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Curves from Lines

The Curves from Lines poster set

The Parabola poster The Ellipse poster The Hyperbola poster The Astroid poster The Deltoid poster

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

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.

“Mathematics is crystallized clarity, precision personified, beauty distilled and rigorously sublimated.”—C O Oakley

The Curves from Lines background notes

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

The Curves from Lines backround notes page 1 The Curves from Lines backround notes page 2 The Curves from Lines backround notes page 3 The Curves from Lines 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’.

Sections of a cone

The parabola as conic section

The parabola, ellipse and hyperbola are conic sections—curves obtained when a cone is cut by a plane. For the parabola the plane is parallel to a slant length of the cone.


The astroid 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 astroid and deltoid are roulettes obtained when one circle rolls inside another.

The Deltoid

The deltoid as a rotor for the astroid

The deltoid is a rotor for the astroid—a deltoid will ‘roll’ inside an astroid while remaining in touch with all four arcs of the astroid, 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

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“I have just received the posters you produced – all were enthusiastically put up by my department, they really are good!”

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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.