This page contain printed references on plane curves. Books are grouped into the following sections.
For online resources, see Websites on Plane Curves.
General References on Plane Curves
Fun Books on Geometry
Books on History Of Mathematics
Undergraduate Texts and Journal Articles on Curves
Undergraduate Texts on Geometry
Graduate Texts on Geometry
A star bullet ★ means the book is recommended. Most of the time the reason it is recommened is because the book is written for relatively beginning students of the topic. For example, it is very rare to find books on modern Algebraic Geometry aimed for undergraduate math student. So, when there is such a book, it is likely to win a star ★ from me. Many modern math topic lacks introductory books. What I have collected here are especially with self-study in mind.
General References on Plane Curves
★ Title: Curves and their Properties amazon Author: Robert C Yates Publisher: The National Council of Teachers of Mathematics Date: 1952. 1974, repr. 1974; Reprinted by University Microfilms International, Ann Arbor, MI, 1992 Notes: out of print. Comment: Written at beginning college math level, this book gather most of the interesting properties of curves with a brief account of their history. Excellent illustration. If you read plane curves for fun, this is the book to get. Most information on this site is from this book. Includes bibliography. As of 1998, it is not in print, but should be available at most university libraries.
This book is now available on CD. See my CD page for info.
★ Title: The Trisection Problem amazon Author: Robert C Yates Publisher: The National Council of Teachers of Mathematics Date: 1971 Comment: Yates expounds noteworthy “solutions” from ancient to new. Well written with many quality illustrations. It contains discussions of many plane curves and mechanical devices as “solutions”. As of 1998, it's not in print.
Title: The Trisectors amazon Author:Underwood Dudley Publisher: Mathematical Association of America Date: 1994-12 Comment: Exposition of the many angle-trisectors in history. This is a very well written fascinating book. The author takes us thru the many trisetors that he personally had been contacted or visited. He tells us who they are, what they do, and what kind of ilk are they, if any. One'd be surprised the numerousness of them besiege unisersities's math department, even today.
It is interesting to note, that I myself have actually been contacted by a angle-trisector in 2001, and asked by him to do some illustration for him. The person actually hoped that I could help him broadcast his “great discovery”. I tried to convince him that it is not possible, and during the first meeting, realized that this persuasion is impossible. It is with this incident when I found this book by Underwood, and in fact, this trisector in particular is accounted in the book. I contacted Underwood and confirmed the identity. (I did, in fact, actually agreed to do drawing for the trisector and got paid for it)
Title: A Book of Curves amazon Author: E H Lockwood Publisher: Cambridge University Press. Date: 1961 Comment: This book teachs practical methods of drawing curves. It even indicates the appropriate paper size to carry out the drawing. The methods of constructions rely on curve's properties, and many proofs, formulas, and bits of history are given along the way. This book is very similiar in nature to Robert C Yates's “Curves and their Properties”. You'll have fun figuring it out. Includes bibliography of 20 items.
This book is reprinted in 2007.
Title: A Catalog of Special Plane Curves amazon Author: J D Lawrence Publisher: Dover Date: 1972 Comment: This book is a good reference on curve's mathematical formulas. It contains formulas of curve's Cartesian, parametric, polar, bipolar, and other intrinsic equations. Information on a curve's intercept, extrema, extent, discontinuity, symmetry, asymptote, loop, node, volume and surface of revolution are also given. However, proofs are not given. A chronological bibliography of 95 items is included. This is a dry but worthy references. The bad points about this book are: It's a bit outdated. Most formulas given in the book can be generated by computer algebra systems such as Mathematica or Maple. The techniques used in the book is not modern. The illustrations look very primitive with respect to today's (1998) plotting softwares.
★ Title: Encyclopedic Dictionary of Mathematics amazon Author: Edited by Shokichi Iyanaga, Yukiyosi Kawada Publisher: MIT Press. Date: 1993 (2nd edition) Comment:
If you are looking for a printed version of the most comprehensive encyclopedia of math, this is it.
