Symmetry

If an image or curve is the mirror reflection of itself with respect to a line, the curve is said to be symmetric with respect to that line (the line of symmetry). A curve has a $\theta$-angle rotational symmetry with respect to a point if it is unchanged by a rotation through angle $\theta$ about that point.

A curve $f(x,y)=0$ may be symmetric about many lines, but this chapter focuses on symmetry about the coordinate axes and about the origin:

• Symmetric about the $y$-axis if $f(x,y)=f(-x,y)$ for all $(x,y)$ — equivalently, if $(x,y)$ is on the curve then so is $(-x,y)$. (Placing a mirror on the $y$-axis, the two halves of the curve coincide.)
• Symmetric about the $x$-axis if $f(x,y)=f(x,-y)$ for all $(x,y)$ — equivalently, if $(x,y)$ is on the curve then so is $(x,-y)$. …