Interactive Design and Optics-Based Visualization of Arbitrary Non-Euclidean Kaleidoscopic Orbifolds

Our work follows recent research in mathematics visualization, such as quaternions [15], knots and links [33], and branched covering spaces [30].

Orbifolds are a modern mathematical concept [32]. The notion of orbifolds has found applications in texture synthesis [3, 4, 2]. Nieser and Polthier provide visualization of analytic functions over the complex plane [23], while Roy et al. [30] visualize the notion and properties of branched covering spaces with respect to $N$-way rotational symmetry ($N$-RoSy) fields [28]. All of this work focuses on the visualization and processing of orbifolds on two-dimensional surfaces. Moreover, the orbifolds that they address only possess rotational symmetries, which are a subclass of reflectional symmetries. In this paper, we address orbifolds that are generated by reflections, which include not only reflectional symmetries but also rotational and translational symmetries.

Conway et al. [9] explain the concepts and results related to planar orbifolds by using popular artwork containing textures with symmetries. Their approach focuses on Euclidean orbifolds. In our work, we provide a system to generate a room that matches any given two-dimensional kaleidoscopic orbifold, even when its universal cover is a non-Euclidean space. Furthermore, we make use of the mirror metaphor to leverage real-life experience with mirrors, which provides a complementary approach to the texture-based visualization of orbifolds.

Our mirror metaphor turns a two-dimensional orbifold into a three-dimensional room, which can also be considered as a three-dimensional orbifold that is the product of the two-dimensional orbifold (floor and ceiling) with a line segment (the height of the room). There has been some past research on visualizing three-dimensional orbifolds [7, 24], with a focus on the three-dimensional sphere $\mathbb{S}^{3}$ and three-dimensional hyperbolic space $\mathbb{H}^{3}$. In these spaces, the geodesics are either a circular arc or a hyperbola. In contrast, the geodesics in our product spaces are spirals, which makes ray-triangle intersection different from those in $\mathbb{S}^{3}$ and $\mathbb{H}^{3}$. Moreover, past research often focuses on using some famous orbifolds in $\mathbb{S}^{3}$ and $\mathbb{H}^{3}$ such as the Poincaré sphere and the mirror dodecahedron. In our paper, we allow the visualization of any arbitrary two-dimensional orbifolds.

In this section, we review necessary mathematical background on orbifolds used in this paper which include the concepts of groups and group actions [18], orbifolds [21, 11], and non-Euclidean spaces [12]. For a rigorous definition of these concepts, we refer our readers to the aforementioned references.

An orbifold $O$ is a topological space $X$ paired with a discrete symmetry group $G$ such that $X$ locally resembles a Euclidean disk under the action of $G$. To better illustrate this, consider the space $L=\{(x,y)|x,y\geq 0\}$ (Figure 4 (right)). For each point in the first quadrant, we can find a small enough disk-shaped neighborhood. However, for a point on the positive $Y$-axis, there is a neighborhood of the shape of a half-disk which corresponds to a full disk in the Euclidean plane (Figure 4 (left)) under the reflection across the $Y$-axis. In other words, the union of the half disk in the first quadrant and its mirror reflection form a full disk. Finally, the origin has a quarter-disk-shaped neighborhood (Figure 4 (right)) which corresponds to a full disk in the Euclidean plane (Figure 4 (left)) when being combined with its reflection across the $X$-axis (in the fourth quadrant), the reflection across the $Y$-axis (in the second quadrant), and the rotation by $\pi$ around the origin (in the third quadrant). Thus, $L$ is an orbifold.

Globally, we can see that $L$ is the range of the following function $\tau(x,y)=(|x|,|y|)$, which introduces a map from $\mathbb{R}^{2}$ to $L$ with the symmetry illustrated as follows. The letter $p$ (Figure 4 (right)) corresponds to the letter $q$ in the second quadrant (Figure 4 (left)) through the reflection across the $Y$-axis and the letter $b$ in the fourth quadrant through the reflection across the $X$-axis. In addition, it corresponds to the letter $d$ in the third quadrant through a rotation of $\pi$ around the origin, which is a composition of the two aforementioned reflections. Thus, the symmetry induced by the map $\tau$ leads to a symmetry group of four elements: the identity, two reflections, and one rotation. The group is the Dihedral group of order $2$, i.e. $\mathbb{D}_{2}$, which, when acted on $\mathbb{R}^{2}$, leads to the orbifold $L$. It has a corner point at the origin and two mirror lines (the positive $X$-axis and the positive $Y$-axis).