This is a 2 volume set. The writing is aimed at professional mathematicians, in a extremely condensed, summarized fashion. First published in 1954, so it's slightly dated, but still is far above all other printed math references in scope and depth.
It has a entry on plane curves (7-pages), as well entries on coordinate systems, differential geometry, algebraic geometry, projective geometry, their subfields and concepts, … and so on. Let's say you just want to own 5 math books in your life, this should be one of them.
Fun books on interesting Geometry
This section are easy-reading books on geometry.
★ Title: The Penguin Dictionary of Curious and Interesting Geometry amazon Author: David Wells Publisher: Penguin Group. Date: 1991 Comment: A fun compendium of interesting geometry and geometrical objects: curves, surfaces, tilings, polyhedras, knots…etc. A short description of the special properties or why they are interesting along with accurate and pleasing illustrations accompanying each entry.
Title: Shadows of the Circle : Conic Sections, Optimal Figures and Non-Euclidean Geometry amazon Author: Vagn Lundsgaard Hansen Publisher: World Scientific Pub Co Date: 1998-11 Comment:
Probably at high-school level.
Here's from the publisher:
The aim of this book is to throw light on various facets of geometry through development of four geometrical themes. The first theme is about the ellipse, the shape of the shadow east by a circle. The next, a natural continuation of the first, is a study of all three types of conic sections, the ellipse, the parabola and the hyperbola. The third theme is about certain properties of geometrical figures related to the problem of finding the largest area that can be enclosed by a curve of given length. This problem is called the isoperimetric problem. In itself, this topic contains motivation for major parts of the curriculum in mathematics at college level and sets the stage for more advanced mathematical subjects such as functions of several variables and the calculus of variations. Here, three types of conic section are discussed briefly. The emergence of non-Euclidean geometries in the beginning of the nineteenth century represents one of the dramatic episodes in the history of mathematics. In the last theme the non-Euclidean geometry in the Poincare disc model of the hyperbolic plane is developed.
History of Math Books
Most famous plane curves are of historical interest.
Significant part of classical math deal with curves.
Books on history of mathematics often include information on various curves sporadically.
The following are some excellent math history books I've used. Easy reading.
Title: Mathematical Thought from Ancient to Modern Times amazon Author:Morris Kline Publisher: Oxford University Press Date: 1972 Notes: 3 volumes. Comment: Perhaps the most ambitious history of mathematics book that is aimed at general math reader. Contents are as accurate as historians can claim. Easy reading but not trivial. This is a acclaimed book.
Title: An Introduction to the History of Mathematics; 6th ed. amazon Author:Howard Eves Publisher: Saunders College Publishing. Date: 1990 Comment: This book is in the form of a typical modern text book, complete with photographs of mathematicians, artifacts, and old writings, some ancient geographic maps, illustrations on math, and one quarter of the book is devoted to annotated exercises that lets the student get a feeling of how math in different periods are like. Each chapter covers a period, followed by a fairly complete bibliography.
★ Title: A History of Mathematics, 2nd ed. amazon Author: by Carl B Boyer, Uta Merzbach (Contributor) Publisher: John Wiley ＆ Sons, Inc. Date: revised by Uta C. Merzbach.1989. First published 1968. Comment: A well-written book…
Highly recommended by Tristan Needham (author of Visual Complex Analysis). It appears that this book differs from most math history texts in that it takes a view point of math subjects, as opposed to going thru the development of math in time periods.
Ancient and Classic math history
If you want to study Greek Mathematics, the following classics are your most readily available source. Sir Thomas L Heath is a authority on ancient mathematics. Famous curves invented and investigated in Greek times include the conic sections, cissoid of Diocles, Archemede's spiral, and conchoid of Nicomedes. By the way, you'll find Euclid's Elements on-line by David Joyce.
