Locus

In geometry, a locus is a collection of points which share a property. For instance a circle can be defined as the locus of points in a plane at a fixed distance from the given point.

A locus can alternatively be described as the path through which the point moves to fulfill the given condition or conditions. So, for instance, a circle can also be defined as the locus of a point moving so as to remain at the given distance from the fixed point.

Sometimes the curve can be defined by the set of conditions rather than by an equation, though an equation can be derived from the given conditions. Then the curve in question would be the locus of all the points which fit the conditions. For instance a circle can be said to be the locus of all points in a plane that is a fixed distance from a fixed point. A straight line can be defined as locus of all the points in a plane equidistant from 2 fixed points. The method of expressing a set of conditions in the analytical form gives an equation. A locus is the set of all points (forming a curve or surface) satisfying some condition. For instance, the locus of points in plane equidistant from a given point is a circle, and the set of points in 3-space equidistant from the given point is a sphere.

The locus of a point equidistant from 2 fixed points is the perpendicular bisector of the segment joining the two points. Any point on the perpendicular bisector of the line segment joining 2 given points is equidistant from them. We can say "the locus of all the points at distance R from the center point is a circle of radius R". In other words, we use the word locus to mean the shape formed by the set of points. An odd thing is that you can just drop the word locus, and it still makes sense: "The set of all the points distance R from a central point forms the circle".

Different rules will make different shapes. Many geometric objects have alternate definitions by using the concept of locus. For instance:

• Straight line "The locus of all the points equidistant from 2 given points".
• Ellipse "The locus of all the points where the sum of the distance to 2 fixed points is a constant.

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