A New Approach to Visualizing Algebraic Curves in Parallel Coordinates

Title: A New Approach to Visualizing Algebraic Curves in Parallel Coordinates

Research Question: How can we create a more effective representation of algebraic curves in parallel coordinates to overcome the "over-plotting" problem?

Methodology: The authors propose a new approach to visualizing algebraic curves in parallel coordinates. They introduce a transformation that provides a direct representation of a curve, called the dual, which is an algebraic curve of degree at most n(n-1) in the absence of singular points. This approach is based on algebraic geometry using resultants and homogeneous polynomials. They provide an algorithm that constructs the dual image of the curve.

Results: The authors show that conics map into conics, and they provide an example of a cubic curve that is transformed into a cubic curve. They also discuss the "trade-off" price for obtaining a planar representation of multidimensional algebraic curves and surfaces, which is the higher degree of the image's boundary.

Implications: The new approach to visualizing algebraic curves in parallel coordinates has several implications. First, it overcomes the "over-plotting" problem, making it easier to see and understand the curve. Second, it provides a more direct representation of the curve, which may be useful in various applications. Third, the method has potential generalizations to multi-dimensional algebraic surfaces and their approximation.

Keywords: Visualization, Parallel Coordinates, Algebraic Dual Curves, Approximations of Algebraic Curves, Surfaces.

AMS: 76M27 ACM: F.2.1, I.1.1

Authors: Z. Zur

Link to Article: https://arxiv.org/abs/0403005v1 Authors: arXiv ID: 0403005v1