# Locus

Definition: A shape created by the set of points whose position satisfies a given set of rules.

Try this In the figure below, the small gray dots represent all the possible points on the plane. They are drawn far apart so you can see them. If we select all the points that are a given distance from the point C ( larger, blue dots), the result is a circle. If you drag the orange dot on the circle, the selected points form a circle whose radius varies.

To understand the concept of locus, imagine that the plane is composed of an infinite number of points packed closely together. Then, select only those points whose location meet certain rules. The selected points then form a shape, perhaps a line or curve.

In the figure above, the rule used is that the point is a certain given distance from another fixed point. The result is a set of points forming a circle, with the given fixed point as its center. All the points closer or further from the center point are not part of the locus. The ones that are the correct distance form a circle.

## Word usage

We can say "the locus of all points on a plane at distance R from a center point is a circle of radius R". In other words, we tend to use the word locus to mean the shape formed by a set of points. An odd thing is that you can often just drop the word locus, and it still makes sense: "The set of all coplanar points distance R from a central point forms a circle".

## Why is the circle above "jagged"?

In the figure above the circle is not exactly round, but has jagged corners in some places. This is because the software draws the points far apart so you can see what is going on. At each position around the circle, it highlights the point closest to the one that forms a circle. If the points were infinitely small and packed tightly, the circle would appear perfectly smooth.

## Other shapes

Different rules will create different shapes. Many geometric object have alternate definitions using the concept of locus. For example:

• Straight line "The locus of all points equidistant from two given points".
• Ellipse "The locus of all points where the sum of the distance to two fixed points is a constant." See Ellipse definition.

## Motion

Sometimes the idea of locus has a slightly different explanation. If you think of a point moving along some path, we sometimes say that the path is the locus of the point. So for example a point that moves a fixed distance from another point draws out a circle. So we could say

"The locus of a point moving at a fixed distance from a center point is a circle".