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Index of Classic Plane Curves and Surfaces

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We invite students and faculty to contribute a favorite animation to the NCB Family of Curves.
Apple of Discord
Anamorphic Tilings
Apple of Discord
Bessel Functions
Bhaskara Proof: Unit Circle
Bow Tie
Bowditch or Lissajous

     Brachistochrone Part II
Brachistochrone Part III
Brachistochrone Part IV
Cannonball Curves

Cartesian Ovals
Cartesian Parabola (Newton)
Cassinian Ovals

Circle of Curvature
Cissoid of Diocles
Conchoid of Nicomedes
Cone of Claveria:  Oblique Cone
Conic Sections
Conic Sections with Maplesoft®
Conic Sections with MATHEMATICA®
          Dandelin Ellipse
Dandelin Hyperbola
          Dandelin Parabola

Constant Width Cycloid
Cycloid Family with Maplesoft®Code
Coeur de Cye
Descartes' Method of Tangents
Devil's Curve
Electric Motor Curve
Euler Circuit, Euler Path Graphs
Fibonacci-Mondrian Spirals
Feynman Diagrams
Flower or Rose Curves

Folium of Descartes
Folium of de Judicibus

Gamma Pulse Yin-Yang
Gaussian Distribution
General Cissoid
General Conchoid
GeoGebra Curves
Hamilton Circuit
Harriot's Cannonball Curve
Hippopede of Proclus

Helen of Geometry
History of Curves and Graphing
Hyperbola of Fermat
Hyperbolic Spiral
Klein Bottle with Maplesoft® Code
Koch Snowflake with Maplesoft® Code
La galande (Folium of Descartes)
Laplace Transform for Voltage Circuit
Lemniscate of Bernoulli
Lemniscate with Maplesoft®Code
Limaçon with Maplesoft®Code
Limaçon of Pascal
Logarithmic Spiral
Menger's Sponge
Möbius Strip in  Maple and Java
Möbius Strip in  MATHEMATICA®
Neile's Parabola
Nephroid with MATHEMATICA® Code
Normal or Frequency Distribution
Cone of Claveria:  Oblique Cone
Ogive-Ogee Curves
Osculating Circle

A collection of famous curves in POVRAY

A collection of famous curves in

Parabola of Fermat
Pearls of Sluze
Phelps on Phillips Curve
Platonic Solids with Hamilton Circuits
Pursuit Curves
Pursuit Curves - Lunar
Pursuit JAVA Applete
Quadratrix of Hippias
Reuleaux Triangle
Reuleaux Polygons, Part II
Rose or Flower Polar Curves
Rose - Eight Petals
Rose - Four Petals
Rose - Three Petals
Double Rose - Three Petals
Roman Surfaces
Semi-cubical Parabola
Siluroid Folium
Sinusoidal Spiral
Spiral of Archimedes
Spirals in Matlab
Spirals with MATHEMATICA®Code:
             Euler, Cornu, Lituus
             Bernoulli and others.
Spirals - Polynomials
Spirals - Fibonacci
Spirals - Fibonacci
Superconic Spirals
Steiner Roman Surfaces
Tricurves: Unique Tessellations
Tschirnhausen Cubic
Unicursal and Multicursal Graphs
Waldman's Heart
Wankel Engine
Witch of Agnesi
Witch of Agnesi - 1748

A collection of famous Calculus
animations in

[Note:  Terms related to analyses of a system of plane curves are given in orange.]
Historical sketch of one curve . . . .

Notice the animated tangent to the parabola in the upper right hand corner.  The parabola is undoubtedly the most studied curve in the history of mathematics.  A treatise, Conic Sections, written by Euclid (ca. 300 B.C.), has been lost but is thought to have provided a foundation for Apollonius' first four books of the same name.     Both the scholar and student can trace this work through the Alexandrian school.  Hypatia (d. 415 A.D.), the first woman in the history of mathematics known by name, is said to have made the translation. 

