## 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

 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.

 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 (xh)2 + ( yk)2r2 = 0 or (xh)2 + ( yk)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.

 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.

 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

 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

# Problems & Solutions

{exa4.1} Example 4.1.

Solution

4) Points Where Tangent Is Horizontal Or Vertical

Solution

Solution

Direction.

Solution

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.