Differences

This shows you the differences between two versions of the page.

--- en:mappingscomplexspace [2026/03/03 17:09] – frankbrennecke
+++ en:mappingscomplexspace [2026/03/08 11:30] (current) – frankbrennecke
@@ Line 1: / Line 1: @@
 {{symbol:icon_AbbildungenImKomplexenRaum.png?64x64}}
-====== Complex space: Images in complex space ======
+====== Complex space: Mappings in complex space ======
-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.
+This graphic is the most versatile and comprehensive mapping 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 magnitude 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 ℝ<sup>3</sup> 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.
+Strictly speaking, the space in which the drawing is made is not spanned on a complex plane; it is a normal ℝ<sup>3</sup> 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 complex space.
-{{screen_EN:scr_AbbildungenRaum.png}}
+{{screen_EN:scr_FiguresSpaceC.png}}
 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.
@@ Line 13: / Line 13: @@
 CGRAPH then varies the variables u and v according to the number of support points specified in the [[3D graphics|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 n<sup>2</sup> 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 [[flaechenimraum|surfaces in space]]. The speed of the 3D engine used limits the maximum possible number of support values here.
+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 n<sup>2</sup> 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 [[flaechenimraum|surfaces in space]]. The speed of the 3D engine used limits the maximum possible number of support values here.
+For each of the three functions, you can choose whether their **//real part//**, their **//imaginary part//** or their **//value//** should be plotted on the respective coordinate axis.
 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<sub>i</sub> to the next support value w<sub>i+1</sub>. See the three examples below.
-After clicking the **//Create graphic//** button, CGRAPH switches to the **//Display//** tab and displays the graphic.
+After clicking the **//Create graphic//** button, CGRAPH switches to the **//Graphics//** tab and displays the graphic.
-{{:screen:scr_AbbildungenRaumsample1.png?300x260}} {{:screen:scr_AbbildungenRaumsample2.png?300x260}} {{:screen:scr_AbbildungenRaumsample3.png?300x260}}
+{{:screen:scr_AbbildungenRaumCsample1.png?300x260}} {{:screen:scr_AbbildungenRaumCsample2.png?300x260}} {{:screen:scr_AbbildungenRaumCsample3.png?300x260}}
-The three graphics show the same image with the functions f<sub>x</sub>=u*cos(v) and f<sub>y</sub>=u*sin(v) – together, these form a circular surface with u as the radius and v as the angle – and the function f<sub>z</sub>=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.
+The three graphics show the same image with the functions f<sub>x</sub>=w*cos(v)*sin(u) and f<sub>y</sub>=w*sin(v)*sin(u) – both together form a surface circle – and the function f<sub>z</sub>=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, resulting in a number of stacked circular planes (here 5, because the number of support values has been set to 5). If, as in the middle image, w is increased with v, a screw is created. In the third image, w is increased with u, resulting in paraboloids.
 This automatic increase is merely a simplification. It can also be achieved by using u and v as variables within the function f<sub>z</sub>.
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 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 [[BedienungParameterR|notes on changing the parameter in the graphic display]].
-{{:screen_en:scr_AbbildungenRaumbaum.png}}
+{{:screen_en:scr_AbbildungenRaumCbaum.png}}
 Double-clicking on the text next to the coloured symbol allows you to [[Stroke3DExt|set the colour and appearance]] of the graph. Clicking on the coloured symbol itself hides or shows the graph.
@@ Line 37: / Line 39: @@
 [[OperatingGraphics3D|Notes on operating the graphic display]]
-[[complexSpace3D|Back to overview]]
+[[complexspace3d|Back to overview]]