg_finish_compress(&cinfo); // JPEG jpeg_destroy_compress(&cinfo); // fclose(outfile); 59
61 [1], OpenGL CG, (2003). [2],, mk/labo/text/wave.pdf (2008). [3],, mk/labo/text/heat-fdm-0.pdf (2008). [4], Java, 2001, mk/labo/report/pdf/2001-mitsui.pdf (2002). [5],, 2004, mk/labo/report/open/ 2004-nakanishi.pdf (2005). [6] James D. Foley, Andres van Dam, Steven K. Feiner, John F. Hughes,,, (2001). 60
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演算増幅器
コンピュータグラフィックス 2 前回は GLUT を使った簡単な 2 次元グラフィックスについて習った 今週は以下の項目について 補足していく イベント駆動型プログラムの動作について コンピュータグラフィックスの座標系 イベント駆動型プログラム従来のプログラムとの違いこれまでに学習してきたプログラムは上から下に順次実行され 条件分岐や繰り返し処理によって プログラムの流れ (flow: フロー )
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/* drawing function */ function_graph();// drawing graph glflush(); int main(int argc, char **argv ) glutinit( &argc, argv ); glutinitdisplaymode( GLU
OpenGL ( ) #include #include #define PI 3.14159265 void function_graph() int j; float x, y; glbegin( GL_LINE_STRIP );// sine curve by line glcolor3f( 0.0f, 1.0f, 1.0f );// line color
3D グラフィックス処理の一般過程 1. 3D グラフィックス処理の一般過程
3. 3D ビューイング 1. 3Dグラフィックス処理の一般過程 2. 射影と射影変換 3. ビューボリュームとクリッピング 4. 陰面処理とデプスバッファ 5. ビューポート変換 6. 3Dグラフィックスを描く 7. モデルビュー変換 3D グラフィックス処理の一般過程 1. 3D グラフィックス処理の一般過程 3D グラフィックス処理の一般過程 1. モデリング変換 座標系の異なる複数のオブジェクトを仮想世界に配置し,
x h = (b a)/n [x i, x i+1 ] = [a+i h, a+ (i + 1) h] A(x i ) A(x i ) = h 2 {f(x i) + f(x i+1 ) = h {f(a + i h) + f(a + (i + 1) h), (2) 2 a b n A(x i )
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演算増幅器
スペースインベーダーもどき 1000 行プログラムの参考として スペースインベーダーもどきのプログラムを配布する いくつか習って いないものもあるので 補足の説明を加えていく 文字列の描画 文字の描画は glutbitmapcharacter() を用いる これは以下のようにして利用する int i; char *str = "Display String"; glcolor3f(0.0, 0.0,
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double rx[natom], ry[natom], rz[natom]; 原子の座標 速度 力 ポテンシャルエ double vx[natom], vy[natom], vz[natom]; ネルギーを受ける配列を準備 double fx[natom], fy[natom], fz[natom
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GLUI による MD の GUI 化 前提条件 :GLUI のプログラミング環境が整っていること 3 原子の MD コード ( 下図 ) viewer ウィンドウ内のマウス左クリックで MD 開始 右クリックで MD 停止 control パネルは solid/wireframe を切り替えるチェックボタン 球の滑らかさと半径を決める窓 ( スピナー ) オブジェクトを回転 移動 拡大縮小させるコントローラ
新版明解Ｃ言語 実践編
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第7章 レンダリング
7 April 11, 2017 1 / 59 7.1 ( ) CG 3 ( ) 2 / 59 7.2 7.2.1 ( ) 3 (rendering) 1 / (hidden line/surface calculation) a (outer normal algorithm) b Z (Z-buffer algorithm) c (scan-line algorithm) 2 (shading)
第7章 レンダリング
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2. 2 次元粒子法シミュレーション (+ 少しだけ OpenGL) 茨城大学工学部 教授乾正知 準備 計算結果を可視化するために OpenGL を 利用する. 2 OpenGL 3 次元コンピュータグラフィックス用の標準的なライブラリ. 特に CAD やアート, アニメーション分野 ( ゲーム以外の分野 ) で広く利用されている. OpenGL は仕様がオープンに決められており, 企業から独立した団体が仕様を管理している.
