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SPECIAL PLANE CURVES

LEMNISCATE
Equation in polar coordinates:
r² = a²cos2θ

Equation in rectanular coordinates:
(x² + y²)² = a²(x² - y²)

Angle between AB' or A'B and x axis = 45^o

Area of one loop = a²/2

CYCLOID
Equations in parametric form:


Area of one arch = 3πa²

Arc length of one arch = 8a

This is a curve described by a point P on a circle of radius a rolling along x axis.

HYPOCYCLOID WITH FOUR CUSPS
Equation in rectangular coordinates:
x^2/3 + y^2/3 = a^2/3

Equations in parametric form:


Area bounded by curve = 3πa²/8

Arc length of entire curve = 6a

This is a curve described by a point P on a circle of radius a/4 as it rolls on the inside of a circle of radius a.

CARDIOID
Equation: r = a(1 + cosθ)

Area bounded by a curve = 3πa²/2

Arc length of a curve = 8a

This is the curve described by a point P of a circle of radius a as it rolls on the outside of a fixed circle of radius a.The curve is also a special case of the limacon of Pascal.

CATENARY
Equation:
y = a(e^x/a + e^-x/a)/2 = acosh(x/a)

This is a curve in which a heavy uniform chain would hang if suspended vertically from fixed points A and B.

THREE-LEAVED ROSE
Equation: r = acos3θ

The equation r = acos3θ is a similar curve obtained by rotating the curve counterclockwise 30^o or π/6 radians.

In general r = acosnθ or r = asinnθ has n leaves if n is odd.

FOUR-LEAVED ROSE
Equation: r = acos2θ

The equation r = asin2θ is a similar curve obtained by rotating the curve counterclockwise through 45^o or π/4 radians.

In general r = acosnθ or r = asinnθ has 2n leaves if n is even.

EPICYCLOID
Parametric equations:


This is a curve described by a point P on a circle of a radius b as it rolls on the outside of a circle of radius a.The cardioid is a special case of an epicycloid.

GENERAL HYPOCYCLOID
Parametric equations:


This is a curve described by a point P on a circle of a radius b as it rolls on the outside of a circle of radius a.

If b = a/4, the curve is hypocycloid with four cusps.

TROCHOID
Parametric equations:


This is a curve described by a pint P at distance b from the center of a circle of radius a as the circle rolls on the x axis.
If b < a, the curve is as shown on Fig.11-10 and is called curtate cycloid.
If b > a, the curve is as shown on Fig.11-11 and is called a prolate cycloid.
If b = a, the curve is a cycloid.

TRACTRIX
Parametric equations:


This is a curve described by endpoint P of a taut string PQ of length a as the other end Q is moved along the x axis.

WHITCH OF AGNESI
Equation in rectangular coordinates: y = 8a³/(x² + 4a²)

Parametric equations:


In the figure the variable line OA intersects y = 2a and the circle of radius a with center (0,a) at A and B respectively. Any point P on the "which" is located by constructing lines parallel to the x and y axes through B and A respectively and determining the point P of intersection.

FOLIUM OF DESCARTES
Equation in rectangular coordinates:
x³ + y³ = 3axy

Parametric equations:


Area of loop 3a²/2

Equation of asymptote: x + y + a = 0.

INVOLUTE OF A CIRCLE
Parametric equations:


This is a curve described by the endpoint P of a string as it unwinds from a circle of radius a while held taut.

EVOLUTE OF AN ELLIPSE
Equation in ractangular coordinates:
(ax)^2/3 + (by)^2/3 = (a² - b²)^2/3

Parametric equations:

This curve is the envelope of the normals to the ellipse x²/a² + y²/b² = 1.

OVALS OF CASSINI
Polar equation: r⁴ + a⁴ - 2a²r²cos2θ = b⁴.

This is the curve described by point P such that the product of its distances from two fixed points[distance 2a apart] is a constant b².

The curve is as in the figures according as b < a or b > a respectively.

If b = a, the curve is a lemniscate

LIMASCON OF PASCAL
Polar equation: r = b + acosθ

Let OQ be a line joining origin O to any point Q on a circle of diameter a passing through O.Then the curve is the locus of all points P such that PQ = b.

The curve is as in the figures below according as b > a or b < a respectively. If b = a, the curve is a cardioid.

CISSOID OF DIOCLES
Equation in rectangular coordinates: y² = x³/(2a - x)

Parametric equations:


This is the curve described by a point P such that the distance OP = distance RS. It is used in the problem if duplication of a cube, i.e. finding the side of a cube which has twice the volume of a given cube.

SPIRAL OF ARCHIMEDES
Polar equation: r = aθ