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4 , 3 , 1 . . 3 , (-1) , 0 . 1. A=(xz+1,yz,xy) B=rotA B=(0,x-y,y) B=(x-y,x-z,z) B=(x-y,x-y,0) B=(x,y,0) B=(3/2x-5y, -3/2y-3z, x2-y2) B=(3/2x-5y,3/4y2+3,x2) B=(x2+3y,y2+3x,3z) B=(x3+3xy,3y-3x2 y,-3yz) B=(-2/3xz, 4/3yz, z2-x2/3-2/3y2) B=(-2/3xz, -4/3yz, z2-x2/3-2/3y2) B=(y, x+z, 2y) B=(2y,x+z, 2y) F (r )=const drF(r )=0 dr F( r)=0 F (r )=r rot

2.

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8. n=0

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n=1

9. n=0

lg 24/lg 5 ln 5 5/ 3 2

n=1

n=2

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a 1 1 2 1

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121 1 2 2 2 : 14. 14 121
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:

0

1

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17. 1 5 1 : 0 1 0 18. « » 19. 0, x c F x 1, x c 20.

0 1 0 21. 1 4 1 : 0 1 0

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.
1. «» ) . . . . ) .

f ( z) z 2 C , f ( z) z 3 C , f ( z ) z 3 Cz, f ( z ) e z b, f ( z ) c1 sin z c2 .
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1 2 2 S1 S 2 . 8

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: T i 1,2, ... , M 1 , j 1,2, ... , M
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. OPENGL.
OpenGL - Open Graphics Library, . "" - . OpenGL (API) . OpenGL 1992 , IRIS GL, Silico n Graphics. OpenGL ( OpenGL, GLU,GLUT,GLAUX ), . OpenGL

: , , .. (z-) : , , (blending), (antialiasing).

, OpenGL , . Windows Unix (WGL GLX ), GLAUX GLUT, .

. . http://ani.cs.msu.su

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, OpenGL , :

, OpenGL , ( GL_) , . , . , . OpenGL : type glCommand_name[1 2 3 4][b s i f d ub us ui][v](type1 arg1,...,typeN argN) , : gl , : OpenGL, GLU, GLUT, GLAUX gl, glu, glut, aux . Command_name [1 2 3 4] [b s i f d ub us ui ] : b GLbyte ( char \++), f GLflo at ( float), i GLint( int) . gl.h GLUT GL Ut ilit y Toolkit, . , Windows, Linux . GLUT :

(callback) 42

. . http://ani.cs.msu.su

. . , . . , . . « »

(, .)

glutInit(int *argcp, char **argv) argcp argc main(), argv , , . , GLUT . : glutInitWindowPosition (int x, int y) glutInitWindowSize (int width, int height) glutInitDisplayMode (unsigned int mode) , , "" (|): GLUT_RGBA RGBA. , GLUT_RGBA GLUT_INDEX. GLUT_RGB , GLUT_RGBA. GLUT_INDEX ( ). GLUT_RGBA. GLUT_SINGLE . . GLUT_DOUBLE . GLUT_SINGLE. GLUT_DEPTH . , . , - , , . z- . GLUT - . , , , , , . : , , , , , "" (idle) , . , . :

vo id glutDisplayFunc (vo id (*func) (vo id)) vo id glutReshapeFunc (vo id (*func) (int width, int height)) vo id glutMouseFunc (vo id (*func) (int button, int state, int x, int y)) vo id glutIdleFunc (void (*func) (vo id))

. glutDisplayFunc() , . , , vo id glutPostRedisplay(vo id) glutReshapeFunc() , .
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glutMouseFunc() , glutIdleFunc() , , . vo id glutMainLoop(void) , GLUT. , :

vo id glVertex[2 3 4][s i f d](type coords) vo id glVertex[2 3 4][s i f d]v(t ype *coords)

: x, y, z, w, (x,y) (x,y,z) , : z=0, w=1. , , . , (0,0) , x , y- , z- . , , . - , , . OpenGL , , , , . :

vo id glBegin(GLenum mode) vo id glEnd(vo id)

mode , :

GL_POINTS . GL_LINES ; , . GL_LINE_STRIP . GL_LINE_LOOP , , . GL_TRIANGLES ; , . GL_TRIANGLE_STRIP . GL_TRIANGLE_FAN ( ). 44

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. . , . . , . . « »

GL_QUADS ; , . GL_QUAD_STRIP n 2n-1, 2n, 2n+2, 2n+1. GL_POLYGON .

