Topic 7. 3D graphics. Animation

In 3D space, coordinate systems are widely used: Cartesian, cylindrical, spherical, and others.

There are types of surface equations:

• obvious;
• implicit (canonical);
• parametric, etc .;
• polar

Surfaces of the 2nd order (quadrics)

The canonical equation of the surface of the 2nd order is quadratic (Table).

Cylindrical surfaces

Cylindrical surfaces are formed by the displacement of a direct (creature) parallel to a certain vector and cross a fixed curve (guideline).

The equation defines a cylindrical surface with a creature parallel to the axis.

Operators of surface construction

The Maple core includes only two commands for building 3D graphics - plot3d and smartplot3d.

To construct with options, the plot3d command is used in the formats:

• plot3d (expr1, x = a..b, y = c..d, options) ;
• plot3d ([expr1, expr2, expr2], u = a..b, v = c..d, options) .

Побудова кількох поверхонь, що визначаються виразами expr1, expr2, ..., залежними від змінних var1[a, b], var2[c, d], здійснюється за допомогою команди:

plot3d({expr1, expr2, ..., exprn}, var1=a..b, var2=c..d, options).

Функції тривимірної графіки:

• contourplot(expr, var1 = a..b, var2 = c..d, options) - виведення ліній рівня функції (expr) змінних (var1, var2), яка ідентична plot3d з опцією style=CONTOUR.
• cylinderplot(expr, var1 = a..b, var2 = c..d, options) - поверхня, яка визначається виразом expr, в циліндричних координатах,
• implicitplot3d(expr=g, var1=a..b, var2=c..d, var3=e..f, options) - зображення поверхні expr від var1, var2, var3.
• matrixplot(M) - поверхня, заданої таблицею M.
• pointplot([[x1, y1, z1], ... [xn, yn, zn]], options) - виведення точок, заданих координатами x, y, z.
• polygonplot3 d ([pnt1, ..., pntn]) - output of the three-dimensional n-angle given by the points with coordinates pnt1 , ..., pntn (the first point connects to the last one).
• spacecurve ([exprx, expry, exprz], var = a..b, options) - construction of the curve in the space given parametrically exprx ( X- co-ordinate), expry ( Y ) and exprz ( Z ). Expressions exprx , expry , exprz must depend on one variable var .
• sphereplot (expr, var1 = a..b, var2 = c..d, options) - construction of a surface in spherical coordinates. The expression expr is the radius depending on two variables var1 and var2 .
• surfdata ([[x1, y1, z1], ... [xn, yn, zn]], options) - construction of a surface that passes through given points (given by its coordinates in space ( x, y, z )).
• tubeplot (exprx, expry, exprz, var = a..b, radius = r) is the construction of a surface determined by the parametric curve (given by the expressions exprx (X-coordinate), expry (Y) and exprz (Z) and radius r ). Expressions exprx , expry , exprz must depend on one variable var .
• setoptions3 d (options1 = exp1, ..., optionsn = expn) - control of the options of three-dimensional graphics accepted by default.
• display3 d ([pic1, pic2, ..., picn]) - construction of n three-dimensional graphic structures in the common axes of coordinates.
• coords = opt - type of coordinates ( CARTESIAN, SPHERICAL or CYLINDRICAL ),
• title = string - the output of the header contained in the string string ,
• axes = opt - the type of coordinate axes ( FRAME, NORMAL, BOXED, NONE ),
• scaling = opt - type of zoom ( UNCONSTRAINED, CONSTRAINED ),
• orientation = [angle1, angle2] - angles of angle in degrees,
• view = az .. bz or view = [ ax .. bx, ay .. by, az .. bz] - part of the image is displayed,
• projection = n - projection type, given by the number n [0,1] ,
• style = opt - output style ( POINT - points, LINE lines, HIDDEN - grid with removal of invisible lines, PATCH - filler, WIREFRAME - grid with conclusion of invisible lines, CONTOUR - line level, PATCHCONTOUR - level lines);
• shading = opt - the function of the filler density ( Z- axis Z , XY - depending on the coordinates XY, XYZ, ZGREYSCALE - on the Z- axis, gray halves , ZHUE , NONE ),
• grid = [ i 1, i 2] is the number of nodes on the X and Y axes to calculate the surface ( grid = [25,25] ),
• numpoints = n - the number of points deduced, is equivalent to grid = [,],
• color = opt - the color of the surface ( red, blue , ..),
• light = [ angl1, angl2, numr, numg, numb] - the brightness function in the angles angl1 , angl2 is determined by the percentage of red ( numr ), green ( numg ) and blue ( numb ), varying from 0 to 1,
• ambientlight = [ numr, numg, numb] - brightness in each direction,
• labels = [ str1, str2, str3] - inscriptions on the axes of coordinates given str1 , str2 , str3 .

Package plottools

The plottools package includes functions: [annulus, arc, arrow, circle, cone, cuboid, curve, cutin, cutout, cylinder, disk, dodecahedron, ellipse,ellipticArc, exportplot, extrude, getdata, hemisphere, hexahedron, homothety, hyperbola, icosahedron, importplot, line, octahedron, parserpipes, a connection, point, polygon, prism, project, rectangle, reflect, rotate, scale, sector, semitours, sphere, stellate, tetrahedron, torus, transform, translate ].

The chart the plot plottools is for construction graphicprimitives on a plane and in space - segments of straight lines, circles, cones, etc.

Most functions have a clear syntax. For example, a cone is constructed using the primitive cone (c, r, h ...), where c is the list of coordinates of the center, r is the radius of the base, h is the height, etc.