Introduction to Examples Using Polar Coordinates  Circles
In order to get a good idea of how the polar coordinate system works, let’s look at the graphs of some familiar objects. Circles, ellipses, spirals, and other figures whose equations are complicated in Cartesian coordinates can often be expressed much more simply in polar coordinates. In general, the polar direction θ is expressed in radians. In the examples that follow, the “rad” abbreviation is eliminated, because it is understood that all angles are in radians.
Circle Centered At Origin
The equation of a circle centered at the origin in the polar plane is given by the following formula:
r = a
where a is a realnumber constant greater than 0. This is illustrated in Fig. 103.
Circle Passing Through Origin
The general form for the equation of a circle passing through the origin and centered at the point ( θ _{0} , r _{0} ) in the polar plane (Fig. 104) is as follows:
r = 2 r _{0} cos ( θ – θ _{0} )
Ellipse Centered At Origin
The equation of an ellipse centered at the origin in the polar plane is given by the following formula:
r = ab /( a ^{2} sin ^{2} θ + b ^{2} cos ^{2} θ ) ^{1/2}
where a and b are realnumber constants greater than 0.
In the ellipse, a represents the distance from the origin to the curve as measured along the “horizontal” ray θ = 0, and b represents the distance from the origin to the curve as measured along the “vertical” ray θ = π/2. This is illustrated in Fig. 105. The values 2 a and 2 b represent the lengths of the semiaxes of the ellipse; the greater value is the length of the major semiaxis , and the lesser value is the length of the minor semiaxis .
Hyperbola Centered At Origin
The general form of the equation of a hyperbola centered at the origin in the polar plane is given by the following formula:
r = ab /( a ^{2} sin ^{2} θ − b ^{2} cos ^{2} θ ) ^{1/2}
where a and b are realnumber constants greater than 0.
Let D represent a rectangle whose center is at the origin, whose vertical edges are tangent to the hyperbola, and whose vertices (corners) lie on the asymptotes of the hyperbola (Fig. 106). Let a represent the distance from the origin to D as measured along the “horizontal” ray θ = 0, and let b represent the distance from the origin to D as measured along the “vertical” ray θ = π/2. The values 2 a and 2 b represent the lengths of the semiaxes of the hyperbola; the greater value is the length of the major semiaxis , and the lesser value is the length of the minor semiaxis .
Lemniscate
The general form of the equation of a lemniscate centered at the origin in the polar plane is given by the following formula:
r = a (cos 2 θ ) ^{1/2}
where a is a realnumber constant greater than 0. This is illustrated in Fig. 107. The area A of each loop of the figure is given by:
A = a ^{2}

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