In this section most features of surf are explained. Many of these features can be invoked from the graphical user interface. All features can be invoked through surf's command language. Command language features are only explained if not accessible through the GUI. For a complete reference to the command language, have a look at the next section.
To draw a plane curve, enter the equation into surf's text
window preceded by curve= and followed by a semicolon.
Then press the button draw curve.
Some seconds later the curve will show up in the window titled
color image.
By default the curve is drawn inside the rectangle
-10.0 <= x,y <= 10.0and is clipped at a circle with radius 10.0. The x-axis is horizontal pointing to the right, the y-axis is vertical and points upwards. By default the image size is 200 x 200 pixels. The image size can be altered by setting width and height in the main window.
The view can be altered in the position window: A different origin can be specified by setting origin x and origin x. A rotation with center at (0,0) can be specified by setting rotation about z-axis. The curve may be scaled by setting scale factor x and scale factor y. The appearance of the curve can be altered in the curve window.
The clipping area can be specified in the clip window. For a curve the only reasonable values are sphere and none.
An arbitrary color can be given to the curve by setting curve red,curve green and curve blue to appropriate values in the curve window. The curve width can be set by changing curve width. A high value of curve gamma sharpens the curve, whereas a low value blurs the curve.
To draw a surface, enter its equation into surf's text window
preceded by surface= and followed by a semicolon. Then press the button
draw surface.
Some more seconds later the surface will appear. By default, the surface
is calculated inside the cube
-10.0 <= x,y,z <= 10.0and clipped at a sphere of radius 10.0. The x-axis is horizontal pointing to the right, the y-axis is vertical and points upwards. The z-axis points to you. The spectator is located at (0,0,25) by default.
Changing the view can be done by altering the settings in the position window. A different origin may be specified by setting origin x, origin y and origin z. To rotate the surface one can set rotation about x-axis, rotation about y-axis and rotation about z-axis to appropriate values. Rotation is performed on the following order: y-axis, x-axis, z-axis. To scale the surface set scale factor x, scale factor y and scale factor z to desired values. It is also possible to switch from central perspective to parallel perspective.
Illumination and color can be altered in the light window. The direction of the normal vector given by the gradient of the surface equation defines one side of the surface which is regarded as outside. You can specify a color for this side by setting surface red, surface green and surface blue. The other side of the surface (inside) can be given a different color by specifying inside red, inside green and inside blue.
Currently only the Phong illumination model is implemented. Therefore the intensity of the surface in one point consists of four components which are calculated separately:
These four light components are added with weights ambient, diffuse, reflected and transmitted.
The number of light sources is limited to nine. For every light source, the position, the color and the intensity can be specified.
The clip window allows to specify a different clipping area. Here the center and radius of the clipping area may be specified. Additionally a front and a back clipping plane may be specified.
To draw one or more hyperplane sections of an algebraic surface,
just specify the hyperplane by setting the global variable
plane to its equation.
The section is drawn when the command cut_with_plane is interpreted.
For example:
rot_x=0.3; // a nice rotation
rot_y=0.2;
surface=x^2*y^2+y^2*z^2+z^2*x^2-16*x*y*z;
clear_screen; // draw the steiner roman surface
draw_surface;
curve_red=0;
curve_green=255;
curve_blue=0;
curve_width=5;
curve_gamma=1.2;
plane=x+y+z; // draw a green hyperplane section
cut_with_plane;
plane=x+y+z+4.0; // draw another one
cut_with_plane;
curve_red, curve_green and curve_blue.
The width of the section is altered by setting curve_width
to any suitable value. A high value of curve_gamma
(eg. 10.0) makes the curve
look very pixelized, whereas a small value (eg. 1.0) makes the section
look blurred.
