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Curves from Lines

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

The Curves from Lines poster set

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

• The Parabola
• The Ellipse
• The Hyperbola
• The Astroid
• The Deltoid

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:
• Curves from Circles
• Tilings
• Polyhedra
• Fractals

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

• Envelopes
• Sections of a cone
• Roulettes
• 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’.

Sections of a cone

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.

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

The Deltoid

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.

© A K Jobbings 2004-2009