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Curve


There are no fewer than three distinct notions of curve throughout mathematics.

In topology, a curve is a one-dimensional continuum (Charatonik and Prajs 2001).

In algebraic geometry, an algebraic curve over a field K is the zero locus of some polynomial f(X,Y) of two variables which has its coefficients in K.

In analytic geometry, a curve is continuous map from a one-dimensional space to an n-dimensional space. Loosely speaking, the word "curve" is often used to mean the function graph of a two- or three-dimensional curve. The simplest curves can be represented parametrically in n-dimensional space as

x_1=f_1(t)
(1)
x_2=f_2(t)
(2)
|
(3)
x_n=f_n(t).
(4)

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

 f(x_1,x_2,...)=0.
(5)

When discussing curves from the standpoint of analytic geometry, care must be exhibited to maintain the important distinction between the curve itself and its image within its codomain. For example, the curves gamma_i:[0,1]->R, i=1,2, defined, respectively, by

 gamma_1(t)=t
(6)

and

 gamma_2(t)=t^2
(7)

are unique as curves even though both functions have the interval [0,1] as their image in R. This distinction is especially important due the fact that unique curves may possess drastically different geometric behavior in terms of self-intersection, etc., despite having identical images.

From the perspective of analytic geometry, the term curve is usually preceded by any of a number of terms to designate certain geometric behavior, e.g., closed curve, simple curve, smooth curve, etc.


See also

Algebraic Geometry, Algebraic Curve, Closed Curve, Continuum, Plane Curve, Simple Curve, Smooth Curve, Space Curve, Spherical Curve Explore this topic in the MathWorld classroom

Portions of this entry contributed by Christopher Stover

Portions of this entry contributed by Matt Insall (author's link)

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References

Charatonik, J. J. and Prajs, J. R. "On Local Connectedness of Absolute Retracts." Pacific J. Math. 201, 83-88, 2001.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.Gallier, J. H. Curves and Surfaces for Geometric Design: Theory and Algorithms. New York: Academic Press, 1999.Oakley, C. O. Analytic Geometry. New York: Barnes and Noble, 1957.Rutter, J. W. Geometry of Curves. Boca Raton, FL: Chapman and Hall/CRC, 2000.Shikin, E. V. Handbook and Atlas of Curves. Boca Raton, FL: CRC Press, 1995.Seggern, D. von CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, 1993.Smith, P. F.; Gale, A. S.; and Neelley, J. H. New Analytic Geometry, Alternate Edition. Boston, MA: Ginn and Company, 1938.Walker, R. J. Algebraic Curves. New York: Springer-Verlag, 1978.Weisstein, E. W. "Books about Curves." http://www.ericweisstein.com/encyclopedias/books/Curves.html.Yates, R. C. The Trisection Problem. Reston, VA: National Council of Teachers of Mathematics, 1971.Zwikker, C. The Advanced Geometry of Plane Curves and Their Applications. New York: Dover, 1963.Zwillinger, D. (Ed.). "Algebraic Curves." §8.1 in CRC Standard Mathematical Tables and Formulae, 3rd ed. Boca Raton, FL: CRC Press, 1996.

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Curve

Cite this as:

Insall, Matt; Stover, Christopher; and Weisstein, Eric W. "Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Curve.html

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