Parallel Algebraic Curves

Intro

An Algebraic Curve is the set of coordiantes which evaluate a polynomial to zero (i.e it's roots). To be more formal, a polynomial defines an Algebraic Curve by: .

How do Algebraic Curves look like, when the coordinates axes are put in parallel?

Transforming a point in Euclidean R^2 to parallel coordinates yields a 2d-line:

source: ref [3]

How does a circle look like after transforming each of its points to parallel coordinates? Well, it looks like a bunch of lines. We can represent the circle by the Envelope of these lines! Which yields:

Code Description

Algebraic Curves translated to Parallel Coordiantes.

Creates an animated plot of an Algebraic Curve next to its dual in Parallel Coordinates, given by the Inselberg Transformation.

To create your own curve, run curve_animation notebook and change the curve parameters.

A Quick Showoff

References

1. Inselberg A., Dimsdale B. (1987) Parallel Coordinates for Visualizing Multi-Dimensional Geometry. In: Kunii T.L. (eds) Computer Graphics 1987. Springer, Tokyo
2. B. Dimsdale. Conic Transformations and Projectivities – IBM LASC Tech. Rep. G320-2753. IBM LA Scientific Center, 1984.
3. Siirtola, H., & Räihä, K. J. (2006). Interacting with parallel coordinates. Interacting with Computers, 18(6), 1278-1309.