Visualizing hyperbolic space: unusual uses of 4x4 matrices
Visualizing hyperbolic space: unusual uses of 4x4 matrices
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可视化双曲空间：4x4 矩阵的不寻常用途
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发表时间：
1992
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通讯作者：
Charles G. Gunn
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作者：
M. Phillips;Charles G. Gunn
我们简要地讨论双曲几何，其中一个最有用和最重要的种类非欧几里德几何。双曲空间的刚体运动可以用4 × 4齐次变换来表示，其方式与欧几里得空间的刚体运动完全相同。对于我们这些有兴趣可视化双曲空间中生活的人来说，这是一个令人高兴的情况，因为这意味着我们可以使用现有的图形硬件和软件库来制作双曲空间中的动画场景。我们提出的公式计算反射，平移和旋转的双曲空间。这些比欧几里得几何的相应公式更复杂一些，这强调了我们对允许完全任意的4 × 4变换的图形库的需求。使用4 × 4变换来表示双曲空间的等距并不是什么新鲜事;自从19-世纪发现非欧几里德几何以来就一直在使用。我们工作的新部分是将这一理论应用于实时3D计算机图形技术，这是有史以来第一次允许数学家交互式地探索双曲几何。几何中心是由国家科学基金会，能源部，明尼苏达州技术公司，和明尼苏达大学。作者可以在：几何中心，1300南第二街，明尼阿波利斯，明尼苏达州55407。(612)626-0888电子邮件：mbpcllgeom. umn。edu，gnnn@ geom. nmn. edu.允许免费复制本材料的全部或部分，前提是复制品不是为了直接的商业利益而制作或分发的，ACM版权声明和出版物的标题及其日期出现，并通知复制是由计算机协会许可的。复制或重新发布，需要付费和/或特定许可。a 1992 ACM 0-89791-471-6/92/0003/0209.$ 1.50介绍使用4 × 4矩阵表示仿射变换的欧几里德3空间是众所周知的计算机图形学。大多数图形语言包括指定4x 4变换的规定，并且大多数交互式图形工作站具有在硬件中乘以4x 4矩阵的能力。这些功能是根据欧几里得几何设计的，因为我们认为我们生活的空间是欧几里得三维空间。然而，在数学和物理学研究和教育中，还有一些令人感兴趣的几何替代系统。其中最重要的是双曲几何。在三维流形的研究和分类中，双曲空间比欧几里得几何更自然地出现。它也经常在几何入门课程中教授，因为它在某种意义上是最简单和最优雅的非欧几里德几何类型。学习双曲几何迫使人们挑战许多通常被认为是理所当然的假设，在这个过程中加强一个人的几何推理技能。双曲几何的“空间”由R3中单位球的内部组成;球的边界，单位球面，是“在无穷远处”。距离被重新定义为当我们靠近这个球体时接近无穷大。因此，从双曲的观点来看，我们永远不可能真正到达边界球。我们可以认为双曲空间是由点、线、平面、曲面等组成的，就像在欧几里得空间中一样。然而，在双曲空间中，一些几何规则是不同的。具体地说，欧几里得的第五公设是无效的：在双曲平面上有许多通过一个给定点的直线不与给定直线相交。另一个非欧几里德性质是平面多边形中的角度之和总是小于180度。例如，有可能有一个“正五边形”（所有五条边都相等，所有五个角都是90度）。图1示出了超像素的镶嵌（平铺）。
We briefly discuss hyperbolic geometry, one of the most useful and important kinds of non-Euclidean geometry. Rigid motions of hyperbolic space may be represented by 4 x 4 homogeneous transformations in exactly the same way as rigid motions of Euclidean space. This is a happy situation for those of us interested in visualizing what life in hyperbolic space might be like, because it means we can use existing graphics hardware and software libraries to animate scenes in hyperbolic space. We present formulas for computing reflections, translations, and rotations in hyperbolic space. These are a bit more complicated than the corresponding formulas for Euclidean geometry, which emphasizes our need for graphics libraries which allow completely arbitrary 4 X 4 transformations. The use of 4 x 4 transformations to represent isometries of hyperbolic space is not new; it has been used since the discovery of non-Euclidean geometry in the 19-century. The new part of our work is the application of this theory to real-time 3D computer graphics technology, which for the first time ever is allowing mathematicians to interactively explore hyperbolic geometry. The Geometry Center is funded by the National Science Foundation, the Department of Energy, Minnesota Technology, Inc., and the University of Minnesota. The authors may be reached at: The Geometry Center, 1300 South Second Street, Minneapolis, MN 55407. (612) 626-0888. Email: mbpcllgeom.umn. edu, gnnn@geom. nmn. edu. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. a 1992 ACM 0-89791-471-6/92/0003/0209...$1.50 Introduction The use of 4 x 4 matrices to represent affine transformations of Euclidean 3-space is well-known in computer graphics. Most graphics languages include provisions for specifying 4 x 4 transformations, and most interactive graphics workstations have the ability to multiply 4 x 4 matrices in hardware. These capabilities were designed with Euclidean geometry in mind, because we think of the space in which we live as Euclidean 3-space. There are, however, alternate systems of geometry which are of interest in mathematics and physics research and education. One of the most important of these is hyperbolic geometry. Hyperbolic space arises naturally, even more so than Euclidean geometry, in the study and classification of 3-manifolds. It is also frequently taught in introductory geometry courses because it is in some sense the simplest and most elegant type of non-Euclidean geometry. Learning hyperbolic geometry forces one to challenge many assumptions which are usually taken for granted, in the process strengthening one’s geometric reasoning skills. The “space” of hyperbolic geometry consists of the interior of the unit ball in R3; the boundary of the ball, the unit sphere, is “at infinity”. Distance is redefined to approach infinity as we move closer to this sphere. From a hyperbolic point of view, therefore, we can never actually reach the boundary sphere. We can think of hyperbolic space as consisting of points, lines, planes, surfaces, etc, just as in Euclidean space. In hyperbolic space, however, some of the rules of geometry are different. Specifically, Euclid’s fifth postulate is not valid: in the hyperbolic plane there are many lines through a given point which do not intersect a given line. Another non-Euclidean property is that the sum of the angles in a planar polygon is always less than 180 degrees. It is possible, for example, to have a “regular right pentagon” (all five sides are equal and all five angles are 90 degrees). Figure 1 shows a tesselation (tiling) of hyper-