Isoparametric

Isoparametric refers to mathematical objects defined by equal parameters or by a parameterization that uses a common set of parameters. The term is used in several contexts, most prominently in differential geometry and numerical analysis, to describe structures that share a unified parameter scale.

In differential geometry, an isoparametric hypersurface in a Riemannian manifold is a hypersurface on which the principal curvatures are constant along the surface. Equivalently, the eigenvalues of the shape operator are constant. The concept began with Elie Cartan, and in spheres it leads to a finite number g of distinct principal curvatures with specific multiplicity relations. Associated isoparametric functions have the property that both their gradient norm and their Laplacian depend only on the function value, so their level sets are isoparametric hypersurfaces.

In finite element analysis, an isoparametric element uses the same shape functions to interpolate both geometry and field variables over a curved element. The mapping from a reference element to a physical element is defined by these shape functions, which allows accurate representation of curved boundaries and interfaces. Subtypes include subparametric elements, where geometry is approximated with lower order interpolation, and superparametric elements, where geometry uses higher order interpolation than the approximated field.

The term isoparametric can also describe curves or surfaces parameterized by a common parameter set, yielding equal parameterization within a given family.