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- In geometry, a generalized circle, also referred to as a "cline" or "circline", is a straight line or a circle. The concept is mainly used in inversive geometry, because straight lines and circles have very similar properties in that geometry and are best treated together. Inversive plane geometry is formulated on the plane extended by one point at infinity. A straight line is then thought of as one of the circles that passes through the asymptotic point at infinity. The fundamental transformations in inversive geometry, the inversions, have the property that they map generalized circles to generalized circles. Möbius transformations, which are compositions of inversions, inherit that property. These transformations do not necessarily map lines to lines and circles to circles: they can mix the two. Inversions come in two kinds: inversions at circles and reflections at lines. Since the two have very similar properties, we combine them and talk about inversions at generalized circles. Given any three distinct points in the extended plane, there exists precisely one generalized circle that passes through the three points. The extended plane can be identified with the sphere using a stereographic projection. The point at infinity then becomes an ordinary point on the sphere, and all generalized circles become circles on the sphere. (en)
- 广义圆是近代几何中的一个概念，表示直线和圆的集合。广义圆的概念主要出现在反演几何里。圆和直线的反演有着相似的性质，因此在反演几何里可以将两者合并为一类，以方便研究。 平面上的反演几何假设平面是由普通的欧几里得平面和一个无穷远点组成。在反演几何中，直线的定义是欧几里得直线加上无穷远点，成为一个经过无穷远点，半径为无穷大的圆。 (zh)
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- 广义圆是近代几何中的一个概念，表示直线和圆的集合。广义圆的概念主要出现在反演几何里。圆和直线的反演有着相似的性质，因此在反演几何里可以将两者合并为一类，以方便研究。 平面上的反演几何假设平面是由普通的欧几里得平面和一个无穷远点组成。在反演几何中，直线的定义是欧几里得直线加上无穷远点，成为一个经过无穷远点，半径为无穷大的圆。 (zh)
- In geometry, a generalized circle, also referred to as a "cline" or "circline", is a straight line or a circle. The concept is mainly used in inversive geometry, because straight lines and circles have very similar properties in that geometry and are best treated together. Inversions come in two kinds: inversions at circles and reflections at lines. Since the two have very similar properties, we combine them and talk about inversions at generalized circles. Given any three distinct points in the extended plane, there exists precisely one generalized circle that passes through the three points. (en)
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- Generalised circle (en)
- 广义圆 (zh)
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