Focus

In geometry, the term center point is closely related to the concept of the geometric center of gravity . Last but not least, it is used in the following contexts:

In the case of a segment , a circle , a sphere or in general an n-dimensional sphere , the center point is the point that has the same (minimum) distance from all points of this sphere . This definition can generally be made in (complete) metric spaces .

In the case of conic sections and the second-order surfaces described by quadrics (e.g. ellipsoids or cones ), the center points are the fixed elements of a reflection , which transforms the given figure into itself. All conic sections with the exception of the parabolas have exactly one center; a surface of the second order can have no, exactly one or a whole straight line or plane of centers. If it has exactly one center, it is referred to as a center quadric.

Description by coordinates

route

Is the end point and the starting point of a route is known, it can be the coordinates of the center point on the relationships , or in addition at a distance in space with determined. ${\ displaystyle \ mathrm {x_ {m} = {\ frac {x_ {1} + x_ {2}} {2}}}}$ ${\ displaystyle \ mathrm {y_ {m} = {\ frac {y_ {1} + y_ {2}} {2}}}}$ ${\ displaystyle \ mathrm {z_ {m} = {\ frac {z_ {1} + z_ {2}} {2}}}}$

Circle / sphere

If a circular equation of the form is given, the coordinates of the center point can be specified directly via . In a ball, the equation is added to the Z-axis: . The center is thus . ${\ displaystyle \ mathrm {r ^ {2} = (xa) ^ {2} + (yb) ^ {2}} \,}$ ${\ displaystyle \ mathrm {M (a, b)} \,}$ ${\ displaystyle \ mathrm {r ^ {2} = (xa) ^ {2} + (yb) ^ {2}} + (zc) ^ {2} \,}$ ${\ displaystyle \ mathrm {M (a, b, c)} \,}$