21.3 Polar coordinates, circles and spirals

I have used polar coordinates in a previous post. But that post contained mathematics that was more much difficult than is needed here. So, I am going to begin with a simple description of polar coordinates. Usually we describe the position of a point in space using Cartesian coordinates; in two dimensions the position of a point P is defined by a distance, x, along and x-axis, and a distance y, along a y-axis. These axes intersect at an origin, O, and, in an orthogonal Cartesian coordinate system, they are at a right-angle.

In a polar coordinate system, we describe the position of P by its distance, r, from O and the angle, θ, that OP makes with the x-axis; an anti-clockwise value of θ is defined to be positive. The picture above shows how the two coordinate systems can be used to describe the position of P. We can use the definitions of sine and cosine to convert polar coordinates (r,θ) into Cartesian coordinates (x,y) with the result that

x = rcosθ   and   y =rsinθ.                    (1)

And we can use Pythagoras’ theorem and the concept of an arctangent (the angle that has a tangent of a given value) to convert Cartesian coordinates to polar coordinates with the result that

r = (x2 + y2)1/2   and   θ = arctan(y/x)                    (2)

noting that when we raise a number to the power ½, we are finding its square root. If you find the arctangent confusing, it’s only another way of writing tanθ = (y/x).

Polar coordinates can make the description of some shapes very simple. For example, a circle of radius a has r = a for all values of θ, so, in polar coordinates it is described by the equation

r = a.                    (3)

Now let’s convert equation 3 into Cartesian coordinates using the first part of equation 2. The result is that

a = (x2 + y2)1/2   ⇒   a2 = x2 + y2.                     (4)

The right-hand version of equation 4 is the equation of a circle that I learnt at school.

The simplest spiral (the Archimedian spiral) has the equation

r = aθ.                    (5)

This means that r increases as θ increases from an initial value of zero. The picture above shows this spiral when a = 1, and θ is measured in radians. Equation 5 is very simple but if you use equations 1 and 2 to obtain the equation of the spiral in Cartesian coordinates, the result will be much more complicated. Note that equation 5 describes a spiral in which r increases as θ increases in an anti-clockwise direction; if a is negative, r increases as θ increases in an clockwise direction.

Some spirals are three-dimensional; like the examples shown in the picture above. Three-dimensional spirals are chiral – they are not identical to their mirror image. In a right-handed three-dimensional Cartesian coordinate system, the z-axis points in the direction of your thumb when you curl your fingers to point from the x– to the y-axis, as shown below. This picture also shows that the green spiral (shown above) turns in the same direction as your fingers. So, the green spiral is right-handed; its mirror image (which turns in the opposite direction) is left-handed.

These spirals move upwards from the origin, as θ increases; r increases with increasing θ (as for a two-dimensional spiral. So equation 6 describes a three-dimensional spiral

r = aθ,   z = bθ                    (6)

Here b is a constant for a given shape of spiral; the higher its value, the more steeply the spiral rises.

The topics in this post not just mathematical curiosities. In later posts, I hope to use these ideas the describe helices and logarithmic spirals – both of these shapes appear in natural systems.

Related posts

17.3 Three-dimensional vectors

Follow-up posts

21.5 Logarithmic spirals
21.6 The helix

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