,

vo id glNormal3[b s i f d](type coords) vo id glNormal3[b s i f d]v(t ype coords)

glBegin() glEnd() , glVertex..(), glCo lor..()glNormal..(), glRect..(), glMaterial..() glTexCoord..(). , void glRect[s i f d]( GLt ype x1, GLt ype y1, GLt ype x2, GLt ype y2 ), vo id glRect[s i f d]v( GLt ype *v1, GLt ype *v2 ) z=0 (x1,y1) (x2,y2), v1 v2. , . . , , . void glFrontFace(GLenum mode) mode GL_CW, GL_CCW. vo id glPo lygonMode(GLenum face, Glenum mode)
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mode , , face , :

GL_FRONT GL_BACK GL_FRONT_AND_BACK GL_POINT .

mode :

GL_LINE . GL_FILL .

, , . glEnable(GL_CULL_FACE), void glullFace(GLenum mode) GL_FRONT , GL_BACK- ( ). , , glEnable(GL_DEPTH_TEST). , glVertex..(), , vo id glVertexPointer( GLint size, GLenum t ype, GLsizei stride, vo id *ptr ) . size ( 2, 3, 4), type ( GL_SHORT, GL_INT, GL_FLOAT, GL_DOUBLE). , stride ; stride , , . ptr , . , ,

vo id NormalPo inter(GLenum t ype, GLsizei stride, vo id*po inter) vo id Co lorPointer(GLintsize, GLenum t ype, GLsizei stride, vo id *pointer)

, , vo id glEnableClientState(GLenum array) GL_VERTEX_ARRAY, GL_NORMAL_ARRAY, GL_COLOR_ARRAY . vo id glDisableClientState(GLenum array)
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array. vo id glArrayElement(GLint index) OpenGL , index. glCo lor..(...), glNormal..(...), glVertex..(...) c . vo id glDrawArrays(GLenum mode, GLint first, GLsizei count) count , mode, first first+count-1. glArrayElement() . , . vo id glDrawArrays(GLenum mode, GLsizei count, GLenum t ype, vo id *indices) indices , , type : GL_UNSIGNED_BYTE, GL_UNSIGNED_SHORT, GL_UNSIGNED_INT, count . . . , ,

, (displa y list) :

vo id glNewList(GLuint list, GLenum mode) vo id glEndList()

, list, mode , :

GL_COMPILE GL_COMPILE_AND_EXECUTE ,

, , vo id glCallList(GLuint list) list . , vo id glCallLists(GLsizei n, GLenum t ype, const GLvoid *lists) n lists, type. GL_BYTE, GL_UNSIGNED_BYTE, GL_SHORT, GL_INT, GL_UNSIGNED_INT > . vo id glDeleteLists(GLint list, GLsizei range) ID list <=ID<= list+range-1.
. . http://ani.cs.msu.su

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OpenGL : , . z: , . x y . gluPerspective(), glOrtho(), , . . OpenGL , : , . 4x4. , , . ( ), . , , , vo id glMatrixMode(GLenum mode) mode GL_MODELVIEW, GL_PROJECTION, GL_TEXTURE , . , . vo id glLoadMatrix[f d](GLt ype *m) m 16 float double , , , . vo id glLoadIdent ity(vo id) . ,

vo id glPushMatrix(vo id) vo id glPopMatrix(vo id)

, . 32, 2. vo id glMultMatrix[f d](GLtype *m) m . , , 4x4 . glLoadIdent it y(). 48

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. . , . . , . . « »

, : => => => => . , . : (x', y', z', 1)T =M * (x, y, z, 1)
T

M . . :

vo id glTranslate[f d](GLtype x, GLtype y, GLt ype z) vo id glRotate[f d](GLtype angle, GLt ype x, GLtype y, GLtype z) vo id glScale[f d](GLt ype x, GLtype y, GLt ype z)

glTranlsate..() , . glRotate..() angle ( ) ( x,y,z ). glScale..() ( ), . , . , , , , glPushMatrix(), glRotate..() , , , glPopMatrix(). , . vo id gluLookAt (GLdouble eyex, GLdouble eyey, GLdouble eyez, GLdouble centerx, GLdouble centery, GLdouble centerz, GLdouble upx, GLdouble upy, GLdouble upz)
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(eyex,eyey,eyez) , (centerx, centery, , , (upx,upy,upz ) , , , , (0,1, (0,-1,0) .

