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Locus

In geometry, a locus (Latin for place, plural loci) is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point.

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

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

The locus of a point equidistant from two fixed points is the perpendicular bisector of the segment joining the two points. Any point (or every point) on the perpendicular bisector of the line segment joining two given points is equidistant from them. We can say the locus of all points 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 points distance R from a central point forms a circle".

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.
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