and His Sidekick Astroid

The study of conic sections is common in high school Algebra
II and Analysis and, indeed, is a Georgia Quality Core Curriculum
requirement for those courses. This page offers resources for
an extension or enrichment of the topic of ellipses.

The general form of equations discussed here are known as Lame' curves after the mathematician Gabriel Lame' (1795-1870), though their study was most advanced by Julius Plucker (1801-1866). As is often the case, Lame's name became associated with these curves because he introduced the notation for their representation. In their modern form, Lame' curves are described by the equation

where n is any positive real number. The standard form for
an ellipse is immediately recognizable when n = 2.

Find more about Gabriel Lame' **here**

When n = 2 and a = b, it should be obvious that the curve represents
a circle with radius a (=b), so for our purposes, we consider
the circle as just a special case of the ellipse.

As part of the usual study of ellipses, the roles of a and b are discussed as the x- and y-intercepts of the ellipse. We will not concern ourselves with those roles here. What is seldom considered in a high school course is the role of the exponent

What if *n* takes on some value other than 2? An easy
and appropriate tool for investigating this question is Graphing
Calculator for the Macintosh (also available with reduced capabilities
for the IBM-compatable as NuCalc from Pacific
Tech ) or the freeware program for the IBM, Winplot
from Rick Parris at Philips Exeter Academy.

If you are using a Macintosh computer, it already has Graphing Calculator as part of its software. Click anywhere on the image below to open Graphing Calculator and explore how changing the exponent changes the graph of as n ranges from 1 to 20.

As you see, the members of this family of curves are all "ellipse-like", but only the red curve is a true ellipse. For n > 2, the curves are known as super ellipses and have as their limiting case the rectangle formed by the lines |x| = 2 and |y| = 1. The faint lavender curve in the figure above is the graph of .

The term super ellipse was coined by the Danish artist, author, designer, and mathematician Piet Hein in the 1970's. He used a super ellipse with n = 2.5 as the design for tabletops and windows. His preferred choices for a and b in the Lame' equation were in the ratio 4:3. The graph is shown here:

Piet Hein is as well known in Scandinavia as is Hans Christian Andersen and has streets, parks and buildings named for him in several major cities. When asked to design a plaza for the city of Stockholm, he used his super ellipse as its basis, enhancing traffic flow as well as creating an estheticly pleasing fountain.

Hein also marketed a three dimensional version of his super
ellipse, which he called a super egg. The super egg is formed
by rotating a super ellipse about its longer axis and is interesting
because of its remarkable stability when balanced on its
end. In the 1970's wooden models were popular in Europe and today,
items such as this *Stress Egg *are available.

In the Lame' equation, if the exponent is 2/3, the resulting curve is known as an astroid (NOT