These manuscripts can then be traced to Western Europe via Arab conquest, first through North Africa, and then into Spain.  Toledo, Cordova and Seville were outstanding centers of learning from the 9th to the llth centuries. 

Later, Galileo (1564 - 1642) discovered that a cannonball follows a parabolic path.  Scientists, monarchs and military leaders immediately took great interest!  The aiming of a cannon became a function of measuring the precise angle of a trajectory, the "throw weight" and its momentum.  With these experiments and especially the famous dropping of objects from the top of the Leaning Tower of Pisa, Galileo revolutionized science.  He introduced the "scientific method" as a permanent contribution to civilization.  Later his heliocentric theory of the universe almost cost him his life.

Descartes, in writing La Géométrie (1637), chose the parabola to illustrate his innovative analytic geometry.  At this time in publishing history, all math figures were difficult to create and to print.

In 1992, R. A. Marcus of the California Institute of Technology won the Nobel Prize in Chemistry for his work showing that parabolic reaction surfaces can be used to calculate how fast electrons travel in molecules.  His most famous theoretical result, an inverted rate-energy parabola, predicts electron transfer will slow down at very high reaction free energies.

Millions of students in recent centuries will remember the parabola as the introductory curve leading into study of the Calculus.

An Informal Glossary of Common Terms:
Caustic: A caustic curve is the envelope of light rays emitted from a point, after reflection or refraction by a given curve. 
The caustics are either catacaustic as a result of reflection or diacaustic as a result of refraction.
Cusp: If a curve is traced by a moving point, a cusp-point is one where the moving point reverses its direction.  See cusp-point  and  curve sketching.
Envelope: An envelope is a curve, or curves, touching every member of a system of lines or curves.
See the illustration of an envelope and also the animation of Neile's Parabola.
Evolute: The envolute of a plane curve is the locus of centers of curvature of another curve, and therefore tangent to all its normals; so called because the other curve (called the involute) can be traced by the end of a string gradually being unwound from it. Moreover, the evolute may also be throught of as the envelope of the normals to the curve.     The family of intersecting normals on the interior of a parabola form the evolute of the parabola and is commonly called a cissoid.  Also, see the animation of Neile's Parabola.
Hermit Point:
Please see the illustration under curve sketching.
Involute: One may think of the involute as the inverse operation of the evolute.  Alfred Gray writes, "...the evolute is related to the involute in the same way that differentiation is related to indefinite integration."
Thus, the original parabola becomes the involute of its evolute.  See the example of an involute of a circle.
Node: Please see the illustration under curve sketching.
Normal: The term "normal" has many usages in mathematics.  In curve tracing, a normal is the perpendicular to a tangent of a curve drawn at the point of tangency.  Also, see the animation of Neile's Parabola.
and its
A pedal curve represents the locus of the feet of perpendiculars let fall from a given point to tangent(s) on a given curve or curved surface.
Singularity: Please see the illustrations under curve sketching and the "hermit" point.

Useful Links and Books
We recommend the following books as having excellent, but varied information on curves and surfaces.
Some of these texts are certain to be in your library and all are available through interlibrary loan.
Eves, Howard.  AN INTRODUCTION TO THE HISTORY OF MATHEMATICS, 6th ed., Saunders College Publishing, Harcourt Brace Jovanovich, 1990. 
Frost, Percival.  An Elementary Treatise on CURVE TRACING, 5th ed., Revised by R. J. T. Bell, Chelsea Publishing Company, 1960.
Gibson, C. G.  Elementary Geometry of Algebraic Curves: An Undergraduate Introduction.  Cambridge University Press, 1998.
Gray, Alfred.  Modern Differential Geometry of Curves and Surfaces with MATHEMATICA® ,2nd ed., CRC Press, 1998.
Hilton, Harold.  Plane Algebraic Curves,  Oxford University Press, 1932.
Lawrence, J. D.  A Catalog of Special Plane Curves,  Dover Publications, 1972.
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.