Title: History of Greek Mathematics: From Thales to Euclid amazon Author:Thomas L Heath Publisher: Dover Date: 1981 Notes: 2 volumes Comment: Thomas L Heath is a well-known author on Greek mathematics.
Title: Thirteen Books of Euclids Elements amazon Author: Thomas L Heath (editor) Publisher: Dover Date: 1956 Notes: 3 volumes Comment: The ancient classic math text with authoritative comments by the expert Thomas Heath.
Title: The Mathematical Works of Issac Newton amazon Author: Issac Newton, Derek Whiteside (Editor) Publisher: Johnson Reprint Corp. N.Y.-London. Date: 1962 Comment: vol. 2, p.135-161 contain classification of cubic curves. Newton's original writings. Not to be read unless you read 15th century English fluently. You may want to browse it to see what mathematics in Newton's era looks like on paper. Newton is the first to classify 3rd degree polynomial curves. The book contain some fairly accurate drawings. If you know a translation of Newton's classification in modern English, please let me know.
Title: Apollonius : Conics Books V to VII: The Arabic Translation of the Lost Greek Original in the Version of the Banu Musa amazon Author: Appollonius, (edited by C. J. Toomer) Publisher: Springer Verlag Date: 1990 Comment: I've looked at this book at a book store. It appears to be a modern translation of a Arabic translation of Appollonius' work, where the editor C. J. Toomer tried to render it as authentic as possible. The 'original' Arab text is also included side by side. The book would be most valuable to math historians or serious conics enthusiasts. The book has two volumes, covering Apollonius's book V and VII.
Title: Apollonius of Perga; Conics, Books I-III amazon Author: Appollonius, (edited by Dana Densmore) Publisher: Green Lion Press Date: 1998 Comment:
Translation by R Catesby Taliaferro. Edited by Dana Densmore. See also Wikipedia: Apollonius of Perga.
Undergraduate Texts and Journal Articles on Curves
Articles on plane curves appear throughout math journals. Here is a compilation of over 100 literatures on plane curves. (not necessarily including other titles listed in this page) curveBibliography.html
The following books are mostly advanced undergraduate level text books that deal with plane curves directly or indirectly. For example, topics like differential geometry, algebraic geometry, and projective geometry. Most of them I haven't read. I've only read prefaces, intros, and scaned chapters. Because I learn math mostly on my own, I often select books that's suitable for self study, which basically means well-written self-contained undergraduate texts. I believe the following are good books, and my comments may serve as a guide for those of you amateur mathematicians to-be.
For the following books, one should at least be comfortable in calculus. In general, the following books assume that you have some familiarity with basic modern math concepts like sets, vector spaces, differentiation; someone who are familiar with math courses offered in the first 2 years of US colleges, which often means multi-variable calculus, linear algebra, and differential equations. Though, don't let these requirements scare you. If you have never heard of the terms, then you might be concerned about the book's suitability. I've listed the books roughly in order of difficulty.
Title: Geometry of Curves amazon Author: John W Rutter Publisher: CRC Press Date: 2000 Comment: A very readable book written for the lower undergraduate. Here is a excerpt from the back cover:
Integrating the three main areas of curve geometry-parametric, algebraic, and projective curves-Geometry of Curves offers a unique approach that provides a mathematical structure for solving problems, not just a catalog of theorems. Almost entirely self-contained, this book begins with the basics then takes readers on a fascinating journey from conics, higher algebraic and transcendental curves. It proceeds through the standard properties of parametric curves, the classification of limacons, and a account of envelopes of curve families, and finally to projective curves, their relationship to algebraic curves, and their application to asymptotes and boundedness.