centerz) . 0),

, , . OpenGL () .

vo id glOrtho(GLdouble left, GLdouble right, GLdouble bottom, GLdouble top, GLdouble near, GLdouble far) vo id gluOrtho2D(GLdouble left, GLdouble right, GLdouble bottom, GLdouble top)

( ) . (left, bottom, -near) (right, top, -near), . near far (0,0,0) . , , near far 1 1 . vo id gluPerspect ive(GLdouble angley, GLdouble aspect, GLdouble znear, GLdouble zfar) . angley 0 180. x aspect, . zfar znear . zfar/znear, , `' 0 1 (. ). (clip) , (xc, yc, zc, wc)T [-1,1]. : (xn, yn, zn)T=(xc/wc, yc/wc, zc/wc)
T

, : vo id glViewPort(GLint x, GLint y, GLint width, GLint height) ( x,y ) . , (0,0) .
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glViewPort(), (ox, oy) ox=x+width/2, oy=y+height/2. px=width, py=height, : (xw, yw, zw)T = ( (px/2) xn+ ox , (py/2) yn+ oy , [(f-n)/2] zn+(n+f)/2 )
T

n f 0 1 . (z-), . n f vo id glDepthRange(GLclampd n, GLclampd f) glViewPort() , glutReshapeFunc(), , , . , , . , , , . , . , OpenGL.

vo id glMaterial[i f](GLenum face, GLenum pname, GLtype param) vo id glMaterial[i f]v(GLenum face, GLenum pname, GLt ype *params) , . pname:

, , param,

GL_AMBIENT params RGBA, ( ). : (0.2, 0.2, 0.2, 1.0). GL_DIFFUSE params RGBA, . :(0.8, 0.8, 0.8, 1.0). GL_SPECULAR params RGBA, . : (0.0, 0.0, 0.0, 1.0). GL_SHININESS params 0 128, . : 0.
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GL_EMISSION params RGBA, . : (0.0, 0.0, 0.0, 1.0). GL_AMBIENT_AND_DIFFUSE glMaterial..() pname GL_AMBIENT GL_DIFFUSE params. , , glMater . ; ial[i f]() .

face , GL_FRONT, GL_BACK GL_FRONT_AND_BACK. , , glEnable() c GL_COLOR_MATERIAL, vo id glColorMaterial(GLenum face, GLenum pname) face , pname . , pname ( ) glColor..(), glMaterial..() . vo id glLight[i f](GLenum light, GLenum pname, GLfloat param) vo id glLight[i f](GLenum light, GLenum pname, GLfloat *params) light , GL_LIGHTi, i 0 GL_MAX_LIGHT, . , glMaterial..(). ( , ): GL_SPOT_EXPONENT param 0 128, . . : 0 ( ). GL_SPOT_CUTOFF param 0 90 180, . , . : 180 ( ). GL_AMBIENT params RGBA, . : (0.0, 0.0, 0.0, 1.0).
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GL_DIFFUSE params RGBA, . : (1.0, 1.0, 1.0, 1.0) LIGHT0 (0.0, 0.0, 0.0, 1.0) . GL_SPECULAR params RGBA, . : (1.0, 1.0, 1.0, 1.0) LIGHT0 (0.0, 0.0, 0.0, 1.0) . GL_POSITION params . 0.0, (x,y,z), (x,y,z,w). : (0.0, 0.0, 1.0, 0.0). GL_SPOT_DIRECTION params , . : (0.0, 0.0, -1.0, 1.0). : glLight..() ( glLookAt() ), , . , . , , , , . glEnable (GL_LIGHTNING), glEnable(GL_LIGHTn). OpenGL , : , , . , glScale..(), . , , glEnable(GL_NORMALIZE). vo id glLightModel[i f](GLenum pname, GLenum param) vo id glLightModel[i f]v(GLenum pname, const GLtype *params) pname , : GL_LIGHT_MODEL_LOCAL_VIEWER param . FALSE, z, x . TRUE,
. . http://ani.cs.msu.su

