Point reflection

The point symmetry, also central symmetry in geometry is a property of a figure. A figure is point-symmetric if it is imaged by the reflection at a symmetry point to itself.

Definition

A (plane ) geometric shape (eg a square ) is called point-symmetric if there is a point reflection that maps this character to be. The point at which this reflection is performed is referred to as center of symmetry.

Point reflection as rotation

A reflection point of the geometric figure corresponding to a rotation by 180 °. Thus, the symmetry point is a special case of rotational symmetry.

Examples

• When a square is point symmetry ( in itself ) if and only present if it is a parallelogram. The center of symmetry is then the intersection of the diagonals. As special cases of the parallelogram rectangle rhombus square and point symmetry.
• Each circle is ( in itself ) point symmetry with respect to its center.
• Two circles with the same radius are point-symmetrical to each other. The center of symmetry is the center of the connecting line of the two circular centers.
• Several centers of symmetry, there can be only when the character is not limited. The simplest example is the straight line. She even infinitely many centers of symmetry.
• A triangle is never point symmetry. But it may be point-symmetrically to one another, two triangles.

Point symmetry of function graphs

Overview

A frequent in school mathematics task is to prove that the graph of a given function with the domain and the real numbers is point-symmetric as range of values.

Is there a point so that the equation for the function

Applies to all, then the function is point symmetric to the point, the condition is called with

Equivalent, such as the substitution shows. In the special case of point symmetry about the origin of this equation simplifies to

If it is valid for all x, is point symmetry prior to the coordinate origin. Then you call the function odd function.

Examples

Point symmetry about the coordinate origin

If the function is Then:

Thus, the function graph is point symmetric about the origin (0,0).

Point symmetry to the point (0,2)

Consider the function Select and Then:

Consequently, the function graph is point symmetric to the point and it is

To determine the point of symmetry, this method does not help. Mostly, however, it is sufficient to sign the function graphs and derive a conjecture concerning the symmetry point.