The author has a web page at http://www.liv.ac.uk/~jwrutter/curves/
Title: Conics and Cubics: A Concrete Introduction to Algebraic Curves amazon Author: Robert Bix Publisher: Springer Verlag Date: 1998 Comment: A valuable book for any amateur mathematician who whishes a intro to algebraic geometry. Here's a excerpt from the book's back cover: «Conics and Cubics is an accessible introduction to algebraic curves. Its focus on curves of degree at most three keeps results tangible and proofs transparent. Theorems follow naturally from high school algebra and two key ideas, homogeneous coordinates and intersection multiplicities. The book is a text for a one-semester course. The course can serve either as the one undergraduate geometry course taken by mathematics majors in general or as a sequel to college geometry for prospective or current teachers of secondary school mathematics. The only prerequisite is first-year calculus.»
Title: Elementary Geometry of Differentiable Curves amazon Author: Chris G Gibson Publisher: Cambridge Univ Press Date: 2001-08 Comment:
This book is a undergraduate introduction to differential treatment of plane curves. You should know your calculus well to read this book. Also, you shoud have some understanding of complex numbers.
This book isn't particularly easy to read or friendly, despite the author might want to claim otherwise. It is written by a English mathematician.
Title: Elementary Geometry of Algebraic Curves: an undergraduate introduction amazon Author: C G Gibson Publisher: Cambridge Univ Press Date: 1998-11 Comment:
Written by a English mathematician of University of Liverpool, England. The writing style is a bit pompous, and because of that it reads funny.
If you had abstract algebra, you'll get along with this book fine. If you never had abstract algebra, it'll be problematic for a self-study.
Title: Modern Differential Geometry of Curves and Surfaces, 2nd ed. amazon Author: Alfred Gray Publisher: CRC Press Date: 1998 Comment: This book now has a web page at http://math.cl.uh.edu/~gray/, which contains 6 long pages of images of various curves and surfaces. Like most modern differential geometry text book, curves in the plane are discussed cursorily as a steppingstone to surfaces and manifolds. The book integrates Mathematica codes for generating surfaces and calculating complex formulas. If you are a Mathematica user, then this is a vary valuable book. If you are interested in a good intro book to differential geometry, ther are better books.
Title: Glimpses of Algebra and Geometry amazon Author: Gabor Toth Publisher: Springer Verlag Date: 1998 Comment: I've looked at this book. This appears to be a very accessible book for junior math majors who wishes to have a very basic glimpse of modern algebra, geometry, and their interconnection among other fields as well.
Title: Curves and Singularities : A Geometrical Introduction to Singularity Theory amazon Author: J. W. Bruce, P. J. Giblin (Contributor) Publisher: Cambridge Univ Press Date: 1992 Comment: This book is a bit difficult for self-study. It appears to be a text that covers some singularity theories that is accessible to undergraduate who had advanced calculus. Singularity theory seems to be relatively new branch of math, which treats some of the classical geometry problems such as envelops.
Undergraduate Texts on Geometry
These text books are not exactly curve related, but on modern geometries. These are carefully picked text books on advanced undergraduate level, usually that means you have had some basic linear algebra and multi-variable calculus.
Title: Modern Geometries: Non-Euclidean, Projective, and Discrete Geometry amazon Author: Michael Henle Publisher: Prentice Hall Date: 2001-01 Comment: Introduction to modern geometries with analytic approach. A highschool student can get much out of this book. The book introduces bits of mobius geometry, hyperbolic, elliptic geometry, and projective geometry and solid geometry. Out of necessacity it is shallow, and the writing style is easy going, informal, and a bit slack. Nevertheless it is a very valuable book introducing geometries to beginning students by using coordinates. There does not seems to have competing books. The author does not have much respect for synthetic approaches. The pre-requisite for this book is a good understanding of analytic geometry, which is often taught before or together with calculus.
Title: Geometry amazon Author: David A. Brannan, Matthew F. Esplen, Jeremy J. Gray Publisher: Cambridge Univ Press Date: 1999-4 Comment:
Looks like a great book for self-stuty. Easily readable. Does not assume advanced knowledge. Some text books claims to be accessible by undergraduate, but in fact is quite esoteric even to other mathematicians.
