Special techniques for height surfaces, real and complex functions

Special techniques for height surfaces, real and complex functions#

The 3D Viewer#

The formatting of 3D bitmap graphics is done with the following app:

../_images/App3D.png

More info to follow.

General Background#

Rod Stephens:

WPF 3d: Three-Dimensional Graphics with WPF and C

Herausgeber: CreateSpace Independent Publishing Platform

Erscheinungstermin: 8. Februar 2018

ISBN-10: 1983905968

ISBN-13: 978-1983905964

Github: WriterRod/WPF-3d-source

3D Bitmaps: Export to JPG and PNG#

These formats are natively supported.

Axes in 3D#

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Positioning 3D objects#

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Resizing 3D objects#

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Rotating 3D objects#

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Translating 3D objects#

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Solid colors#

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Wireframes#

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Textures#

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Transparency#

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Wpf figures as standard display of complex functions#

This is a description of the standard display of complex functions, using the square function as an example:

The figures below are showing the real part (left figure), imaginary part (middle figure) and absolute value with color-coded phase (right figure) of the complex function \(z = \mathrm{sqr}(x + iy)\) with \(-2 \le x \le 2\) (blue axis), \(-2 \le y \le 2\) (red axis), \(-10 \le z \le 10\) (green axis).

01a_TestSquare_re1 \(\quad\) 01b_TestSquare_im1 \(\quad\) 01c_TestSquare_abs1

Note

Although the range of the green axis is stated in the form \(z_{\text{min}} \le z \le z_{\text{max}}\), with \(z_{\text{min}} \ne 0\) in general, this applies only for the figures showing the real and imaginary part. For the figure showing the absolute value we have always \(z_{\text{min}} = 0\). This note is omitted from the standard text describing complex functions in this manual for better readability.

Truncation vs loglog transformation#

TestTruncated \(\quad\) TestLogLog

Left figure: Surface plot without cutting of the branch cut. Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Right figure: Surface plot with cutting of the branch cut. Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Displaying branch cuts#

TestWithoutCut \(\quad\) TestWithCut

Left figure: Surface plot without cutting of the branch cut. Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

Right figure: Surface plot with cutting of the branch cut. Camera angles are \(\theta=135^\circ\) and \(\phi = -12^\circ\).

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