Real space: mappings in 3D space
This graphic offers the most comprehensive mapping options in real 3D space. For each of the coordinate axes x, y and z, a separate mapping function can be defined that depends on three variables u, v and w. This allows, for example, plane families or overlapping three-dimensional figures to be generated.
To emphasise the independence of the variables from the coordinate axes, only the names u, v, w and param are possible here. Enter the desired term in the input fields for the functions, or select one of the existing functions 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, resulting in a total of 16 times as much data 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 wi to the next support value wi+1. See the three examples below.
After clicking on the Create graphic button, CGRAPH switches to the Graphics tab and displays the graphic.
The three graphics show the same image with the functions fx=u*cos(v) and fy=u*sin(v) – together, these form a circular surface with u as the radius and v as the angle – and the function fz=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 (two in this case, because the number of support values is fixed at two). 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 fz.
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.