Science

Circles

Conventionally, a circle is described by its centre C (a, b) and radius (r). The distance of a point P (x, y) from the centre of the circle is given by Which is based on Pythagoras' Theorem.

• ♦: d = 0, P = C, it is at the centre of the circle.
• ♦: 0 < d < r, P is within the circle, but not at the centre.
• ♦: d = r, P is on the edge ("circumference") of the circle.
• ♦: d > r, P lies outside the circle.

The problem

Most graphics programs either allow you to draw circles by marking the centre, then dragging to set the radius, or they draw them to fit an invisible box dragged out by the cursor. Sometimes, I may want a circle in an image to act as a border, and cross through several features that should be at the edge of an image, so neither of the common ways of describing circles to a graphics program were particulally useful. Yes, you can keep re-drawing and dragging circles until one fits, but doesn't that seem a bit tedious?

What are lines and circles, mathematically?

A line can be described by two points that it passes through. It can also be described with a point on the line, and a direction that the line leaves/enters the point in:

A circle can also be described by three points that it's circumference passes through.
As all three of our points lie on the circumference, they will all have a distance from the centre of the circle, equal to it's radius. Therefore, given three points, the centre of the circle can be anywhere that is at the same distance ("equidistant") from each of the three points. Usually, there is only one such place that is equidistant from the three given points, so we will get one circle and only one.

Parallax and the bisector of two points

With two points (that aren't overlapping), the path formed by points ("locus") that are equidistant from them lies along a line called the "bisector". This line is at exactly 90 degrees (a "right angle") to a line joining the two points, and halfway along it, ie. halfway between either point. With three points, we can draw three bisectors. For this problem though, we only need two.

• If two lines are "parallel", they never* meet (ie. don't "intersect"), unless they also overlap:
• If the lines are parallel and overlap ("colinear"), then they are always touching - they never separate.
• In all other cases ("parallax"), they meet only once, at one point (the "intersection").

* = not quite true, but we can assume so for this problem. Technically, parallel lines meet at infinity (). We can never have parallel lines in this problem (unless they're also colinear), as they have to meet at their common point (the point that they both pass through - we have two lines from three points, so they must share a point). If the lines were colinear, this would give us a circle with an infite radius though - which just turns out to be a straight line through the points. Since we want a circle, we can ignore this possibility.

Three points and some bisectors

We start with three points that aren't colinear. First, we number them 1, 2 and 3 and draw some lines to connect them:

We now draw the perpendicular* bisectors of the connecting lines and label these. Below, I've labelled the bisector of points 1 and 2 "A" and the bisector of points 2 and 3 "B":
* perpendicular = at a right angle (90 degrees)

Finding the centre and radius

Now time to apply some logic:

• Any point on A is always an equal distance d_1,2 from points 1 and 2 (see "bisector" above).
• Any point on B is always an equal distance d_2,3 from points 2 and 3.
• Therefore, where A and B meet, at d_1,2 = d_2,3, there is a point that is an equal distance d from all three points.
• As the circumference of a circle is the locus of points at a distance r from its centre, r = d and the centre of the circle is at the intersection of the bisectors.

Or put in simpler terms: where these bisectors meet, we have the centre of our circle! The radius of the circle (distance from the centre to the edge) is the distance between any of the points and the centre:

As we have a centre and a radius, we can now draw the circle:

Animated method

The whole process in stages is shown in the animation below:

This requires an SVG-capable browser such as Opera. It will almost certainly NOT work in Internet Explorer, and Chrome doesn't seem too happy with animated SVG either.

The algebra

To describe this mathematically, vectors are very useful. They aren't essential, but they do reduce the amount of writing considerably, and simplify it.