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Curves from Circles

A set of five laminated posters with background notes, from The Beauty of Mathematics poster collection.

The Curves from Circles poster set

The posters illustrate five curves which are all envelopes of systems of circles:

• The Cardioid
• The Limaçon
• The Nephroid
• The Lemniscate of Bernoulli
• The Right Strophoid

Each poster describes how to draw the curve and also gives brief mathematical or historical information, such as the origin of the curve’s name, or where the curve occurs in nature.

See below for more information about the background notes as well as samples from the notes. Larger images of the posters may be seen by going to the Poster Gallery.

• Poster Gallery
• Order
• Other poster sets:
• Tilings
• Polyhedra
• Curves from Lines
• Fractals

“Noli turbare circulos meos. [Do not disturb my circles!]”—Archimedes

The Curves from Circles background notes

A descriptive booklet accompanies the posters, explaining the mathematics behind the images in the posters and related topics.

The four pages of background notes give more information about the five curves and about envelopes in general. The topics covered are:

• Envelopes
• Roulettes
• Caustics
• Properties of each curve
• Construction methods for each curve
• Formulae
• Glossary
• References

The notes help interested students and teachers explore further, and provide ideas for activities connected to the theme of the posters.

Extracts from the background notes

Envelopes

An envelope is a curve touching every member of a system of lines or curves, or, equivalently, the boundary of the region filled by the system. ‘Touching’ means ‘tangent to’; another name for a ‘system’ is a ‘family’.

Roulettes

A roulette is the path traced by a point attached to a curve, as the curve rolls without slipping along another fixed curve. The cardioid, limaçon and nephroid are roulettes obtained when one circle rolls along another.

Caustics

Light rays reflected from a curved mirror form a family of straight lines; the envelope of this family is a caustic curve. Everyday examples of caustics may be seen when light is reflected by familiar objects like mugs and cups.

The cardioid and the nephroid are both caustics of a circle: if the light source is on the circle, a cardioid results; if the rays are parallel, a nephroid is formed.

The Lemniscate of Bernoulli

The lemniscate appears as a section of a torus with ‘holes’ of equal radii, as shown.

© A K Jobbings 2004-2009