Calculus Of One Real Variable By Pheng Kim Ving
Chapter 13: Plane Curves Section 13.2.2: Sketching Polar Curves


13.2.2
Sketching Polar Curves

 

 

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1. Same Relation As Cartesian Or Polar Equation

 

 

{1.1} Example 4.1.

 

Fig. 1.1

 

 

Fig. 1.2

 

 

 

In this section we discuss the sketching of polar curves, which are curves of polar equations.

 

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2. Sketching By Using Cartesian Equations

 

To sketch a polar curve given with its polar equation, we can first try to find its Cartesian equation. If the Cartesian equation is
a function y = f(x), or if it's a relation f(x, y) = 0 which is recognized (eg (x h)2 + ( y k)2 r2 = 0 or (x h)2 + ( y k)2 =
r2 is recognized as the circle with centre (h, k) and radius r), then we can sketch the curve.

 

Example 2.1

 

 

Solution

 

Fig. 2.1

 

Curve For Example 2.1.

EOS

 

Example 2.2

 

 

Solution

 

Fig. 2.2

 

Circle With Centre (0, a/2) And Radius a/2.

EOS

 

When determining the direction of the curve, we rely on the original polar equation.

 

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3. Properties Of Polar Curves

 

There are cases where the Cartesian equation is of the form f(x, y) = 0 that isn't recognized (eg (x2 + y2 + ax)2 = a2(x2 + y2), where a is a positive constant). There are also cases where we're asked to use the polar equation itself directly without finding
its corresponding Cartesian equation. A table of values of course is helpful. Also there are properties of polar curves that can
help us in sketching a polar curve with more accuracy than by using a table of values alone.

Symmetry

 

Symmetry About The x-Axis

 

 

If one replacement yields the same equation, we can conclude that the curve is symmetric about the x-axis, without having to

Fig. 3.1

 

Symmetry About The x-Axis.

 

bother about the other replacement.

 

For a curve symmetric about the x-axis, getting its part above (or below) the x-axis is enough to sketch the entire curve, as the
part below (or above) the x-axis is simply the reflection about the x-axis of the part above (or below).

 

Symmetry About The y-Axis

 

 

Fig. 3.2

 

Symmetry About The y-Axis.

 

Symmetry About The Origin

 

 

Fig. 3.3

 

Symmetry About The Origin.

 

Tangent

 

Slope Of Tangent

 

 

 

 

 

Tangents At The Origin

 

 

Fig. 3.4

 

Tangents At The Origin.

 

 

 

 

Angle From Ray To Tangent

 

 

Fig. 3.5

 

Angle From Ray To Tangent.

 

 

 

 

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4. Sketching By Using Polar Equations

 

General Procedure

 

 

{4.1} Part 1.

 

Example 4.1 The Cardioid

 

 

Solution

 

the slope of the tangent at the origin at (0, 0) is:

 

 

 

Fig. 4.1

 

The Cardioid.

EOS

 

The origin of the name cardioid is clear from the shape of the curve.

 

 

Fig. 4.2

 

An Incorrect Sketch Of The Cardioid At The Origin.

 

 

Solution

 

 

Fig. 4.3

 

 

Direction.

 

EOS

 

 

Example 4.3 The Four-Leaved Rose

 

 

Solution

 

 

 

Fig. 4.4

 

The Four-Leaved Rose.

EOS

 

Example 4.4 The Lemniscate

 

 

Solution

 

4) Points Where Tangent Is Horizontal Or Vertical.

 

 

Fig. 4.5

 

The Lemniscate.

 

Direction.

EOS

 

Example 4.5 The Spiral Of Archimedes

 

 

Solution

 

Fig. 4.6

 

The Spiral Of Archimedes.

EOS

 

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5. Intersections Of Polar Curves

 

 

The Origin

 

 

{fig4.1} Fig. 4.1

{exa4.1} Example 4.1

{fig4.3} Fig. 4.3
{exa4.2} Example 4.2

 

Fig. 5.1

 

The Cardioid.

 

Fig. 5.2

 

 

Points Other Than The Origin

 

 

{fig4.4} Fig. 4.4

{exa4.3} Example 4.3

 

Fig. 5.3

 

A Circle And A Four-Leaved Rose.

 

 

Finding Points Of Intersection

 

 

 

separately, where k is an arbitrary integer.

 

 

Example 5.1

 

 

Solution

For The Origin. For the circle r = a/2, r can never be 0. So the origin isn't a point of intersection.

 

For Points Other Than The Origin.



EOS

 

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Problems & Solutions

 

 

{exa4.1} Example 4.1.

 

Solution

 

 

 

4) Points Where Tangent Is Horizontal Or Vertical

 

 

 

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Solution

 

 

 

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Solution

 

 

 

 

Direction.

 

 

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Solution

 

 

 

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Solution

 

For The Origin. For the circle r = a, r can never be 0. So the origin isn't a point of intersection.

 

For Points Other Than The Origin.

 

 

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