Bash Basharat

Posted on Monday, 19 April, 2004 - 07:49 pm:

Can anyone explain to me the relationship of transformations between polar curves? Is there one?

For example - is it possible to describe the nature of the transformation from say $r = \cos (2 θ)$ to $r = 2 + 4 \cos (2 θ)$ ? It looks like a stretch in both the $x$ and $y$ axes but that doesn't quite work.

Matthew Smith

Posted on Friday, 23 April, 2004 - 12:16 pm:

Well, part of your transformation, from

r = \cos (2 θ)

r = 4 \cos (2 θ)

, is just a stretch in both axes. It's multiplying

r

, the distance of each point from the origin, by 4, which is exactly the same as multiplying both the

x

and

y

co-ordinates by 4. For example, the point (1,1) is mapped to (4,4), the point (2,3) is mapped to (8,12) and, in general, the point

(x, y)

is mapped to

(4 x, 4 y)

. This doesn't change

θ

because the direction represented by the ratio

y / x

doesn't change.

However, the other part of your transformation, from $r = 4 \cos (2 θ)$ to $r = 2 + 4 \cos (2 θ)$ , isn't quite so simple to think of in Cartesian co-ordinates, as it maps $(x, y)$ to $(x + 2 x / \sqrt{x^{2} + y^{2}}, y + 2 y / \sqrt{x^{2} + y^{2}})$ , which isn't very helpful.

The most natural way to describe your transformation is in the language of polar co-ordinates: it is a stretch by a factor 4 in the radial direction, followed by a step outwards of 2 in the radial direction. A step in the radial direction, however, isn't quite like a shift in the $x$ or $y$ axes, as it doesn't leave the graph undistorted.

Matthew.