Plane curve

(mathematics) Any curve lying entirely within a plane.

The locus of points in the euclidean plane that satisfy some geometric or algebraic definition. Not all sets of points deserve to be called a curve, but the distinction is somewhat arbitrary. For most of this article, a curve is considered to be the locus of a set of points that satisfy an algebraic or transcendental equation in two variables.

The most interesting geometric properties are those preserved by linear transformations, especially translations, rotations, reflections, and magnifications. Useful geometric properties include the number of branches into which the curve is divided; the number and degree of nodes, cusps, isolated points, and flex points; the number of loops; symmetries; branches that go to infinity; and asymptotes.

These terms can be defined informally as follows. A branch is a maximal smooth continuous portion of the curve. A multiple point is a point in the plane that lies on two or more branches; its degree is the number of branches involved. A node is a multiple point where the branches cross. A cusp is a multiple point where the branches meet but do not pass; that is, each of the branches ends at that point. An isolated point is a point of the curve through which no branches pass. Multiple and isolated points are collectively termed singular points. Any point that is not singular is termed ordinary. A flex (or point of inflection) is a point on the curve whose tangent cuts the curve. A smooth closed branch forms a loop. A curve is symmetric about a line L if every line perpendicular to L intersects the curve at equal distances from L on opposite sides of L; that is, portions of the curve form mirror images about L. The curve is symmetric about a point P if every line through P intersects the curve at equal distances from P in opposite directions. An asymptote is a line toward which a branch approaches as it moves to infinity from the origin; the curve and line are said to intersect at infinity.

In addition to these geometric properties, the form of the defining equation is of interest. This can be an algebraic (polynomial in x and y) or transcendental equation. In the former case, quadratic, cubic, and quartic equations are of special interest.

Once a coordinate system has been chosen, and the defining equation is known (in any of the forms, though the parametric form is usually the most useful), various properties of the curve can be defined in terms of the equation. These include the locations of x- and y-intercepts, local maxima and minima, flexes, nodes, and cusps. See also Analytic geometry.

The illustration shows some of the many plane curves of sufficient historical interest to have received names. See also Cardioid; Cycloid; Lemniscate of Bernoulli; Rose curve.

Right strophoid. (b) Trident of Newton. (c) Cardioid. (d) Deltoid. (e) Devil on two sticks. (f) Lemniscate of Bernoulli. (g) Epitrochoid. (h) Rhodona. (i) Bowditch curve. (j) Fermat's spiral. (k) Logarithmic spiral. (l) Cycloid.">
Plane curves. (a) Right strophoid. (b) Trident of Newton. (c) Cardioid. (d) Deltoid. (e) Devil on two sticks. (f) Lemniscate of Bernoulli. (g) Epitrochoid. (h) Rhodona. (i) Bowditch curve. (j) Fermat's spiral. (k) Logarithmic spiral. (l) Cycloid.

This article is in need of attention from an expert on the subject. WikiProject Mathematics or the Mathematics Portal may be able to help recruit one. (November 2008)

In mathematics, a plane curve is a curve in a Euclidian plane (cf. space curve). The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves.

A smooth plane curve is a curve in a real Euclidian plane R² and is a one-dimensional smooth manifold. Equivalently, a smooth plane curve can be given locally by an equation ƒ(x,y) = 0, where ƒ : R² → R is a smooth function, and the partial derivatives ∂ƒ/∂x and ∂ƒ/∂y are never both 0. In other words, a smooth plane curve is a plane curve which "locally looks like a line" with respect to a smooth change of coordinates.

An algebraic plane curve is a curve in an affine or projective plane given by one polynomial equation ƒ(x,y) = 0 (or ƒ(x,y,z) = 0, where ƒ is a homogeneous polynomial, in the projective case.)

Algebraic curves were studied extensively in the 18th to 20th centuries, leading to a very rich and deep theory. The founders of the theory are Isaac Newton, Bernhard Riemann et al., with some main contributors being Niels Henrik Abel, Henri Poincaré, Max Noether, et al. Every algebraic plane curve has a degree, which can be defined, in case of an algebraically closed field, as number of intersections of the curve with a generic line. For example, the circle given by the equation x² + y² = 1 has degree 2.

An important classical result states that every non-singular plane curve of degree 2 in a projective plane is isomorphic to the projection of the circle x² + y² = 1. However, the theory of plane curves of degree 3 is already very deep, and connected with the Weierstrass's theory of bi-periodic complex analytic functions (cf. elliptic curves, Weierstrass P-function).

There are many questions in the theory of plane algebraic curves for which the answer is not known as of the beginning of the 21st century.

See also

• Smooth manifolds
• Differential geometry
• Algebraic curve
• Algebraic geometry
• Projective varieties

References

• Coolidge, J. L. (April 28 2004), A Treatise on Algebraic Plane Curves, Dover Publications, ISBN 0486495760 .
• Yates, R. C. (1952), A handbook on curves and their properties, J.W. Edwards , ASIN B0007EKXV0.

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