Calculus Of One Real Variable By Pheng Kim Ving
Chapter 13: Plane Curves Section 13.2.1: The Polar Coordinate System


13.2.1
The Polar Coordinate System

 

 

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1. The Polar Coordinate System

 

 

Fig. 1.1

 

Defining A Unique Point.

 

To setup a polar coordinate system, select a point O in the plane and draw a ray from O. See Fig. 1.2. Conventionally for
convenience the ray is horizontal and to the right of O.

 

Definition 1.1 The Polar Coordinate System

 

A point O and a ray emanating from it to the right form the polar coordinate system. The point O is called the pole and
the ray is called the polar axis. The pole O is the origin of this system. Any point P in the plane then has its position in the
polar coordinate system determined by:

 

 

 

Fig. 1.3 displays some points with their polar coordinates.

 

Fig. 1.2

 

Polar Coordinate System.

 

Fig. 1.3

 

Some Points With Their Polar Coordinates.

 

 

Angle

 

 

Negative Distance

 

 

Fig. 1.4

 

Negative Distance.

 

Infinitely Many Polar Coordinates For A Point

 

 

Property 1.1 Polar Coordinates

 

For any real r > 0 and for all integer k:

 

 

 

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2. Cartesian And Polar Coordinate Systems


 

Property 2.1 Cartesian And Polar Coordinates

 

 

 

Fig. 2.1

 

Cartesian And Polar Coordinate Systems.

 

Eqs. [2.1] express the Cartesian coordinates in terms of the polar ones, and Eqs. [2.2] and [2.3] express the polar coordinates
in terms of the Cartesian ones. These equations can be used to transform the coordinates of points from one coordinate
system to the other.

 

Example 2.1

 

 

Solution

EOS

 

 

Example 2.2

 

 

Solution

EOS

 

 

The quadrant where the point lies is important. The coordinates of the point, Cartesian or polar, must mean that the point lies
in the correct quadrant.

 

Example 2.2 is the reverse of Example 2.1. The reverse task maps a single ordered pair of Cartesian coordinates back to the
original ordered pair of polar coordinates plus maps it to infinitely many other such ordered pairs. Of course the sign of the
distance depends on whether the given point lies on the ray determined by the angle (positive) or its extension (negative).

 

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3. Polar Equations And Polar Curves

 

 

Fig. 3.1

 

Polar Equation.

 

 

Definition 3.1 Polar Equations And Polar Curves

 

 

 

A Cartesian equation generally represents some curve in the Cartesian coordinate system. Similarly, a polar equation generally
represents some curve in the polar coordinate system.

 

Level Coordinate Curves

 

 

Fig. 3.2

 

Level Coordinate Curves Of Polar System.

 

Cartesian And Polar Equations

 

Equations in Property 2.1 can be utilized to transform the Cartesian representation of a curve into its polar representation and
vice versa.

 

Example 3.1

 

Transform the Cartesian equation y = 2x 1 of a line to a polar equation.

 

Solution

EOS

 

Example 3.2

 

Find the polar representation of the ellipse with Cartesian representation x2 + 3y2 = 5.

 

Solution

EOS

 

 

Example 3.3

 

 

Solution 1


EOS

 

Solution 2


EOS

 

 

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Problems & Solutions

 

 

Solution

 

 

Note

 

For part c, as r = 0, it's the origin, so there's no need to do any calculation.

 

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Solution

 

 

Note

 

 

{p&s 1} Problem & Solution 1.

 

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3. Find the polar equation of each of the following curves with the given Cartesian equation.
a. x = c.
b. x2y + y3 = 4.

 

Solution

 

 

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4. Determine the Cartesian equation of each of the following curves with the given polar equation. Identify the curve.

 

Solution

 

 

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Solution

 

 

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