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 = 45o
Area of one loop = a2/2
CYCLOID
Equations in parametric form:
x = a(f - sinf)
y = a(1 - cosf)
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:
x2/3 + y2/3 = a2/3
Equations in parametric form:
x = a cos3q
y = a sin3q
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(ex/a + e-x/a)/2 = a cosh(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 30o 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 45o 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:
x = a(ln cotœf - cosf)
y = a sinf
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:
X = 2a cot q
Y = a(1 – cos 2q)
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:
x = a(cos f + f sin f)
y = a(sin f - f cos f)
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:
ax = (a2 – b2) cos3q
by = (a2 – b2) sin3q
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
