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.

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Ray-marching Thurston 几何

我们描述了使用光线行进生成八个 Thurston 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，与所涉及的几何形状无关。除了几何 $X$ 内的场景，我们还考虑商流形和 orbifolds $X / \Gamma$ 内的场景。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度的计算，它与测地球的面积密度有关。我们还为每个几何图形提供了广泛的实用细节。