Some Advancements to Visualizing Constrained Functions and Inequalities of Two Variables

Abstract

Based on the problem of visualizing the potential energy of a two-degree-of-freedom spring-mass system restrained by an elastic string, several advancements to visualizing functions with constraints and inequalities of two variables are introduced. These innovations include logarithmically spacing level curves (either mapped on the surface or projected on the bottom plane) and the possibility of truncating the portions of the function surface that exceed above and below the bounding box—both allowing better detailing of certain regions of the function surface, in particular the minimum and maximum areas. By selectively displaying the surface patches that either intersect or not the top and/or bottom planes of the bounding box in a truncated representation, sets of inequalities of two variables can be represented graphically in a suggestive manner. Also proposed are a new approach to producing the gradient of the function as an arrow field mapped on the bottom of the plot box that uses a finite-difference scheme applied to the 2D image-space nodes of the function surface rather than the original 3D data, and a new way of displaying the color scale in 3D surface plots.