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Geometry of Curves

John W. Rutter
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Chapman Hall/CRC Mathematics Series 17
[Reviewed by
P. N. Ruane
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I’ll risk being corrected in saying that no other book fulfils the aims of the one described in this review. Its contents are readily accessible to students with a basic knowledge of vectors, complex numbers, coordinate geometry and first year university calculus. It integrates the three main areas of plane curve geometry, described by the author as parametric, algebraic and projective curves, and it uses a wide range of mathematical techniques in the process.

John Rutter’s book forms ideal preparation for subsequent courses on algebraic and differential geometry; but it could stand alone purely and simply as part of an overall mathematical education. It is suitable both for self-study and as a basic text for a two-semester course.

The book begins with a review of various coordinate systems (Cartesian, polar, complex and parametric). It considers the geometry of lines, circles and conics, together with their classical geometric construction, historical commentary and notes on applications.

But the wide range of curves considered in this book is impressive. They can be grouped under the two headings of algebraic curves and transcendental curves, which are organised into very many sub-groups, such as cubics, Watt’s curves, limaçons, spirals and roulettes etc. These are represented algebraically and parametrically in R2 or by use of complex numbers in the Argand plane. Eventually, many of the curves are considered as affine views of projective curves in RP2.

The main analytic themes of the book are those of contact order and curvature, and curves are thereby classified according to the nature of their inflexions, undulations or singularities. Central to this process are the three chapters on curvature, the penultimate chapter on singular points of algebraic curves and the final chapter on projective curves.

Other major themes are introduced, and they are used to determine additional properties of curves and to establish relations between various types. In this respect, there are chapters on evolutes, parallels and involutes; and there is there is a lengthy chapter on envelopes that includes discussion of singular-set envelopes, orthotomics and caustics etc.

Overall, the book is very readable, well illustrated and clearly organised. In classic textbook mould, it contains very many worked examples and graded exercises. There are more extended exercises intended for cooperative problem solving amongst small groups of students. Also, the use of traditional curve sketching techniques is encouraged, as well of computer curve plotting technology such as Maple and MATLAB.

So, if your department offers courses in differential or algebraic geometry, then give due consideration to this book as the basis of a preparatory course for such subjects.

You won’t regret it!

In years gone by, Peter Ruane enjoyed many an hour doing practical activities on the theme of envelopes (curve-stitching) with primary school children.


Lines, Circles and Conics

Conics: General Position

Some Higher Algebraic and Transcendental Curves

Parameters, Tangents, Normals. Contact, Inflexions


Cusps, Non-Regular Points


Curvature Applications

Circle of Curvature



Parallels, Involutes



Singular Points of Algebraic Curves

Projective Curves