Abstract

We describe algorithms that produce accurate real-time interactive in-space views of the eight Thurston geometries using ray-marching. We give a theoretical framework for our algorithms, independent of the geometry involved. In addition to scenes within a geometry X, we also consider scenes within quotient manifolds and orbifolds $X/\Gamma$. We adapt the Phong lighting model to non-euclidean geometries. The most difficult part of this is the calculation of light intensity, which relates to the area density of geodesic spheres. We also give extensive practical details for each geometry.

中文翻译：

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射线行进瑟斯顿几何

摘要

我们描述了使用光线行进生成八个瑟斯顿几何图形的准确实时交互式空间视图的算法。我们为我们的算法提供了一个理论框架，与所涉及的几何无关。除了几何X中的场景，我们还考虑商流形和轨道中的场景$X/\Gamma$. 我们将 Phong 照明模型调整为非欧几何。其中最困难的部分是光强的计算，这与测地球的面积密度有关。我们还为每个几何体提供了广泛的实用细节。