ENTRY CURVES Authors: Oliver Knill, Andrew Chi, 
2003 
Literature: www.mathworld.com, www.2dcurves.com


+------------------------------------------------------------
| astroid
+------------------------------------------------------------

An astroid is the curve 
t mapsto (cos^3(t),a sin^3(t)) with a>0. 
An asteroid is a 4-cusped hypocycloid. 
It is sometimes also called a tetracuspid, 
cubocycloid, or paracycle. 


+------------------------------------------------------------
| Archimedes spiral
+------------------------------------------------------------

An Archimedes spiral is a curve described as
the polar graph r(t) = a t where a>0 is 
a constant. In words: the distance r(t) to the 
origin grows linearly with the angle. 


+------------------------------------------------------------
| bowditch curve
+------------------------------------------------------------

The bowditch curve is a special Lissajous curve 
r(t)=( a sin(nt+c), b sin(t)). 


+------------------------------------------------------------
| brachistochone
+------------------------------------------------------------

A brachistochone is a curve along which a particle
will slide in the shortest time from one point to
an other. It is a cycloid.


+------------------------------------------------------------
| Cassini ovals
+------------------------------------------------------------

Cassini ovals are curves described by
((x+a) +y^2)((x-a)^2+y^2) = k^4, where k^2<a^2 are
constants. They are named after the Italian astronomer 
Goivanni Domenico Cassini (1625-1712). Geometrically
Cassini ovals are the set of points whose product to 
two fixed points P=(-a,0),Q=(0,0) in the plane is the 
constant k^2. For k^2=a^2, the curve is called a 
Lemniscate.


+------------------------------------------------------------
| cardioid
+------------------------------------------------------------

The cardioid is a plane curve belonging to the class of 
epicycloids. The fact that it has the shape of a heart gave
it the name. The cardioid is the locus of a fixed point P 
on a circle roling on a fixed circle. In polar
coordinates, the curve given by r(phi) = a (1+cos(phi)).


+------------------------------------------------------------
| catenary
+------------------------------------------------------------

The catenary is the plane curve which is 
the graph y = c cosh(x/c). It was discovered by Jacques Bernoulli.
It has the shape of a uniform flexible chain hung from two points.


+------------------------------------------------------------
| catenoid
+------------------------------------------------------------

The catenoid is the surface obatained by rotating
the catenary about the x-axis. The minimal surface
bounded by two coaxial rings can be a catenoid.


+------------------------------------------------------------
| circle
+------------------------------------------------------------

A circle in the plane is the curve
r(t)=(r cos(t),r sin(t)) where the radius r
is a constant. It is the set of points which have
a fixed distance r from the origin. 
More generally, a circle is the set of points in a metric space 
which have a fixed distance from a given point. 


+------------------------------------------------------------
| cissoid
+------------------------------------------------------------

A cissoid is a plane curve given in Euclidean coordinates by
y^2 (2a-x) = x^3. In polar coordinates, it satisfies
r(t) = 2 a tan(t) sin(t) or in Euclidean coordinates
r(t)= (2a sin^2(t),2a sin^3(t)/cos(t)).
The curve has a cusp at the origin. It was first mentioned by 
Diocles in 180 B.C. 


+------------------------------------------------------------
| conic section
+------------------------------------------------------------

A conic section is a nondegenerate curve generated by intersecting a
plane with one or two nappes of a cone.  The three congruence
classes of conic sections are the ellipse, the parabola, and the
hyperbola.


+------------------------------------------------------------
| ellipse
+------------------------------------------------------------

An ellipse is the locus of all points in the plane the sum of
whose distances from two fixed points is a positive constant.  It is
also the conic section which results from a plane which intersects
only one nappe of the cone.  The general formula for an ellipse up
to rotation and translation is
fracx^2a^2 + fracy^2b^2 = 1.


+------------------------------------------------------------
| hyperbola
+------------------------------------------------------------

A hyperbola is the set of points in the plane for which 
the difference of the distances from two fixed points is a constant.
It is also the conic section which results from a plane
intersecting a cone.  The general formula for a
hyperbola up to rotation and translation is
fracx^2a^2 - fracy^2b^2 = 1.


+------------------------------------------------------------
| curve
+------------------------------------------------------------

A curve is a continuous map from the real line to an
the plane or to space. The word "curve" is often
used to mean the image of this map. Curves can be 
represented parametrically by r(t)=(x(t),y(t)) or 
implicitely as f(x,y)=0. 


+------------------------------------------------------------
| Airy function
+------------------------------------------------------------

The Airy function is commonly found as a solution to boundary value
problems in quantum mechanics and electromagnetism. It is the
solution to the differential equation:
y''=xy. The two independent solutions are (without constants):
 Ai(x) = int_0^infinity cos ( fract^33 + xt ) dt
 Bi(x) = int_0^infinity ( e^-fract^33 + xt 
         + sin ( fract^33 + xt ) ) dt 


+------------------------------------------------------------
| algebraic curve
+------------------------------------------------------------

A plane curve is an algebraic curve if it is given by
g(x,y)=0 where g is algebraic a polynomial in x and y. 
An algebraic curve with degree greater than 2 is called a higher plane curve.
The circle g(x,y) = x^2+y^2-1=0 is an example of an algebraic curve, the 
catenary g(x,y) = y - c cosh(x/c)=0 is an example of a nonalgebraic curve.