C による数値計算法入門 ( 第 2 版 ) 新装版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 新装版 1 刷発行時のものです.
C による数値計算法入門 ( 第 2 版 ) 新装版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009383 このサンプルページの内容は, 新装版 1 刷発行時のものです. i 2 22 2 13 ( ) 2 (1) ANSI (2) 2 (3) Web http://www.morikita.co.jp/books/mid/009383
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Microsoft Word - C.....u.K...doc
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ohp07.dvi
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Microsoft PowerPoint - 04.pptx
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情報工学 212 年度後期第 5 回 [1 月 31 日 ] 静岡大学 創造科学技術大学院情報科学専攻工学部機械工学科計測情報講座 三浦憲二郎 講義日程 第 8 回 11 月 21 日 ( 水 ) CG パート試験 講義アウトライン [1 月 31 日 ] ビジュアル情報処理 1.3.4 投影変換 1.3.5 いろいろな座標系と変換 OpenGL 投影変換 曲線の描画 トロコイド ( 外トロコイドと内トロコイド
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情報工学 217 年度後期第 4 回 [1 月 25 日 ] 静岡大学 工学研究科機械工学専攻ロボット 計測情報講座創造科学技術大学院情報科学専攻 三浦憲二郎 ローカル座標系による移動 講義アウトライン [1 月 25 日 ] ビジュアル情報処理 1.3.4 投影変換 1.3.5 いろいろな座標系と変換 OpenGL 投影変換 曲線の描画 トロコイド ( 外トロコイドと内トロコイド ) 頂点変換の手順
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#define N1 N+1 double x[n1] =.5, 1., 2.; double hokan[n1] = 1.65, 2.72, 7.39 ; double xx[]=.2,.4,.6,.8,1.2,1.4,1.6,1.8; double lagrng(double xx); main
![#define N1 N+1 double x[n1] =.5, 1., 2.; double hokan[n1] = 1.65, 2.72, 7.39 ; double xx[]=.2,.4,.6,.8,1.2,1.4,1.6,1.8; double lagrng(double xx); main](/thumbs/88/114913669.jpg "#define N1 N+1 double x[n1] =.5, 1., 2.; double hokan[n1] = 1.65, 2.72, 7.39 ; double xx[]=.2,.4,.6,.8,1.2,1.4,1.6,1.8; double lagrng(double xx); main")
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p.130 p.198 p.208 2 double weight[num]; double min, max; min = max = weight[0]; for( i= 1; i i < NUM; i++ ) ) if if ( weight[i] > max ) max = weight[i]: if if ( weight[i] < min ) min = weight[i]: weight
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2007 年度メディア情報学実験 1 CG テキスト第 2~4 週 :OpenGL ライブラリを使った 3 次元 CG プログラミング立命館大学情報理工学部メディア情報学科 1. 実験の目的と手順本実験は,(1)OpenGLライブラリを用いたCGプログラミングの手法を学ぶ,(2) プログラミングを通して, CGの基礎技術を体験的に学ぶ,(3) インタラクティブな3 次元 CGアニメーションを作成する方法について学ぶ,
2 1 list0415.c forfor #include int i, j; for (i = 1; i プログラミング方法論 II 第 14,15 回 ( 担当 : 鈴木伸夫 ) 問題 17. x 座標と y 座標をメンバに持つ構造体 Point を作成せよ 但し座標 は double 型とする typedef struct{ (a) x; (b) y; } Point; 問題 18. 問題 17 の
プログラミング方法論 II 第 14,15 回 ( 担当 : 鈴木伸夫 ) 問題 17. x 座標と y 座標をメンバに持つ構造体 Point を作成せよ 但し座標 は double 型とする typedef struct{ (a) x; (b) y; Point; 問題 18. 問題 17 の Point を用いて 2 点の座標を入力するとその 2 点間の距 離を表示するプログラムを作成せよ 平方根は
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ex12.dvi
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slide5.pptx
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