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The Conchoid Family of Curves
Area of loop figure  
Area of the loop . . .

For the Conchoid of Nicomedes....

Conchoid animation

Conchoid animation   
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Let S be any curve and let A be a fixed point.  If a straight line is drawn through A to meet the curve at Q, and if P and P' are points on this line such that PQ = P'Q = a constant term, the locus of the points P and P' is called a conchoid of the curve with respect to the fixed point A.

The Conchoid family of curves is easily entered and modified on a graphing calculator.  
Be sure to use  (1/cos) for the secant term.
Another graph of conchoid    

"The invention of the conchoid ( mussel-shell shape ) is ascribed to Nicomedes by Pappus and other classical authors;  it was a favourite with the mathematicians of the seventeenth century as a specimen for the new method of analytical geometry and calculus.  It could be used (as was the purpose of the invention) to solve the two problems of doubling the cube and of trisecting an angle; and hence for every cubic or quartic problem.  For this reason Newton suggested that it should be treated as a  standard  curve."
E. H. Lockwood  

Historical Sketch

Nicomedes (ca. 225 BC) is credited with being the first to investigate the conchoid, a name from the Greek meaning "shell-like" or "shell form."  He sought to find two mean proportions between given lengths in order to solve the famous Delian problem.  

The Delian problem is also known as duplication of the cube; in other words, finding the edge of a cube having a volume exactly twice that of a given cube.  This is one of the three famous constructions dating from antiquity.  If, contrary to Euclidean assumptions, we permit ourselves to mark a straight edge, the conchoid may, in fact be applied to these famous problems.

With adjustments in various constant terms, the conchoid is modified into a circle, spiral, limaçon, or cardiod.  Moreover, we find equations for the conchleoid and conchal as well as focal conchoids of conic sections in the literature.  But the most famous are the . . . . . .

  • Conchoid of Dürer
  • Conchoid of de Sluze
  • Limaçon of Pascal, father of Blaise Pascal
  • Conchoid of Külp
  • and of course, the oldest, the Conchoid of Nicomedes.

To underscore the historical importance of the conchoid, we have selected Figure #139 of John Colson's translation of Maria Gaetana Agnesi's Instituzioni analitche, now generally recognized as the first book of mathematics written by a woman.  Colson held the prestigious Lucasian chair at Cambridge.  In 1801 he chose to translate Agnesi's widely acclaimed treatise on Calculus written one-half century earlier (1748). Unfortunately, his translation of the name of the curve now known in English speaking countries as the Witch of Agnesi was a slight, but lasting error.  However, his Conchoid of Nicomedes was accurate and an indication of the enduring fame of this popular curve.   Colson's EXAMPLE IV speaks for itself.

Colson's Conchoid in  Instituzioni analitche
Reproduced with permission from the Rare Books Division, Dept. of Rare Books and Special Collections, Princeton University Library.

Please observe that Agnesi  first sketched a circle to generate both her Witch and the conchoid before proceeding to give a proof.

For the modern graphing calculator . . . .
Graphing Calculator Animation
90-270 graph
Equation of Asymptote
Another view of graph

Useful Links and Books
Eves, Howard, An Introduction to the History of Mathematics, 6th ed,. The Saunders College Publishing, 1990.
Gray, Alfred,  Modern Differential Geometry of Curves and Surfaces with MATHEMATICA®, 2nd ed., CRC Press, 1998,  p. 898.
Lockwood, E. H., A Book of Curves, Cambridge University Press, 1961.
Shikin, Eugene V., Handbook and Atlas of Curves, CRC Press, 1995.
Yates, Robert,  CURVES AND THEIR PROPERTIES, The National Council of Teachers of Mathematics, 1952.
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MATHEMATICA® Code and animation contributed by
Gustavo Gordillo,  2006.