Title: Complex Numbers and Geometry amazon Author: Liang-Shin Hahn Date: 1994-04 Comment: A fantastic little book on complex numbers published by Mathematical Association of America. MAA publishes a serious of thin undergraduate books that are almost always excellent, also because they are small and easy reading, excellent material for math enthusiastics who want to learn real mathematics. (as opposed to pop math books for laymen which usually only talks superficially about the subject.) Complex numbers is so important in math, esp pure math. I advice anyone to get familiar with complex numbers as soon as possible.
A complex number is complex in that it has something called Imaginary part, where a imaginary number denoted by ⅈ, is a number such that its square root is -1. When i was learning math, i regard complex numbers as some kind of oddity and i approach it with caution and relunctance. Now having studied math over a decade, my advice for students new to complex number is to embrace it immediately, to simply force youself to love it, to at least pretent that it is the most natural number of all. Soon it will be, and in fact is. Once you do this, few years down the road you will realize that you possess resolutions to all objections of the oddity and imagined “imaginary” aspect of complex numbers. The more you resist it now, the less you will absorb it. It a psychology thing. Trust me.
This is a great book. However it is not my personal favorite. I didn't like it because the English writing style is very informal and sloppy. It particular, it treats math as physicists do, and i hate that. I like math texts to be formal (as in Hilbert's formalism), logical (as in Russell's logicism), and the exposition precise, but don't necessarily like “rigorous” texts, nor completely abstract and generalized approach. (Note: by definition, a good formalism or logicism approach basically makes it as rigorous can any other treatment can practically be.)
Tristan's writing style is chatty and is in fact one of its selling point. Nevertheless, if you are looking for self-study on complex numbers, especially the geometric aspect, this is perhaps the only and best book.
Title: The Real Projective Plane amazon Author:H S M Coxeter Publisher: Springer Verlag Date: 1992 Comment: This book teaches the basics of projective geometry from a abstract and axiomatic approach. The book claims that any bright highschool student should be able to understand it, but it's more likely that a 3rd year undergraduate math major will have difficulty digesting the book by himself. This is because its axiomatic approach, plus it is written extremely succinctly. If you are going to read this book on your own, some experience with modern math and history of geometry is a good pre-requisite. The book by itself is a excellent work, and is valuable to mathematicians. The software that accompanies the book is not of much value. If you want to learn projective geometry with some computer software enhancement, I highly suggest Cabri Geometry II or The Geometer's Sketchpad. Note that projective geometry of two or three dimensions is essentially theory of conics. Anyone who are serious about conics should study projective geometry.
Title: Introduction to Geometry, 2nd Edition amazon Author: H. S. M. Coxeter Publisher: John Wiley ＆ Sons Date: 1989-02 Comment:
This is a classic reprint. As with all classic books, it is aged. I think it is not the best introduction or survey of modern geometries, but is a must-have reference for any geometry fan and text book collectors. Coxeter is a well-known geometer of our time. He wrote a lot geometry books for the amateur mathematics circle. Almost any of his books are valuable.
Title: Geometric Constructions amazon Author: George E. Martin Publisher: Springer Date: 1998 Comment:
Review coming. Seems pretty good. Easy reading.
Title: A History of the Circle: Mathematical Reasoning and the Physical Universe amazon Author: Ernest Zebrowski Publisher: Date: Comment:
I have not checked out this book. From amazon.com comment, it appears to be a good general interest science-popularizing book, but lacks mathematical history of the circle.
Graduate or More Advanced Texts on Geometry
These text books are geometry at graduate level. I selected them based on amazon.com comments. I think they are great books, but i do not know for certain.
Title: Geometry: Euclid and Beyond amazon Author: Robin Hartshorne Publisher: Springer Verlag Date: 2000-01 Comment: