31. The intersection of several lines is the point at infinity in the motion direction, and one constraint on the affine structure. 32. Imagine two copies of the cut complex plane, the cuts extending along the positive real axis from to the point at infinity . 33. A generalized circle is either a circle or a line, the latter being considered as a circle through the point at infinity . 34. Similarly, if the ground field is the real or the complex field, the set of points at infinity is a manifold. 35. The singular point at infinity can be removed ( since this is a curve ) by the normalization ( integral closure ) process. 36. A third property which characterizes Dupin cyclides is that their curvature lines are all circles ( possibly through the point at infinity ). 37. Passing to projective space is a common tool in algebraic geometry because the added points at infinity lead to simpler formulations of many theorems. 38. As a projective space over a field is a smooth algebraic variety, the same is true for the set of points at infinity . 39. This does depend on having a " K "-rational point, which serves as the point at infinity in Weierstrass form. 40. The " points at infinity " in the Teichm�ller compactification consist of geodesic rays ( for the Teichm�ller metric ) starting at a fixed basepoint.
