tripcurves

Tripcurves are a class of mathematical curves used to represent the relationship among three interdependent quantities. In their simplest form, a tripcurve is a parametric curve gamma(t) = (x(t), y(t), z(t)) in three-dimensional space, or its two-dimensional projection for visualization. The term is used in data visualization, design optimization, and motion planning to illustrate how three objectives or variables trade off against one another as a single parameter varies.

Construction and representation: Tripcurves can be defined directly by analytic equations or obtained by projecting higher-dimensional data onto a three- or two-dimensional representation. Endpoints often encode extreme configurations, while the path reflects how the triple changes together. Smooth tripcurves assume differentiable x(t), y(t), z(t).

Variants and shapes: In practice tripcurves appear as planar projections or spatial curves. Shapes include monotone sweeps, loops, or s-curves, depending on parameterization and constraints. Curvature and torsion describe bending and twisting; arc length parameterization is common for comparing different curves.

Computation and analysis: Tripcurves are generated from analytic models or sampled from data. Analysis focuses on curvature, torsion, speed, and acceleration along the curve, as well as proximity to critical regions such as Pareto-optimal bands in multi-objective contexts. Numerical methods and interpolation are used for discrete data.

Applications and history: The concept is used to visualize trade-offs in engineering design, logistics and route planning, and robotics. It provides a compact representation of how three linked quantities evolve together as a control parameter changes. The term arose in discussions of multi-objective visualization and trajectory design in the early 21st century.