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algebraic antigraphy antilogy apse arcs axis base-circle base-line base-point becomes canonical form cardioid centre centroid circumcentre circumcircle clinant co-ordinates coefficients coincide common normal conjugate consider constant corresponds cos2 Cremona transformation cross-ratios curvature curves of flow cusp cylinder denote direction distance double point doublets edges elliptic envelope Euclidean geometry Euclidean group Euclidean space Exercise Fermat points fixed points focal pairing foci focus four points function geometry given gives Hence Hessian points homography homology hyperbolic images interchange intersect inversive group involution Jacobian limacon map-equation meet mid-point motion negative nine-point circle orthocentre orthogonal osculant circle parabola parallel parallelogram parameters perpendicular plane points xi polar polygon positive quadratic radius rational reflexion rhombus right angles rotation self-conjugate equation sends sin2 singular points sink sphere tangents theorem three lines three points touch triangle vanishes vector vertices whence write x-plane