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3. Features

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.

3.1 Plane curves

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 <= 10.0
and 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.

3.2 Surfaces

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 <= 10.0
and 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:

Ambient light is a constant which represents the light a point on the surface receives from the whole environment (the sky, the floor, the lawn ...) but not from the light sources. Diffuse light is the light the point receives from the light sources and which is reflected equally in every possible direction. The amount of diffuse light is independent of the spectator position, it is proportional to the cosine of the angle between the normal vector and the vector from the point to the light source. Reflected light is the light from the light sources which is reflected specular from the surface point. Its amount is proportional to a power of the cosine of the angle between the vector from the point to the spectator and the specular reflection vector from the light source. If a high power of the cosine is taken, the surface will appear shiny, whereas a low power of the cosine lets the surface look rough. Therfore this power is labelled smoothness. Transmitted light comes in if a surface is transparent. A constant called transparence specifies the percentage of light which passes through the surface. Algebraic surfaces are infinitesimally thin. However our eye is not used to such objects, so we pretend that our surfaces have a constant thickness. Specifying a positive thickness for a transparent surface results in a loss of transparency in the places where the surface normal does not point to the spectator.

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.

3.3 Hyperplane sections

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
    clear_screen;                     // draw the steiner roman surface
    plane=x+y+z;                      // draw a green hyperplane section
    plane=x+y+z+4.0;                  // draw another one

The color of the hyperplane section can be set by specifying 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.

3.4 Multiple curves/surfaces

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:

    draw_curve;             // draw a cubic
    draw_curve;             // draw y-axis
    draw_curve;             // draw y-axis

Not that every curve will be drawn just over all curves that have been draw so far.

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
    illumination=ambient_light +     // specify illumination 
                 diffuse_light +     // model
                 reflected_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
    draw_surface;                    // draw the surface
    plane=x+y+z-6.0;                 // draw a green hyperplane section
    surf_nr=1;                       // on the sphere
    plane=x+y+z+4.0;                 // draw a turquoise hyperplane section
    surf_nr=2;                       // on the steiner surface

3.5 Graphs and isolines

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:

    poly f=x^2+y^2; // graph of (x,y)->x^2+y^2
    draw_surface;      // draw the graph
    curve_width=3;     // width of isoline
    cut_with_plane;    // draw isoline f(x,y)=1
    cut_with_plane;    // draw isoline f(x,y)=2  
    cut_with_plane;    // draw isoline f(x,y)=3
    cut_with_plane;    // draw isoline f(x,y)=4
    cut_with_plane;    // draw isoline f(x,y)=5
    cut_with_plane;    // draw isoline f(x,y)=6
    cut_with_plane;    // draw isoline f(x,y)=7
    cut_with_plane;    // draw isoline f(x,y)=8
    cut_with_plane;    // draw isoline f(x,y)=9

If however your function 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.

3.6 Interactive positioning

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.

3.7 Preview

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.

3.8 Anti aliasing surfaces

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.

3.9 Animations

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
    // --------------------------
    double a=-10.0;
    if( a <= 10.0 ) goto loop;

calculates some 200 curves. In a 200x200 window, surf shows me about five frames per second on a sparc 20. However, real time animations of algebraic surfaces are still beyond computation power (or do you call a 200-processor-machine your own?). But you can calculate a series of images with surf and convert this series of images to the movie format of your choice.
    // --------------------------
    // the 4-nodal cubic rotating
    // --------------------------
    height=200;                  // set image size
    double sf=0.3;
    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;
        surface=rotate(cubic,2*Pi/100*i,zAxis); // rotate the cubic
        draw_surface;                           // draw the cubic
        save_color_image;                       // save the image
    if( i < 100 ) goto loop;                    // repeat 100 times

Here some 100 SUN rasterfiles are created. Afterwards you could use some tool to convert these single images to a movie.

3.10 Stereo pictures

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.

3.11 Black & white images

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:

Dithering with blue noise

All three filters are based on the same idea of error distribution. Floyd Steinberg is the simplest one, whereas Stucki differs from Jarvis only by its weights. They tend to produce disturbing patterns if they process large areas of intensity near 0.5. Therefore one can let them proceed in a serpentine fashion, which reduces the patterns. Nearly all patterns disappear if the weights are disturbed randomly. The algorithms are best for use with low resolution printers, typically 300 dpi. Some (most?) 600 dpi laser printers do not like these algorithms, since they do not like isolated pixels.

Dithering with ordered dither

The clustered dot ordered dither is a fast method and produces satisfying results in combination with high resolution printers (600 dpi and more). The second algorithm is for use with low resolution printers. Both perform no error distribution. Depending on the printer resolution and the number of emulated gray levels, one can choose the pattern size:

Hybrid methods

Both algorithms combine clustered dot ordered dither and error distribution. Depending on the printer resolution on can choose the number of barons in a 8x8 matrix to be The barons are the bad guys in a matrix which get all the error left over from the good guys.

The black & white problem

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.

3.12 Algorithms

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.

3.13 Output

surf can store color images in one of several file formats. In the save color image window you can choose between

Additionally the color space can be chosen among The first colormap is just the 6x6x6 colormap Netscape uses. The second one results from an octree algorithm which chooses the most used 216 colors among all colors of the image. Storing an image in True color results in better quality, but bigger file size.

surf can store black and white images in different file formats. We have implemented

For postscript and encapsulated postscript also the resolution may be specified among These settings may be chosen in the save dithered image window. When using postscript, the image will (regardless its size) appear centred on the side (which is assumed to be a4).

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