Oriented projective geometry
Oriented projective geometry
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定向射影几何
DOI：
10.1145/41958.41966
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发表时间：
1987
期刊：
影响因子：
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通讯作者：
J. Stolfi
中科院分区：
文献类型：
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作者：
J. Stolfi
有向射影几何是一种几何计算模型，它结合了经典射影几何的优雅和谈论定向直线和平面、符号角度、线段、凸图形和许多其他经典版本中无法定义的概念的能力。经典射影几何是许多几何计算的隐式框架，因为它是众所周知的齐次坐标表示法的基础。本文认为，有向射影几何及其基于符号齐次坐标的解析模型为计算几何提供了比经典几何更好的基础。
经典版本和定向版本之间的差异很大程度上局限于数学形式主义及其解释。在计算上，这些变化很小，并且不会增加几何算法的成本和复杂性。使用齐次坐标的几何算法可以很容易地转换为面向框架，并且成本很低。必要的更改在很大程度上是注意操作数的顺序和坐标的符号，这在经典框架中经常被忽略或未指定。
Oriented projective geometry is a model for geometric computation that combines the elegance of classical projective geometry with the ability to talk about oriented lines and planes, signed angles, line segments, convex figures, and many other concepts that cannot be defined within the classical version. Classical projective geometry is the implicit framework of many geometric computations, since it underlies the well-known homogeneous coordinate representation. It is argued here that oriented projective geometry — and its analytic model, based on signed homogeneous coordinates — provide a better foundation for computational geometry than their classical counterparts.
The differences between the classical and oriented versions are largely confined to the mathematical formalism and its interpretation. Computationally, the changes are minimal and do not increase the cost and complexity of geometric algorithms. Geometric algorithms that use homogeneous coordinates can be easily converted to the oriented framework at little cost. The necessary changes are largely a matter of paying attention to the order of operands and to the signs of coordinates, which are frequently ignored or left unspecified in the classical framework.