, w ,

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. , . : FALSE. GL_LIGHT_MODEL_TWO_SIDE param , . FALSE, . TRUE, . : FALSE . GL_LIGHT_MODEL_AMBIENT params , . :(0.2, 0.2, 0.2,1.0). , . . : , . . OpenGL Windows DIB , (R,G,B) , . , , GLAUX ( glaux.lib), . AUX_RGBImageRec* auxDIBImageLoad(string file) file *.bmp *.dib. , . . -, , . . , . vo id gluScaleImage(GLenum format, GLint widthin,
. . http://ani.cs.msu.su

, ,

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GL height in, GLenum t ypein, const void *datain, GLint widthout, GLint heightout, GLenum t ypeout, void *dataout) format GL_RGB GL_RGBA, . widthin, height in, widhtout, heightout , typein t ypeout , datain dataout. , GL_UNSIGNED_BYTE, GL_SHORT, GL_INT . , dataout. -, , . , . , . . , 2mx2n max(m,n)+1 , . vo id gluBuild2DMipmaps(GLenum target, GLint components, GLint width, GLint height, GLenum format, GLenum t ype, const void *data) target GL_TEXTURE_2D, components , 1 4 (1- ,2- alpha, 3-, , , 4- ). width, height, data , format type , gluScaleImage(). OpenGL , , target GL_TEXTURE_1D. gluBuild1DMipmaps(), height.
. . http://ani.cs.msu.su

1xN,

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, OpenGL , . , vo id glGenTextures(GLsizei n, GLuint*textures) n , textures. vo id glBindTexture(GLenum target, GLuint texture) target GL_TEXTURE_1D GL_TEXTURE_2D, texture , . , , glBindTexture() c target texture. , glBindTexture() texture, , , . , , , , . , , , . (s,t), s t [0,1]. : vo id glTexParameter[i f](GLenum target, GLenum pname, GLenum param) vo id glTexParameter[i f]v(GLenum target, GLenum pname, GLenum *params) target , , pname , , param params . pname: GL_TEXTURE_MIN_FILTER param , . GL_NEAREST (), GL_LINEAR . : GL_LINEAR. GL_TEXTURE_MAG_FILTER param , () . GL_NEAREST (), GL_LINEAR . : GL_LINEAR. GL_TEXTURE_WRAP_S param s, [0,1]. GL_REPEAT s , . GL_CLAMP : 0 1, , . : GL_REPEAT.

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GL_TEXTURE_WRAP_T t.

,

GL_NEAREST , , GL_LINEAR . , , , , vo id glTexEnv[i f](GLenum target, GLenum pname, GLt ype param) vo id glTexEnv[i f]v(GLenum target, GLenum pname, GLt ype *params) target GL_TEXTURE_ENV, pname GL_TEXTURE_ENV_MODE, . param : GL_MODULATE . GL_REPLACE . GL_BLEND . . : , . vo id glTexCoord[1 2 3 4][s i f d](type coord) vo id glTexCoord[1 2 3 4][s i f d]v(type *coord) glTexCoord2..(type s, type t), . , , . , GLU , , , , . vo id gluQuadricTexture(GLUquadricObj*quadObject, GLboolean textureCoords) textureCoords GL_TRUE, . vo id glTexGen[i f d](GLenum coord, GLenum pname, GLt ype param) vo id glTexGen[i f d]v(GLenum coord, GLenum pname, const GLtype *params) coord GL_S, GL_T; pname GL_TEXTURE_GEN_MODE, GL_OBJECT_PLANE, GL_EYE_PLANE. params , param GL_OBJECT_LINEAR, GL_EYE_LINEAR, GL_SPHERE_MAP.
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, , . , . s_coeffs t_coeffs (1,0,0,1) (0,1,0,1) , : glEnable(GL_TEXTURE_GEN_S); glTexGeni(GL_S, GL_TEXTURE_GEN_MODE, GL_EYE_LINEAR); glTexGendv(GL_S, GL_EYE_PLANE, s_coeffs); t .

