Abstract

We present novel methods for real-time native geodesic rendering of anisotropic geometries ${\mathrm{ℍ}}^2\times ℝ,{\mathbb{S}}^2\times ℝ,\mathbf{Solv}$ and similar geometries, Nil, twisted ${\mathrm{ℍ}}^2\times ℝ$. We also include partial results for the Berger sphere and explain why such real-time rendering of this geometry is difficult. Current approaches are not applicable for rendering complex shapes in these geometries, such as traditional 3D models, because of the computational complexity of ray-based approaches or significant rendering artifacts in older primitive-based approaches. We use tessellations to represent large shapes without numerical precision issues. Our efficient methods for computing the inverse exponential mapping are applicable not only for visualization but for games, physics simulations, and machine learning purposes as well.

摘要

我们提出了各向异性几何体的实时本地测地线渲染的新方法${\mathrm{ℍ}}^{\mathrm{2个}}\times ℝ,{\mathbb{小号}}^{\mathrm{2个}}\times ℝ,\mathbf{求解}$ 和类似的几何形状，无，扭曲 ${\mathrm{ℍ}}^{\mathrm{2个}}\times ℝ$. 我们还包括 Berger 球体的部分结果，并解释为什么这种几何体的实时渲染很困难。当前的方法不适用于渲染这些几何体中的复杂形状，例如传统的 3D 模型，因为基于射线的方法的计算复杂性或旧的基于图元的方法中的显着渲染伪影。我们使用镶嵌来表示没有数值精度问题的大形状。我们计算逆指数映射的有效方法不仅适用于可视化，也适用于游戏、物理模拟和机器学习目的。