- Source: Plane curve
In mathematics, a plane curve is a curve in a plane that may be 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
f
(
x
,
y
)
=
0
{\displaystyle f(x,y)=0}
for some specific function f. If this equation can be solved explicitly for y or x – that is, rewritten as
y
=
g
(
x
)
{\displaystyle y=g(x)}
or
x
=
h
(
y
)
{\displaystyle x=h(y)}
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
(
x
,
y
)
=
(
x
(
t
)
,
y
(
t
)
)
{\displaystyle (x,y)=(x(t),y(t))}
for specific functions
x
(
t
)
{\displaystyle x(t)}
and
y
(
t
)
.
{\displaystyle y(t).}
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
R
2
{\displaystyle \mathbb {R} ^{2}}
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
f
(
x
,
y
)
=
0
,
{\displaystyle f(x,y)=0,}
where
f
:
R
2
→
R
{\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} }
is a smooth function, and the partial derivatives
∂
f
/
∂
x
{\displaystyle \partial f/\partial x}
and
∂
f
/
∂
y
{\displaystyle \partial f/\partial y}
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
f
(
x
,
y
)
=
0
{\displaystyle f(x,y)=0}
(or
F
(
x
,
y
,
z
)
=
0
,
{\displaystyle F(x,y,z)=0,}
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
x
2
+
y
2
=
1
{\displaystyle x^{2}+y^{2}=1}
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
x
2
+
y
2
=
1
{\displaystyle x^{2}+y^{2}=1}
(that is the projective curve of equation
x
2
+
y
2
−
z
2
=
0
{\displaystyle x^{2}+y^{2}-z^{2}=0}
). 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):
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, ASIN B0007EKXV0.
Lawrence, J. Dennis (1972), A catalog of special plane curves, Dover, ISBN 0-486-60288-5.
External links
Weisstein, Eric W. "Plane Curve". MathWorld.
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