The parallel projection is part of the projective mapping between the pencils at formula_244 and formula_245 needed. The intersection points of any two related lines formula_246 and formula_247 are points of the uniquely defined hyperbola. "Remark:" The subdivision could be extended beyond the points formula_161 and formula_163 in order to get more points, but the determination of the intersection points would become more inaccurate. A better idea is extending the points already constructed by symmetry (see animation). "Remark:"  A hyperbola with equation formula_250 is uniquely determined by three points formula_251 with different "x"- and "y"-coordinates. A simple way to determine the shape parameters formula_252 uses the "inscribed angle theorem" for hyperbolas:  Analogous to the inscribed angle theorem for circles one gets the  Inscribed angle theorem for hyperbolas:,:  A consequence of the inscribed angle theorem for hyperbolas is the  3-point-form of a hyperbola's equation:  For a hyperbola formula_264 the intersection points of "orthogonal" tangents lie on the circle formula_265. This circle is called the "orthoptic" of the given hyperbola. The tangents may belong to points on different branches of the hyperbola. In case of formula_266 there are no pairs of orthogonal tangents. Any hyperbola can be described in a suitable coordinate system by an equation formula_58. The equation of the tangent at a point formula_268 of the hyperbola is formula_269 If one allows point formula_268 to be an arbitrary point different from the origin, then  This relation between points and lines is a bijection. The inverse function maps  Such a relation between points and lines generated by a conic is called pole-polar relation or just "polarity". The pole is the point, the polar the line. See Pole and polar. By calculation one checks the following properties of the pole-polar relation of the hyperbola:  "Remarks:" Pole-polar relations exist for ellipses and parabolas, too. Another definition of a hyperbola uses affine transformations:  An affine transformation of the Euclidean plane has the form formula_289, where formula_161 is a regular matrix (its determinant is not 0) and formula_291 is an arbitrary vector. If formula_292 are the column vectors of the matrix formula_161, the unit hyperbola formula_294 is mapped onto the hyperbola  formula_291 is the center, formula_297 a point of the hyperbola and formula_298 a tangent vector at this point.