HYPERBOLIC AND SPHERICAL POWER OF A CIRCLE
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초록
Suppose that a line passing through a given point $P$ intersects a given circle $\mathcal{C}$ at $Q$ and $R$ in the Euclidean plane. It is well known that $|PQ||PR|$ is independent of the choice of the line as long as the line meets the circle at two points. It is also known that similar properties hold in the 2-sphere and in the hyperbolic plane. New proofs for the similar properties in the 2-sphere and in the hyperbolic plane are given.
Abstract
Suppose that a line passing through a given point $P$ intersects a given circle $\mathcal{C}$ at $Q$ and $R$ in the Euclidean plane. It is well known that $|PQ||PR|$ is independent of the choice of the line as long as the line meets the circle at two points. It is also known that similar properties hold in the 2-sphere and in the hyperbolic plane. New proofs for the similar properties in the 2-sphere and in the hyperbolic plane are given.
참고문헌 (3)
- Ákos Horváth, "Malfatti’s problem on the hyperbolic plane", Studia Scientiarum Mathematicarum Hungarica, 51(2), pp.201~212 (2014), Akademiai Kiado Zrt.[DOI]
- Ákos G. Horváth, "Hyperbolic plane geometry revisited", Journal of Geometry, 106(2), pp.341~362 (2014), Springer Science and Business Media LLC[DOI]
- D. M. Y. Sommerville, "The Elements of Non-Euclidean Geometry" (1919), The Open Court Publishing