A note on invertible quadratic transformations of the real plane

A polynomial transformation of the real plane ${\mathbb{R}}^2$ is a mapping $\Bbb R^2\to\Bbb R^2$ given by two polynomials of two variables. Such a transformation is called quadratic if the degrees of its polynomials are not greater than two. In the present paper an exhaustive description of invertible quadratic transformations of the real plane is given. Their application to the perfect cuboid problem is discussed.