+------------------------------------------------------------
| cubic curve
+------------------------------------------------------------

A cubic curve is an algebraic curve of order three. 
Newton showed that all cubics can be generated as projections of the five
divergent cubic parabolas. Examples include the cissoid of Diocles
and ellptic curves.


+------------------------------------------------------------
| ampersand curve
+------------------------------------------------------------

The ampersand curve is a quartic curve with implicit equation
(y^2 - x^2)(x-1)(2x-3) = 4 (x^2 + y^2 - 2x)^2.
It looks like an ampersand.


+------------------------------------------------------------
| bean curve
+------------------------------------------------------------

The bean curve is a quartic curve given by the implicit equation:
x^4 + x^2 y^2 + y^4 = x (x^2 + y^2). It looks like a bean.


+------------------------------------------------------------
| bicorn
+------------------------------------------------------------

The bicorn is the name of a collection of quartic curves studied by
Sylvester in 1864 and Cayley in 1867. It is given by
y^2(a^2 - x^2) = (x^2 + 2ay - a^2)^2. 


+------------------------------------------------------------
| bicuspid
+------------------------------------------------------------

The bicuspid is the quartic curve given by the implicit equation:
(x^2 - a^2)(x-a)^2 + (y^2-a^2)^2 = 0.


+------------------------------------------------------------
| bow
+------------------------------------------------------------

The bow is a quartic curve with the implicit equation:
x^4 = x^2 y - y^3.


+------------------------------------------------------------
| cartesian oval
+------------------------------------------------------------

A cartesian oval is a quartic curve consisting of two ovals.  It
is the locus of a point P whose distances from two foci F_1 and
F_2 in two-center bipolar coordinates satisfy mr pm nr' = k
where m and n are positive integers, k is a positive real, and
r and r' are the distances from F_1 and F_2.  If m=n, then
the oval becomes an ellipse.


+------------------------------------------------------------
| Cassini oval
+------------------------------------------------------------

A Cassini oval is one of a family of quartic curves, also called
Cassini ellipses, described by a point such that the product of its
distances from two fixed points a distance 2a apart is constant
b^2. The shape of the curve depends on b/a.  The Cassini ovals
are defined in two-center bipolar coordinates by the equation 
r_1 r_2 = b^2 where b is a positive constant.


+------------------------------------------------------------
| cruciform
+------------------------------------------------------------

A cruciform is a plane quartic curve also called the cross curve
or policeman on point duty curve.  It is given by the implicit
equation: x^2 y^2 - b^2 x^2 - a^2 y^2 = 0. 


+------------------------------------------------------------
| lemniscate
+------------------------------------------------------------

The lemniscate, also known as the lemniscate of Bernoulli, is a
polar curve whose most common form is the locus of points the product
of whose distances from two fixed points a distance 2a away is the
constant a^2.  The usual polar coordinate form is as follows: 
r^2 = a^2 cos (2 theta).


+------------------------------------------------------------
| natural equation
+------------------------------------------------------------

A natural equation is an equation which specifies a curve
independent of any choice of coordinates or parametrization.  This
arose in the solution to the following problem: given two functions of
one parameter, find the space curve for which the functions are the
curvature and torsion. Often, the natural equation will be in
terms of integrals.


+------------------------------------------------------------
| polynomial curve
+------------------------------------------------------------

A polynomial curve is a curve obtained by fitting polynomials to
a sequence of points. To fit curves better, splines like Bezier curve 
are more suited.


+------------------------------------------------------------
| quadrifolium
+------------------------------------------------------------

A quadrifolium is a rose curve with n=2.  It has polar equation
r = a sin (2 theta).


+------------------------------------------------------------
| sextic curve
+------------------------------------------------------------

A sextic curve is an algebraic curve of degree 6.  Examples include the
atriphtaloid and the butterfly curve y^6 = x^2 - x^6.


+------------------------------------------------------------
| atriphtaloid
+------------------------------------------------------------

The atriphtaloid is a sextic curve also known as atriphtothlassic
curve and given by the equation: x^4(x^2+y^2) - (ax^2 - b)^2 = 0.


+------------------------------------------------------------
| butterfly curve
+------------------------------------------------------------

The butterfly curve is a sextic plane curve given by the 
implicit equation y^6 = x^2 - x^6.


+------------------------------------------------------------
| trifolium
+------------------------------------------------------------

A trifolium is the 3-petalled rose given in polar form as
r(t) = a |cos (3 t)|.


+------------------------------------------------------------
| spiral
+------------------------------------------------------------

A spiral, in general, is a curve with tau(s)/kappa(s) constant for 

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