Some 2D Math Pictures

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Christian Gaier creates colorful geometric-mathematical pictures and videos as hobby. He developed a set of mathematical functions to visualize basic geometric objects (line, circle and arc) in a surrounding colorful field. These functions can be easily combined to create more complex geometric structures. The main idea is that these functions deliver zero values at the position of the geometric object. Therefore, combinations of these functions can be easily realized by simple multiplications. The software units for picture computation are written in the programming language C.

Formule

f = \sqrt[N]{\prod\limits^N_{i=1}\left|\sqrt{(x-x_i)^2+(y-y_i)^2}-r_i\right|}

Cybernetic Kaleidoscope 1

A superposition of many circles. The basic function for one circle is a cone with the apex pointing down and compressed up. The final function for N circles is the geometric mean value of N basic functions.

Auteurs Christian Gaier

Licence CC BY-NC-SA-3.0

Cybernetic Kaleidoscope 2

Licence CC BY-NC-SA-3.0

Cybernetic Kaleidoscope 3

Licence CC BY-NC-SA-3.0

Cybernetic Kaleidoscope 4

Licence CC BY-NC-SA-3.0

Cybernetic Kaleidoscope 5

Licence CC BY-NC-SA-3.0

Cybernetic Kaleidoscope 6

Licence CC BY-NC-SA-3.0

Cybernetic Kaleidoscope 7

Licence CC BY-NC-SA-3.0

Cybernetic Kaleidoscope 8

Licence CC BY-NC-SA-3.0

Cybernetic Kaleidoscope 9

Licence CC BY-NC-SA-3.0

Formule

\sqrt[N]{\prod\limits^N_{i=1}\left(\sqrt{(x-x_{i1})^2+(y-y_{i1})^2}+\sqrt{(x-x_{i2})^2+(y-y_{i2})^2}-\sqrt{(x_{i1}-x_{i2})^2+(y_{i1}-y_{i2})^2}\right)}

Cybernetic Star

A superposition of several lines. The basic function for one line running from point 1 to point 2 consists of two cones with the apex pointing down at the position of point 1 and 2. The final function for N lines is the geometric mean value of N basic functions.

Auteurs Christian Gaier

Licence CC BY-NC-SA-3.0

Cybernetic Flowers 1

Arcs are constructed by conformal mapping of circles on lines and using the line function of previous image. Again, several arcs are combined by taking the geometric mean value of the basic functions for single arcs.

Auteurs Christian Gaier

Licence CC BY-NC-SA-3.0