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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 Astroid Bessel Functions Bhaskara Proof: Unit Circle Bow Tie Bowditch or Lissajous Brachistochrone      Brachistochrone Part II      Brachistochrone Part III      Brachistochrone Part IV      Brachistochrone-Parabola Cannonball Curves Cardioid Cartesian Ovals Cartesian Parabola (Newton)           (Trident) Cassinian Ovals Catacaustics Caustics Catenary Catenary 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 Envelopes Epicycloid Euler Circuit, Euler Path Graphs Evolutes Fibonacci-Mondrian Spirals Feynman Diagrams Flower or Rose Curves
[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. Pedal and its Pole: 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.