Multiple curves can be drawn in script files just by NOT clearing the screen. This works fine for plane curves. Just consider the following example:
do_background=yes;
clear_screen;
curve=y^2-x^2*(x-1);
draw_curve; // draw a cubic
do_background=no;
curve=x;
draw_curve; // draw y-axis
curve=y;
draw_curve; // draw y-axis
Multiple surfaces can be drawn by specifying up to 9 surfaces
in the variables surface, surface2 ...
surface9. Additionally it is possible to draw on every surface
any number of hyperplane sections.
rot_x=0.69; // a nice rotation
rot_y=0.35;
illumination=ambient_light + // specify illumination
diffuse_light + // model
reflected_light +
transmitted_light;
transparence=35; // set transparence for surface no 1
transparence2=35; // set transparence for surface no 2
surface=x^2+y^2+z^2-30; // first surface: a sphere
surface2_red=255; // second surface: a red steiner surface
surface2_green=0;
surface2_blue=0;
surface2=x^2*y^2+x^2*z^2+y^2*z^2-16*x*y*z;
clear_screen;
draw_surface; // draw the surface
curve_width=5;
curve_red=0;
curve_green=255;
curve_blue=0;
plane=x+y+z-6.0; // draw a green hyperplane section
surf_nr=1; // on the sphere
cut_with_plane;
curve_red=0;
curve_green=255;
curve_blue=255;
plane=x+y+z+4.0; // draw a turquoise hyperplane section
surf_nr=2; // on the steiner surface
cut_with_plane;
Given a polynomial function f(x,y) and a set of levels
z1, ... ,zn, surf can visualize the graph
z=f(x,y) and all isoline for the levels
z1, ... ,zn as follows:
rot_x=-0.8;
clear_screen;
poly f=x^2+y^2; // graph of (x,y)->x^2+y^2
surface=z-f;
draw_surface; // draw the graph
curve_width=3; // width of isoline
plane=z-1;
cut_with_plane; // draw isoline f(x,y)=1
plane=z-2;
cut_with_plane; // draw isoline f(x,y)=2
plane=z-3;
cut_with_plane; // draw isoline f(x,y)=3
plane=z-4;
cut_with_plane; // draw isoline f(x,y)=4
plane=z-5;
cut_with_plane; // draw isoline f(x,y)=5
plane=z-6;
cut_with_plane; // draw isoline f(x,y)=6
plane=z-7;
cut_with_plane; // draw isoline f(x,y)=7
plane=z-8;
cut_with_plane; // draw isoline f(x,y)=8
plane=z-9;
cut_with_plane; // draw isoline f(x,y)=9
f is not polynomial, try to expand
calculate its Taylor series. Since the new root algorithms work fine
with polynomials of degree up to 30, you might approximate f
by its Taylor series. If your function is piecewise defined, better
use another program.
The position window provides an interface to adjust the curve/surface position. You can set the 9 buttons into the three modes translate, rotate and scale.
If you try to draw a surface and give the equation to surf, the resulting image normally does not look nice at all. You have to find the right scaling, rotation and so on. Often you want to see immediately what happens if you change some value. But it simply takes surf too long to calculate one image. Here comes the preview in. Setting the preview buttons in the main window to 3x3 has the effect that only every 9th pixel is calculated, setting it to 9x9 only every 81st pixel is calculated. But one can still get an impression of what the image looks like, AND computation is speeded up by the factor 9 resp. 81.
Up to two preview buttons can be pressed at one time. If for example 9x9 and 1x1 are pressed, then the image will be calculated in three steps. First, every 81st pixel, after that every 9th pixel and finally every pixel will be calculated.
Especially in animations aliasing is very disturbing. Therefore if in the display window, antialiasing level is set to a value n > 1, then in a second pass all pixels differing by a value of at least antialiasing threshold from one of their neighbours are refined. Exactly n^2+1 intensity values are calculated. In most cases an antialiasing level of 4 will remove aliasing.
