7.9.1 Symmetry

. . Symmetry Consider the following curves and observe that each of them is having some special properties, called symmetry with respect to a point, with respect to a line. Fig.

. We now formally define the symmetry as follows : If an image or a curve is a mirror reflection of another image with respect to a line, we say the image or the curve is symmetric with respect to that line. The line is called the line of symmetry. A curve is said to have a θ angle rotational symmetry with respect to a point if the curve is unchanged by a rotation of an angle θ with respect to that point.

A curve may be symmetric with respect to many lines. Specifically, we consider the symmetry with respect to the co-ordinate axes and symmetric with respect to the origin. Mathematically, a curve f x y ( , ) = is said to be • Symmetric with respect to the y -axis if f x y x y , ) ∀ x y or if ( , ) x y is a point on the graph of the curve then so is ( , ) − x y . If we keep a mirror on the y -axis the portion of the curve on one side of the mirror is the same as the portion of the curve on the other side of the mirror.

• Symmetric with respect to the x -axis if f x y x y ( , ∀ or if ( , ) x y is a point on the graph of the curve then so is ( , . If we keep a mirror on the x -axis the portion of the curve on one side of the mirror is the same as the portion of the curve on the other side of the mirror. • Symmetric with respect to the origin if f x y x y ∀ or if ( , ) x y is a point on the graph of the curve then so is ( y . That is the curve is unchanged if we rotate it by ° about the origin.

For instance, the curves mentioned above x and y are symmetric with respect to x -axis, y -axis and origin respectively.