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