Geometric place

In elementary geometry , a geometric location (plural: geometric locations ) denotes a set of points that have a certain, given property. In planar geometry , this is usually a curve , for which the word locus or locus is also used. In navigation , however, one speaks of base lines .

Locus lines have been fundamental to geometric constructions since Euclid's Elements : A point is determined by specifying two locus lines whose intersection it forms. In the classic case, where only compasses and rulers are permitted, these are two straight lines , two circles or one straight line and one circle.

Examples

The classic locus lines in plane geometry

The locus of all points that have a fixed distance from a given point is the circle around with the radius . ${\ displaystyle M}$ ${\ displaystyle r}$ ${\ displaystyle M}$ ${\ displaystyle r}$

The locus of all points that are a fixed distance from a given line is the pair of parallels to at a distance . ${\ displaystyle g}$ ${\ displaystyle d}$ ${\ displaystyle g}$ ${\ displaystyle d}$

The locus of all points that points given by two and the same distance have is the mid-vertical on the track . ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle {\ overline {AB}}}$

The locus of all points which are given by two intersecting straight lines and have the same distance is the pair of bisectors to and . ${\ displaystyle g}$ ${\ displaystyle h}$ ${\ displaystyle g}$ ${\ displaystyle h}$

The locus of all points which are of two given parallel straight lines and have the same distance is the median parallel to and . ${\ displaystyle g}$ ${\ displaystyle h}$ ${\ displaystyle g}$ ${\ displaystyle h}$

The locus of all points that lie in a certain direction from a given point is the straight line through this point with the given direction (e.g. bearing ).

Geometric locations that are not local lines

The geometric location of all points whose distance from a given point is less than a fixed number is the open circular disk with the radius . ${\ displaystyle M}$ ${\ displaystyle r}$ ${\ displaystyle M}$ ${\ displaystyle r}$

The geometric location of all points whose distance from a given point is not greater than the distance from another given point is the closed half-plane , which is delimited by the perpendicular above the line and in which it lies. ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle {\ overline {AB}}}$ ${\ displaystyle A}$

etc.

The geometric location of all points that are equidistant from the three corners of a triangle is the circumcenter .

The geometric location of all points that are equidistant from the three sides of a triangle is the center of the circle .

Spatial geometry

The locus of all points of a given point at a fixed distance have is the spherical surface to the radius . Practical examples include inclined distances and positioning with GPS satellites. ${\ displaystyle M}$ ${\ displaystyle r}$ ${\ displaystyle M}$ ${\ displaystyle r}$

The locus of all points from a given point and a given plane at the same distance, forming a paraboloid order . ${\ displaystyle M}$ ${\ displaystyle E}$ ${\ displaystyle M}$

etc.

More examples from planar geometry

The locus of all vertices of right angles , the legs of which go through two given points and , is the Thales circle over the line . ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle {\ overline {AB}}}$

The locus of all points from which two given points and at a certain angle can be seen is the pair of barrel arcs over with the peripheral angle (circumferential angle) . ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle \ varphi}$ ${\ displaystyle {\ overline {AB}}}$ ${\ displaystyle \ varphi}$

The locus of all points for which the sum of their distances from two given points and has the fixed value is the ellipse with the focal points and and the semi-major axis . ${\ displaystyle F_ {1}}$ ${\ displaystyle F_ {2}}$ ${\ displaystyle 2a}$ ${\ displaystyle F_ {1}}$ ${\ displaystyle F_ {2}}$ ${\ displaystyle a}$

The locus of all points for which the difference between their distances from two given points and has the fixed value is the hyperbola with the focal points and and the real semi-axis . ${\ displaystyle F_ {1}}$ ${\ displaystyle F_ {2}}$ ${\ displaystyle 2a}$ ${\ displaystyle F_ {1}}$ ${\ displaystyle F_ {2}}$ ${\ displaystyle a}$

The locus of all points that have the same distance to a given straight line and a given point is the parabola with the focal point and the guideline (guide line) . ${\ displaystyle g}$ ${\ displaystyle F}$ ${\ displaystyle F}$ ${\ displaystyle g}$

The geometrical location of all points for which the quotient of their distances from two given points has a certain value is the circle of Apollonios . ${\ displaystyle \ lambda \ neq 1}$

Application example

In order to draw the tangent to a given circle (with a center point ) that goes through a point specified outside the circle , it is not sufficient to use the ruler to determine a line that goes through and "grazes" as well as possible. Rather, the point of contact located on the circle must first be determined. This results from the intersection of two locus lines: ${\ displaystyle k}$ ${\ displaystyle M}$ ${\ displaystyle P}$ ${\ displaystyle P}$ ${\ displaystyle k}$

The first local line here is the already given circle.
The second local line in this case is the Thaleskreis above the route . ${\ displaystyle {\ overline {PM}}}$

There are two points of intersection, hence two tangents.