taking the appropriate limit; e.g. the cross-ratio of is The cross ratio of four different points is real if and only if there is a line or a circle passing through them. This is another way to show that Möbius transformations preserve generalized circles. Conjugation Two points z1 and z2 are conjugate with respect to a generalized circle C, if, given a generalized circle D passing through z1 and z2 and cutting C in two points a and b, are in harmonic cross-ratio (i.e. their cross ratio is −1). This property does not depend on the choice of the circle D. This property is also sometimes referred to as being symmetric with respect to a line or circle. Two points z, z∗ are conjugate with respect to a line, if they are symmetric with respect to the line. Two points are conjugate with respect to a circle if they are exchanged by the inversion with respect to this circle. The point z∗ conjugate to z when L is the line determined by the vector based eiθ at the point z0 can be explicitly given as The point z∗ conjugate to z when C is the circle of radius r centered z0 can be explicitly given as Since Möbius transformations preserve generalized circles and cross-ratios, they preserve also the conjugation. Projective matrix representations The natural action of PGL(2,C) on the complex projective line CP1 is exactly the natural action of the Möbius group on the Riemann sphere, where the projective line CP1 and the Riemann sphere are identified as follows: Here [z1:z2] are homogeneous coordinates on CP1; the point [1:0] corresponds to the point of the Riemann sphere. By using homogeneous coordinates, many calculations involving Möbius transformations can be simplified, since no case distinctions dealing with are required. Every invertible complex 2×2 matrix acts on the projective line as where Since the above matrix is invertible if and only if its determinant is not zero, this induces an identification of the action of the group of Möbius transformations with the action of PGL(2,C) on the complex projective line. In this identification, the above matrix corresponds to the Möbius transformation This identification is a group isomorphism, since the multiplication of by a non zero scalar does not change the element of PGL(2,C), and, as this multiplication consists of multiplying all matrix entries by this does not change the corresponding Möbius transformation. For