﻿ Locus (mathematics)

# Locus (mathematics)

In elementary geometry refers to the locus (plural: loci geometric ) a set of points that have a certain given property. In the planar geometry, this is a curve in the control, for which is used the word locus or locus. In the navigation one speaks of state lines.

Loci are fundamental for geometric constructions since Euclid's " Elements": A point is determined that two loci are specified, the intersection of which it forms. In the classical case, where only compass and ruler are allowed, are the two straight lines, two circles or a straight line and a circle.

## Examples

### The classical local lines in the plane geometry

• The locus of all points having r of a given point M a fixed distance, the circle M the radius r.
• The locus of all points that have g d of a given line at a fixed distance, the pair of parallels to g at a distance d
• The locus of all points having the same distance from the two given points A and B is on the perpendicular bisector of the line.
• The locus of all points of two given intersecting lines g and h have the same distance, the pair of bisectors to g and h
• The locus of all points g and h have the same distance from two given parallel straight lines is parallel to the means g, and h
• The locus of all points that are of a given point in a certain direction is the line through this point with the given direction (eg bearing ).

### Geometric loci that are not local lines

• The locus of points whose distance from a given point M is less than a fixed number r, the open disk to M with the radius r.
• The locus of points whose distance from a given point A is not greater than the distance from another given point B, the closed half- plane bounded by the perpendicular bisector to the line and is located in the A.
• Etc.
• The locus of all points that are equidistant from the three vertices of a triangle, the circumcenter.
• The locus of all points that are equidistant from the three sides of a triangle is the incenter.

### Spatial geometry

• The locus of all points having r of a given point M a fixed distance, the spherical surface to M with the radius r. Practical examples are slope distances and the location with GPS satellites.
• The locus of all points from a given point M and a given plane E have the same distance, forming a paraboloid of M.
• Etc.

### Other examples from the planar geometry

• The locus of all vertices of right angles, the legs go through two given points A and B, is the Thales circle over the track.
• The locus of all points that can be seen from those given two points A and B at a certain angle, the barrel arc pair on with the peripheral angle ( angle at circumference ).
• The locus of points for which the sum of its distances from two given points F1 and F2, has a fixed value of 2a, the ellipse having the focal points F1 and F2 and the semimajor axis a
• The locus of all points for which the difference of its distances from two given points F1 and F2 has the fixed value 2a, is the hyperbola with foci F1 and F2 and the real half axis a
• The locus of all points that have the same distance from a given line g and a given point F, is the parabola with focus F and Guideline ( directrix ) g
• The locus of all points for which the ratio of its distances from two given points has a certain value, is the circle of Apollonius.

## Example of use

Order k ( with center M ) to draw the tangent to a given circle, which passes through a given point P outside the circle, it is not enough to identify with the ruler, draw a line that passes through P and k as well as possible " touches ". Rather, the first is located on the circle of contact to determine. This was calculated as the intersection of two loci:

• First locus is the circle already given here.
• Second locus in this case is the Thales circle over the track.

This results in two intersections, hence two tangents. (see drawing)

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