Curve

A Continuous Map from a 1-D Space to an -D Space. Loosely speaking, the word ``curve'' is often used to mean the Graph of a 2- or 3-D curve. The simplest curves can be represented parametrically in -D Space as

$\displaystyle x_1$	$\textstyle =$	$\displaystyle f_1(t)$
$\displaystyle x_2$	$\textstyle =$	$\displaystyle f_2(t)$
$\displaystyle \vdots$
$\displaystyle x_n$	$\textstyle =$	$\displaystyle f_n(t).$

Other simple curves can be simply defined only implicitly, i.e., in the form

$\begin{displaymath} f(x_1,x_2,\dots)=0. \end{displaymath}$

See also Archimedean Spiral, Astroid, Asymptotic Curve, Baseball Cover, Batrachion, Bicorn, Bifolium, Bow, Bullet Nose, Butterfly Curve, Cardioid, Cassini Ovals, Catalan's Trisectrix, Catenary, Caustic, Cayley's Sextic, Cesàro Equation, Circle, Circle Involute, Cissoid, Cissoid of Diocles, Cochleoid, Conchoid, Conchoid of Nicomedes, Cross Curve, Cruciform, Cubical Parabola, Curve of Constant Precession, Curve of Constant Width, Curtate Cycloid, Cycloid, Delta Curve, Deltoid, Devil's Curve, Devil on Two Sticks, Dumbbell Curve, Dürer's Conchoid, Eight Curve, Electric Motor Curve, Ellipse, Ellipse Involute, Elliptic Curve, Envelope, Epicycloid, Equipotential Curve, Eudoxus's Kampyle, Evolute, Exponential Ramp, Fermat Conic, Folium of Descartes, Freeth's Nephroid, Frey Curve, Gaussian Function, Gerono Lemniscate, Glissette, Gudermannian Function, Gutschoven's Curve, Hippopede, Horse Fetter, Hyperbola, Hyperellipse, Hypocycloid, Hypoellipse, Involute, Isoptic Curve, Kappa Curve, Keratoid Cusp, Knot Curve, Lamé Curve, Lemniscate, L'Hospital's Cubic, Limaçon, Links Curve, Lissajous Curve, Lituus, Logarithmic Spiral, Maclaurin Trisectrix, Maltese Cross, Mill, Natural Equation, Negative Pedal Curve, Nephroid, Nielsen's Spiral, Orthoptic Curve, Parabola, Pear Curve, Pear-Shaped Curve, Pearls of Sluze, Pedal Curve, Peg Top, Piriform, Plateau Curves, Policeman on Point Duty Curve, Prolate Cycloid, Pursuit Curve, Quadratrix of Hippias, Radial Curve, Rhodonea, Rose, Roulette, Semicubical Parabola, Serpentine Curve, Sici Spiral, Sigmoid Curve, Sinusoidal Spiral, Space Curve, Strophoid, Superellipse, Swastika, Sweep Signal, Talbot's Curve, Teardrop Curve, Tractrix, Trident, Trident of Descartes, Trident of Newton, Trochoid, Tschirnhausen Cubic, Versiera, Watt's Curve, Whewell Equation, Witch of Agnesi

References

Curves

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 71-75, 1989.

``Geometry.'' The New Encyclopædia Britannica, 15th ed. 19, pp. 946-951, 1990.

Gray, A. ``Famous Plane Curves.'' Ch. 3 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 37-55, 1993.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, 1972.

Lee, X. ``A Catalog of Special Plane Curves.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/specialPlaneCurves.html.

Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, 1961.

MacTutor History of Mathematics Archive. http://www-groups.dcs.st-and.ac.uk/~history/Curves/Curves.html.

Oakley, C. O. Analytic Geometry. New York: Barnes and Noble, 1957.

Shikin, E. V. Handbook and Atlas of Curves. Boca Raton, FL: CRC Press, 1995.

Smith, P. F.; Gale, A. S.; and Neelley, J. H. New Analytic Geometry, Alternate Edition. Boston, MA: Ginn and Company, 1938.

von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, 1993.

Walker, R. J. Algebraic Curves. New York: Springer-Verlag, 1978.

Weisstein, E. W. ``Plane Curves.'' Mathematica notebook Curves.m.

Yates, R. C. A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, 1947.

Yates, R. C. The Trisection Problem. Reston, VA: National Council of Teachers of Mathematics, 1971.

Zwillinger, D. (Ed.). ``Algebraic Curves.'' §8.1 in CRC Standard Mathematical Tables and Formulae, 3rd ed. Boca Raton, FL: CRC Press, 1996. http://www.geom.umn.edu/docs/reference/CRC-formulas/node33.html.