On a nifty machine surf is fast enough to provide a real time animation of an algebraic curve of degree < 5. For example
// --------------------------
// animation of a cubic curve
// --------------------------
clear_screen;
double a=-10.0;
loop:
curve=y^2-(x^2-1)*(x-a);
clear_pixmap;
draw_curve;
a=a+0.1;
if( a <= 10.0 ) goto loop;
// --------------------------
// the 4-nodal cubic rotating
// --------------------------
width=200;
height=200; // set image size
double sf=0.3;
scale_x=sf;
scale_y=sf;
scale_z=sf; // set scaling
double Pi=2*arccos(0);
double w2=sqrt(2); // define some constants
poly p=1-z-w2*x;
poly q=1-z+w2*x;
poly r=1+z+w2*y;
poly s=1+z-w2*y; // define tetrahedral coordinates
poly cubic=4*(p^3+q^3+r^3+s^3)-(p+q+r+s)^3; // the cubic
int i=0;
loop:
surface=rotate(cubic,2*Pi/100*i,zAxis); // rotate the cubic
clear_screen;
draw_surface; // draw the cubic
filename="cubic"+itostrn(3,i)+".ras";
save_color_image; // save the image
i=i+1;
if( i < 100 ) goto loop; // repeat 100 times
Have you ever watched one of those films with that red and green glasses? surf tries to accomplish exactly this effect when you set eye distance in the display window to a value greater than zero. The following situation is simulated: The spectator is located at (0,0,spectator z) and the distance between his eyes is eye distance. The surface will appear at the z-coordinate distance from screen. Furthermore it is possible to adjust to specific red-green or red-blue glasses by setting left eye red value, right eye green value end right eye blue value. In particular it is assumed that the right eye wears the red glass.
If a color image of a surface/curve has been calculated, this image can be mapped to a black and white image by pressing the button dither surface or dither curve. The second one is just designed for dithering curves. The appearance of the black and white image can be altered/adjusted in several ways in the dither window. Since the mapping itself is done by dithering, the dithering algorithm can be specified. Currently available are seven algorithms coming in three groups:
The surfaces on black and white images often don't look very impressive; often it is hard to detect the edges of a surface. An algorithm called enhancing the edges avoids this drawback. This algorithm takes a value alpha in [0,1] as input. Best results are achieved with alpha around 0.9.
The intensity of the background on the black and white image can be specified by altering the value background to any value in [0.1]. Here 0 is black whereas 1 means white.
The tone scale adjustment maps intensity values between 0 and 0.1 to 0, values between 0.1 and 0.9 linear to [0,1] and values between 0.9 and 1 to 1. This is used to enhance the contrast of an image. An additional gamma correction can be also performed to correct the linearity of an output device.
By specifying pixel size one can correct the printer pixel size: A value of 50 means that the radius of a pixel is exactly half the distance between two neighbouring pixels. A value of 100 says that the radius of a pixel is exactly the distance between two neighbouring pixels.
The heart of surf is an algorithm which determines all roots of a polynomial in one variable. Currently you can choose between seven methods in the numeric window. The first six methods use a chain of derivatives to determine intervals where the polynomial has exactly one root. They differ by the iteration method which is used to find the roots in these intervals. Some of the iteration methods were just implemented out of academic interest. However, they all work. The last method uses Rockwoods all roots algorithm: the polynomial is converted into a bezier function and the roots of the bezier function are approximated by the roots of the control polygon.
For curves/surfaces of degree less than ten, all methods work. When the degree gets higher, best results are achieved by the bisection, the Newton and the bezier all roots method. At last, for a degree higher than 30 only the bisection methods seems to work (up to degree 50). If a curve has multiple components, the bisection and the Newton method tend to produce the best results.
Moreover it is possible to specify a numerical precision epsilon which is used in all root finders. Additionally the maximal number of iterations of the iteration methods can be specified.
surf can store color images in one of several file formats. In the save color image window you can choose between
surf can store black and white images in different file formats. We have implemented