Lamé Ovals

Abstract

Generalization of the ellipse tois credited to Gabriel Lamé, the 19th-century French physicist and the eponym of this family of curves. Lamé curves have recently been taken out of mathematical limbo due to their appeal to certain designers and architects. Particularly to be mentioned is Piet Hein, the contemporary Danish poet-designer-scientist (and inventor of mathematical games) who rediscovered the curves and has been using “superellipses” (Lamé curves with n > 2, and therefore oval) in objets d’art, furniture, pottery, fabric patterns, and so on. His major achievement to date is a sunken oval shopping plaza, promenade, and pool in the centre of Stockholm; its shape is a superellipse with and . Derivatively, Gerald Robinson, a Toronto architect, has incorporated a superelliptic parking garage into a shopping complex (called a Superblock) in downtown Peterborough, Ontario; the parameters are and n = 2.71828 … (the exponential coefficient). In such urban contexts the two-parameter super-ellipse is efficiently tailorable to the dimensions of the site. It is said to be very elegant.

RNA backbone: Consensus all-angle conformers and modular string nomenclature (an RNA Ontology Consortium contribution)

“…This is done using a superellipsoid, the multidimensional generalization of the superellipse (Gielis 2003) or Lamé oval (Gridgeman 1970). This figure bulges smoothly outward progressively more from an ellipse into the corners of the superscribed rectangle as the exponent increases from 2.…”

An eigenvalue problem in two dimensions for an irregular boundary

“…They are also known as Lamé curves or Lamé ovals [17]. Superellipses can be parametrically described as, x = a cos 2/t φ, and y = b sin 2/t φ. and use the symmetry of the figure to continue to the other quadrants.…”

Approximation on non-tensor domains including squircles, Part III: Polynomial hyperinterpolation and radial basis function interpolation on Chebyshev-like grids and truncated uniform grids

“…The Danish architect and poet Piet Hien incorporated superellipses (unequal axes) into some of his major projects [20]. Olympic Stadium in Mexico City is a famous example.…”

“…We choose Gaussian functions with the univariate kernel φ(r; α, h) ≡ exp −[α/h] 2 r 2 (20) where h is the grid spacing between nearest neighbor grid points on a uniform grid or the average grid spacing if the grid is non-uniform. The constant α is the "shape parameter", which is actually an inverse width relative to the average grid spacing.…”