Orbifolds are a modern mathematical concept that arises in the research of hyperbolic geometry with applications in computer graphics and visualization. In this paper, we make use of rooms with mirrors as the visual metaphor for orbifolds. Given any arbitrary two-dimensional kaleidoscopic orbifold, we provide an algorithm to construct a Euclidean, spherical, or hyperbolic polygon to match the orbifold. This polygon is then used to create a room for which the polygon serves as the floor and the ceiling. With our system that implements M\"obius transformations, the user can interactively edit the scene and see the reflections of the edited objects. To correctly visualize non-Euclidean orbifolds, we adapt the rendering algorithms to account for the geodesics in these spaces, which light rays follow. Our interactive orbifold design system allows the user to create arbitrary two-dimensional kaleidoscopic orbifolds. In addition, our mirror-based orbifold visualization approach has the potential of helping our users gain insight on the orbifold, including its orbifold notation as well as its universal cover, which can also be the spherical space and the hyperbolic space.

翻译：轨道流形是双曲几何研究中的现代数学概念，在计算机图形学与可视化领域具有应用价值。本文以镜面房间作为轨道流形的视觉隐喻，针对任意二维万花筒轨道流形，提出了一种构造匹配该流形的欧几里得、球面或双曲多边形的算法。该多边形被用于构建房间，以其作为地板与天花板。通过实现莫比乌斯变换的系统，用户可以交互式编辑场景并观察编辑对象的反射。为正确可视化非欧几里得轨道流形，我们调整渲染算法以适配这些空间中光线遵循的测地线。该交互式轨道流形设计系统允许用户创建任意二维万花筒轨道流形。此外，基于镜面的轨道流形可视化方法有助于用户理解轨道流形，包括其轨道流形记法及其万有覆盖空间（同样可为球面空间与双曲空间）。