Does "isoparametric" mean what it sounds like?

Question

Please forgive my inexact language.

Given an infinitely dense family of smooth curves on $\mathbb{R}^2$, all of which are parameterized by the same variable such that points with the same parameter value on "adjacent" curves are "contiguous". I want to call the entire set of points determined by a given parameter value an isoparametric curve.

For example, imagine some well behaved 2-dimensional fluid flow. At time $t=0$ a straight line segment of dye marker is emitted roughly perpendicular to the fluid flow, thereby designating those fluid points as having the same time parameter value. The time evolution of that line segment would constitute an isoparametric curve.

Is this a proper use of the term isoparametric?

Answer 1

Yes, that usage is reasonable: lines of constant parameter value on tensor-product subdivision surfaces are commonly called "isoparametric lines," for instance.

Note that there is at least a potential for confusion with the similar-sounding "isoperimetric," so I would avoid "isoparametric" if you are also solving variational problems using your family of curves.

Answer 2

Beware, the word isoparametric is used in differential geometry in a different context (though I think it's not very extended).

A smooth function $f:\mathbb{R}^n\to\mathbb{R}$ (usually required to be non-constant in every open subset of $\mathbb{R}^n$) is said to be isoparametric if $\Delta f=a\circ f$ and $|\nabla f|^2=b\circ f$ for $a\in C^0(\mathbb{R}), b\in C^2(\mathbb{R})$.

An isoparametric manifold is a regular level set of an isoparametric function.

The name is due to historical reasons. Here's a reference to this theory.