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The locus of points in the euclidean plane that satisfy some geometric or algebraic definition is called a curve.
a curve is considered to be the locus of a set of points that satisfy an algebraic or transcendental equation in two variables.
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.

cardioid

A heart-shaped curve generated by a point of a circle that rolls (without slipping) on a fixed circle of the same diameter. In point-wise construction of the curve, let O be a fixed point of a circle C of diameter a, and Q a variable point of C. Lay off distance a along the secant OQ, in both directions from Q. The locus of the two points thus obtained is a cardioid (see illustration). If a rectangular coordinate system is chosen with O for origin initially and y axis tangent to C at O, the cardioid has equation (x² + y² ? ax)² = a²(x² ? y²). The equation in polar coordinates is p = a(1 + cos ?). Its area is 3/2?a², or six times the area of C, and its length is 8a.

The red curve is a cardioid.

parametric equations

$x(t) = 2r \left( \cos t - {1 \over 2} \cos 2 t \right),$

$y(t) = 2r \left( \sin t - {1 \over 2} \sin 2 t \right)$

polar equation

$\rho(\theta) = 2r(1 - \cos \theta). \$

The area of a cardioid with polar equation

?(?) = a(1 - cos?)

is

$A = {3\over 2} \pi a^2$ .

Cycloid

A curve traced in the plane by a point on a circle that rolls, without slipping, on a line. If the line is the x axis of a rectangular coordinate system, at whose origin O the moving point P touches the axis, parametric equations of the cycloid are x = a(? ? sin ?), y = a(1 ? cos ?), when a is the radius of the rolling circle, and the parameter ? is the variable angle through which the circle rolls

Cycloid (red) generated by a rolling circle

Graph of cycloid generated by a circle of radius r=2

The cycloid through the origin, created by a circle of radius r, consists of the points (x,y) with

$x = r(t - \sin t)\,$

$y = r(1 - \cos t)\,$

Area

One arch of a cycloid genereated by a circle of radius r can be parametrized by

$x = r(t - \sin t)\,$

$y = r(1 - \cos t)\,$

with

$0 \le t \le 2 \pi$ .

Since

$\frac{dx}{dt} = r(1- \cos t)$

we find the area under the arch to be

$A=\int_{t=0}^{t=2 \pi} y \, dx = \int_{t=0}^{t=2 \pi} r^2(1-\cos t)^2 \, dt =\left. r^2 \left( \frac{3}{2}t-2\sin t + \frac{1}{2} \cos t \sin t\right) \right|_{t=0}^{t=2\pi} =3 \pi r^2.$

Lemniscate of Bernoulli

A curve shaped like the figure eight (see illustration), referred to by Jacques Bernoulli in 1694. Let F₁, F₂ be points of a plane ? with F₁F₂ = 2 a, a > 0. The locus of a point P of ? which moves so that PF₁ · PF₂ = b₂, where b is a positive constant, is called an oval of Cassini The lemniscate is obtained when b = a. Its equation in rectangular coordinates is (x² + y²)²= a²(x² ? y²) and in polar coordinates ?²= a² cos 2?.

Curve known as lemniscate.

Rose curve

A type of plane curve that consists of loops (leaves, petals) emanating from a common point and that has a roselike appearance. Taking the common point O as the pole of a polar coordinate system (see illustration), these curves have equations of the form ? = a · sin n?, where a > 0 and n is a positive integer (also ? = a · cos n?, with a different choice of the initial line of the coordinate system). The curve is a circle of diameter a for n = 1. It has n or 2n leaves, according as n is an odd or even integer, respectively. The lemniscate is sometimes called a two-leaved rose, though its equation ?² = a ² cos 2? is not of the form given above.

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