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.
[1] William E. Lorensen, Harvey E. Cline. CG vo l.21, no.4, July 1987 [2] Bernardo P. Carneiro, Arie E. Kaufman. SIGGRAPH'96, pp. 205-210 [3] Andre Gueziec , IEEE Transact ions on Visualization and Co mputer Graphics. Vol. 1, Issue 4, pp. 328 - 342, December 1995 [4] Vaclav Skala , Conference on Scient ific Co mputing 2000, pp. 368 - 378. http://www.emis.de/journals/AMUC/_contributed/algo2000/skala.pdf [5] Paolo Cignoni. IEEE Transact ion on visualization and CG, vol.3, no.2 April-June 1997. [6] D. V. Mogilenskikh, I. V. Pavlov. «Color interpolat ion algorithm in visualizing result s of numerical simulat ions». Visualizat ion and imaging in transport pheno mena. Annals o f the New York Academy o f Sciences. v. 97, pp. 43-52, 2002. [7] A. Hilton, A. J. Stoddart, J. Illingworth, T. Windeatt. International Conference on Image Processing, 1996. ftp://ftp.ee.surrey.ac.uk/pub/vision/papers/hilton-icip96.ps.Z [8] Tasso Karkanis , A. James Stewart. IEEE Computer Graphics and Applicat ions, v.21 n.2, p.60-69, March 2001 [9] B. Crespin, P. Guitton, C. Schlick. Proceedings of CSG'98, 1998 [10] Marshall Bern and David Eppstein. Comput ing in Euclidean Geometry Eds. World Scient ific, 1992, pp. 23-90 [11] M. Durst, "Letters: Addit ional Reference to Marching Cubes," Computer Graphics, vo l. 22, no. 2, pp. 72-73, 1988. [12] C. Rocchini, P. Cignoni, F. Ganovelli. International Conference on Shape Modeling & Applicat ions, 2001. http://smi2001.ima.ge.cnr.it/abstracts/47.pdf [13] Yutaka Ohtake, Alexander G. Belyaev. ACM Symposium on So lid Modeling and Applicat ions, pp. 171 - 178, 2002. http://cis.k.hosei.ac.jp/~F-rep/SM02ob.pdf [14] Y. Ohtake, A. G. Belyaev. Proceedings of the International Conference on Shape Modeling & Applications, Page: 74, 2001. http://cis.k.hosei.ac.jp/~F-rep/OhtakeSmi01.pdf [15] Gregory M. Nielso n, Bernd Hamann. IEEE Visualization, Proceedings of the 2nd conference on Visualizat ion '91, pp.83 - 91, 1991 [16] Adriano Lopes, Ken Brodlie. IEEE Transact ions on Visualizat ion and Computer Graphics, pp 16-29, 2002 [17] S. V. Matveev. Proceedings IEEE Visualizat ion '94, pp. 288-292 [18] Andrew P. Witkin, Paul S. Heckbert. Proceedings o f the 21st annual conference on Computer graphics and interact ive techniques, pp: 269 - 277, 1994. www.cs.cmu.edu/~aw/pdf/particles-reprint.pdf [19] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle. SIGGRAPH'92 proceedings, 26(2), pp. 71-78. [20] T. Itoh and K. Ko yamada. «Automat ic Isosurface Propagation using an Extrema Graph and Sorted Boundary Cell Lists» IEEE Transactions on Visualizat ion and Co mputer Graphics 1(4) pp.319-327, 1995

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[21] C. L. Bajaj, V. Pascucci, D. R.Schikore. «Fast Isocontouring for improved Interactivit y» In 1996 Symposium o n Vo lume Visualizat ion Proc., San Francisco (CA), Oct. 28-29, pp.39-46, 1996. [22] H. Shen, C. D. Hansen, Y. Livnat and C. R. Johnos. « Isosurfacing in Span Space wit h Utmost Effeciency (ISSUE)» in Visualisat ion 96' Conf. Proc., San Francisco, Oct. 27-Nov.1, pp. 287-294,1996. [23] J. Wilhelms and A. Van Gelder. «Octrees for faster isosurface generation» ACM Transact ion on Graphics, 11(3) pp.201-227, July 1992. [24] ., .. .. . .: . 1989. 622 . [25] .... . .: . 1989. 488 c. [26] . . . .: . 1990. 192 . [27] .. . .1 . 2. .: . 1982. [28] . . . « ». VII . . 2003. [29] .., .., ... « OpenGL». - , , 2003 . [31] .-. , . . . . .: . 1993. [32] . . . . .: . 2000. [33] . . . . .: . 2004.

[30] . . . : . 2001

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