Plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. Plane curves also include the Jordan curves (curves that enclose a region of the plane but need not be smooth) and the graphs of continuous functions.
Symbolic representation
A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form [math]\displaystyle{ f(x,y)=0 }[/math] for some specific function f. If this equation can be solved explicitly for y or x – that is, rewritten as [math]\displaystyle{ y=g(x) }[/math] or [math]\displaystyle{ x=h(y) }[/math] for specific function g or h – then this provides an alternative, explicit, form of the representation. A plane curve can also often be represented in Cartesian coordinates by a parametric equation of the form [math]\displaystyle{ (x,y)=(x(t), y(t)) }[/math] for specific functions [math]\displaystyle{ x(t) }[/math] and [math]\displaystyle{ y(t). }[/math]
Plane curves can sometimes also be represented in alternative coordinate systems, such as polar coordinates that express the location of each point in terms of an angle and a distance from the origin.
Smooth plane curve
A smooth plane curve is a curve in a real Euclidean plane [math]\displaystyle{ \R^2 }[/math] and is a one-dimensional smooth manifold. This means that a smooth plane curve is a plane curve which "locally looks like a line", in the sense that near every point, it may be mapped to a line by a smooth function. Equivalently, a smooth plane curve can be given locally by an equation [math]\displaystyle{ f(x, y) = 0, }[/math] where [math]\displaystyle{ f: \R^2 \to \R }[/math] is a smooth function, and the partial derivatives [math]\displaystyle{ \partial f/\partial x }[/math] and [math]\displaystyle{ \partial f/\partial y }[/math] are never both 0 at a point of the curve.
Algebraic plane curve
An algebraic plane curve is a curve in an affine or projective plane given by one polynomial equation [math]\displaystyle{ f(x,y) = 0 }[/math] (or [math]\displaystyle{ F(x,y,z) = 0, }[/math] where F is a homogeneous polynomial, in the projective case.)
Algebraic curves have been studied extensively since the 18th century.
Every algebraic plane curve has a degree, the degree of the defining equation, which is equal, in case of an algebraically closed field, to the number of intersections of the curve with a line in general position. For example, the circle given by the equation [math]\displaystyle{ x^2 + y^2 = 1 }[/math] has degree 2.
The non-singular plane algebraic curves of degree 2 are called conic sections, and their projective completion are all isomorphic to the projective completion of the circle [math]\displaystyle{ x^2 + y^2 = 1 }[/math] (that is the projective curve of equation [math]\displaystyle{ x^2 + y^2 - z^2 = 0 }[/math]). The plane curves of degree 3 are called cubic plane curves and, if they are non-singular, elliptic curves. Those of degree 4 are called quartic plane curves.
Examples
Numerous examples of plane curves are shown in Gallery of curves and listed at List of curves. The algebraic curves of degree 1 or 2 are shown here (an algebraic curve of degree less than 3 is always contained in a plane):
| Name | Implicit equation | Parametric equation | As a function | graph |
|---|---|---|---|---|
| Straight line | [math]\displaystyle{ a x+b y=c }[/math] | [math]\displaystyle{ (x,y)=(x_0 + \alpha t,y_0+\beta t) }[/math] | [math]\displaystyle{ y=m x+c }[/math] | 
|
| Circle | [math]\displaystyle{ x^2+y^2=r^2 }[/math] | [math]\displaystyle{ (x,y)=(r \cos t, r \sin t) }[/math] | 
| |
| Parabola | [math]\displaystyle{ y-x^2=0 }[/math] | [math]\displaystyle{ (x,y)=(t,t^2) }[/math] | [math]\displaystyle{ y=x^2 }[/math] | 
|
| Ellipse | [math]\displaystyle{ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 }[/math] | [math]\displaystyle{ (x,y)=(a \cos t, b \sin t) }[/math] | 
| |
| Hyperbola | [math]\displaystyle{ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 }[/math] | [math]\displaystyle{ (x,y)=(a \cosh t, b \sinh t) }[/math] | 
|
See also
- Algebraic geometry
- Convex curve
- Differential geometry
- Osgood curve
- Plane curve fitting
- Projective varieties
- Skew curve
References
- Coolidge, J. L. (April 28, 2004), A Treatise on Algebraic Plane Curves, Dover Publications, ISBN 0-486-49576-0.
- Yates, R. C. (1952), A handbook on curves and their properties, J.W. Edwards.
- Lawrence, J. Dennis (1972), A catalog of special plane curves, Dover, ISBN 0-486-60288-5, https://archive.org/details/catalogofspecial00lawr.
External links
es:Curva plana
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