This graphic is the most versatile and comprehensive imaging option in GGRAPH. For each of the coordinate axes x, y and z, a separate complex mapping function can be defined, which depends on three equally complex variables u, v and w. For each of the functions, it can be specified whether their real part, imaginary part or absolute value is used as a reference for the coordinate axis. This allows even the most complex mappings and three-dimensional figures to be generated.

Strictly speaking, the space in which the drawing is made is not spanned on a complex plane; it is a normal ℝ³ with real number values for the x, y and z coordinates. However, since a function with complex input values is used for each coordinate, this graphic is assigned to the complex space.

To emphasise the independence of the variables from the coordinate axes, only the names u, v, w and param are possible here. The variables may also take complex values; only the parameter is restricted to real numbers. Enter the desired terms in the input fields for the functions, or select one of the existing terms by clicking on the arrow to the right of the input field. All three terms can depend on u, v and w.

The values for the start and end of the variables u, v and w can be set separately. The only requirement is that the start and end must not be identical.

CGRAPH then varies the variables u and v according to the number of support points specified in the settings. The number of support values for the variable w can be specified here – a maximum of 16 iterations are possible.

An explanation for this restriction is provided by the way in which CGRAPH generates the images in space. The mapping over u and v (with n iterations each, i.e. a total of n² calculations) is repeated for each w – for w=16, this means 16 times, so that a total of 16 times as much data is generated as for the surfaces in space. The speed of the 3D engine used limits the maximum possible number of support values here.

For simplicity, the variable w can also be automatically increased with each additional v and u within its own support values. It then increases continuously from the support value w_i to the next support value w_i+1. See the three examples below.

After clicking the Create graphic button, CGRAPH switches to the Display tab and displays the graphic.

The three graphics show the same image with the functions f_x=u*cos(v) and f_y=u*sin(v) – together, these form a circular surface with u as the radius and v as the angle – and the function f_z=w, in which the z-coordinate is simply increased from support value to support value. In the first image, w remains unchanged when u and v are increased – this results in a number of stacked circular planes (here 2, because the number of support values is set to 2). If, as in the middle image, w is increased with v, a helix is created because w is continuously increased with the angle. In the third image, w is increased with u, resulting in cones because the radius is the relevant variable here.

This automatic increase is merely a simplification. It can also be achieved by using u and v as variables within the function f_z.

The default values can be viewed in the tree on the left. Clicking on the (optional) parameter also activates the parameter's adjustability at the bottom. An explanation of this function can be found in the notes on changing the parameter in the graphic display.

Double-clicking on the text next to the coloured symbol allows you to set the colour and appearance of the graph. Clicking on the coloured symbol itself hides or shows the graph.

The default values can be changed later – clicking on one of the values switches CGRAPH back to the tab with the default values. The graphic is adjusted as soon as the Create graphic button is clicked.

Notes on operating the graphic display

Back to overview