Reflection symmetry

Symmetry axis is a property of a character in the geometry. Equivalent descriptions of this property are axial symmetry or axial symmetry. A character is called axially symmetric if it is represented by the vertical axis reflection on its symmetry axis to itself. In the case of a two-dimensional figure, the terms " axis of symmetry " and " mirror symmetry " synonymous. In three-dimensional spaces, the symmetry is referred to as a plane of symmetry of mirror symmetry, as a rule.

Definition

A figure is axisymmetric if it is a straight line, so that there is a further (possibly with identical P ), for every point of the figure point of the figure, so that the link is cut in half at right angles from this line.

Then this line is called the axis of symmetry.

Examples

• As can be seen in the adjacent figure, the square has exactly four axes of symmetry. Rectangles that are not squares, have less or no symmetry axes. A rectangle, for example, still has two axes of symmetry, and that the two perpendicular bisectors and the opposite sides of the isosceles trapezoidal, parallelogram and the Antiparallelogramm still have at least a symmetry axis.
• The circuit even an infinite number of axes of symmetry, as it is symmetrical with respect to each diameter.
• Another character with an infinite number of axes of symmetry is the line. It is infinitely long and thus symmetric with respect to each to their vertical axis, and the lying on her own axis.
• Not only 2-dimensional figures may be axially symmetric. So the ball is with respect to each axisymmetric line through the center. This should not be confused with the plane symmetry. The ball is just symmetric. That is, it is symmetrical with respect to a reflection through a plane containing the center of the sphere.
• Also, the square is axisymmetric.
• The graph of the cosine function is also axisymmetrically. The subject axisymmetric features considered in the following section.

Axis of symmetry of function graphs

Overview

A particularly popular at school task is to detect the axis of symmetry for the graph of a function. Is the y- axis of the coordinate system, the axis of symmetry, it must be shown that the equation

Is satisfied for all x of the domain. Then it is said that the function is symmetrical about the y-axis. Such functions are also called even functions. This condition states that the function values ​​of the arguments and must match.

If you want to examine the general axis of symmetry of a function to a straight line through any points, so must test whether the function of the equation

Met for a firm and for all from the domain. By substituting with yields the equivalent condition

Examples

As an example, the quadratic function

Application of the condition of the axis of symmetry to the y- axis results

The graph ( a parabola ) is thus symmetrical with respect to the y-axis.

One example of the function is given, the graph is not symmetric to the Y- axis but the axis -symmetrical. The function

Is one such example. The claim is that the function is axially symmetric to the vertical. It is therefore necessary and it follows

Thus, the assumption of axial symmetry is confirmed.

Body of revolution

A class axisymmetric body in 3-dimensional space, the rotation body. A three-dimensional object is a body of rotation when rotated to any angle about a fixed axis images the object onto itself. This axis is the axis of symmetry. The simplest example of a body of revolution, the cylinder.

Plane symmetry

Another generalization of the axis of symmetry of the three -dimensional space, the plane of symmetry. A figure is plane-symmetric if there is a level, so that under reflection in this figure is mapped onto itself.

• Euclidean geometry