poly_root_cloud.cpp File Reference

Draw the roots of paramaterized polynomials. More...

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Detailed Description

Draw the roots of paramaterized polynomials.

Author

Mitch Richling http://www.mitchr.me/

Keywords

polynomial root

Standards

C++23

See also

https://github.com/richmit/mraster/

Copyright

Copyright (c) 2025, Mitchell Jay Richling http://www.mitchr.me/ All rights reserved.

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

1. Redistributions of source code must retain the above copyright notice, this list of conditions, and the following disclaimer.
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Details

Plot roots of the polynomial p. Two of the coefficients of p (c1 & c2) are parameterized by polynomials (p2 & p2) over the unit circle.

In pseudo-code: Loop Select sample points on the unit circle for t1 & t2 Plug t1 & t2 into p1 & p2 Set coefficients c1 & c2 in p to the value of p1 & p2 Find the roots of p and plot them Repeat

I made a prototype for this problem in Matlab. In the Matlab code I placed semitransparent filled circles at each root, and allowed color to build up with repeated layers of roots. This results in a "fuzzy" look spreading color more smoothly over the envelope of the root cloud. That code may be found here: https://github.com/richmit/PolyRootCloud/

In this code I wanted to emphasize more directly the way the circles in the parametrization map to the root cloud space. So I'm actually creating a histogram of root hits in the plane with a very fine grid, and using many more sample points on one parameter. This produces a Spirograph'esq effect.

Definition in file poly_root_cloud.cpp.