Arbelos logo

Solutions for Mathematics Enrichment

Home Books Book Gallery Posters Poster Gallery ClubMaths™ ClubMaths™ Gallery EM Resources EM Resources Gallery Testimonials Order Links About Us Contact Us FAQ Privacy & cookies
View Cart

Curves from Circles

The Curves from Circles poster set

The Cardiod poster The Limaçon poster The Nephroid poster The Lemniscate of Bernoulli poster The Right Strophoid poster

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

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.

“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 Curves from Circles backround notes page 1 The Curves from Circles backround notes page 2 The Curves from Circles backround notes page 3 The Curves from Circles backround notes page 4

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

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


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’.


The cardioid as a roulette

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.


The nephroid as a caustic curve

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 as a section of a torus

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

On this page

Clicking a link will scroll the page to the relevant section.


Visit the Poster Gallery to see larger images of the posters.

Poster Gallery

£29.95 per set + delivery

Add to Cart

“Will shortly put in an order for the Tilings and Polyhedra too — really fabulous posters.”

More testimonials
Poster size

All posters are A2 size:

42 cm by 59.4 cm


The front face of each poster is covered with a wipe-clean laminated film.

Booklet size

Each booklet of background notes has four A4 pages and includes many diagrams.