In general, a two-dimensional kaleidoscopic orbifold $O$ is a polygon with a symmetry group induced by reflections across all of its edges. Without causing ambiguity, we also refer to the polygon as $O$. Each edge of the polygon is thus a mirror line, and every vertex of the polygon is a corner point corresponding to the symmetry of $\mathbb{D}_{k}$, the dihedral group of order $k$. Note that $\mathbb{D}_{k}$ consists of $k$ rotations (including the identity) and $k$ reflections. In Figure 5 we show four such orbifolds, whose polygons have a configurations of a square (a), a $60^{\circ}-60^{\circ}-60^{\circ}$ triangle (b), a $90^{\circ}-45^{\circ}-45^{\circ}$ triangle (c), and a $90^{\circ}-60^{\circ}-30^{\circ}$ triangle (d). These orbifolds are given the orbifold notations (a) $*2222$, (b) $*333$, (c) $*244$, and (d) $*236$, respectively. A generic kaleidoscopic orbifold corresponding to an $N$-gon $O$ is given the notation $*k_{1}...k_{N}$ where the $*$ indicates the existence of the mirror and $k_{i}$ implies that the angle of the polygon at the $i$-th corner is $\frac{\pi}{k_{i}}$.

An orbifold (the polygon) and all of its virtual copies through its symmetry group can seamlessly tile a space, which is its universal cover. The aforementioned orbifolds are kaleidoscopic orbifolds whose universal cover is the Euclidean plane, thus Euclidean orbifolds. Each Euclidean orbifold has a translational cover, which, along with its translational copies, form the universal cover. The translational cover of $*2222$ consists of the orbifold, two of its reflections, and one rotation by $\pi$ (Figure 5 (a): any $2\times 2$ subgrid with the letters $q$, $p$, $d$, and $b$). The translations needed to generate the universal cover is the Gaussian integer grid $\mathbb{Z}[\textit{i}]$ [14]. The translational covers of the other Euclidean orbifolds respectively consist of six copies arranged in a hexagon (Figure 5 (b): $*333$), eight copies arranged in an octagon (Figure 5(c): $*244$), and twelve copies arranged in a dodecagon (Figure 5(d): $*236$). The set of translations for $*244$ is also $\mathbb{Z}[\textit{i}]$. On the other hand, the set of translations for $*333$ and $*236$ is the Eisenstein integer grid $\mathbb{Z}[\textit{$\omega$}]$ [14] where $\omega=\frac{-1+\sqrt{3}i}{2}$.

While it may seem that these are the only kaleidoscopic orbifolds and that all kaleidoscopic orbifolds must be triangular or rectangular, there are many more. In fact, given an arbitrary polygon with at least three sides and whose corner angles divide $\pi$ individually, there is an orbifold that corresponds to the polygon. Figure 13 shows a room with three, four, and five mirrors, respectively. However, these orbifolds cannot tile the Euclidean plane as their universal covers are either the unit sphere (spherical orbifolds) or the hyperbolic plane (hyperbolic orbifolds). The hyperbolic plane can be modeled as the upper sheet of the double-sheet hyperboloid $z^{2}-x^{2}-y^{2}=1$ (Figure 6). Like the Euclidean orbifolds, both spherical orbifolds and hyperbolic orbifolds are polygons whose edges follow the geodesics in their universal cover. The geodesics in the unit sphere are the great circles, and the geodesic passing through two mutually distinct points $p$ and $q$ in the hyperbolic space is the intersection of the plane containing $p$, $q$ and the vertex of the lower-sheet of the hyperboloid (Figure 6: the curve passing through $p$ and $q$). In fact, given an orbifold $O=$$*k_{1}...k_{N}$ where $N$ is the number of walls and $k_{i}>1$ ($1\leq i\leq N$), its universal cover is decided by the Euler characteristic of the orbifold as follows:

An orbifold $O$ is spherical, Euclidean, or hyperbolic when $\chi(O)>0$, $\chi(O)=0$, $\chi(O)<0$, respectively.

In the next section, we describe our orbifold design system starting with an enumeration of all two-dimensional kaleidoscopic orbifolds.

While there has been a complete enumeration of spherical and Euclidean orbifolds, to our best knowledge such an enumeration is not explicitly given for hyperbolic orbifolds. In addition, the enumeration for spherical and Euclidean orbifolds is in the form of an exhaustive list. Our orbifold design system is based on the number of walls (the cardinality of the underlying polygon) in the orbifold. Thus, we strive for an explicit enumeration for all two-dimensional kaleidoscopic orbifolds based on the combination of the polygon cardinality and the universal cover.

There are three types of spherical orbifolds: (1) one mirror, (2) two mirrors, and (3) three mirrors. The only one mirror spherical orbifold is $*$, which corresponds to a room that is half of the sphere with its boundary being the mirror. There are no corners. In this case, one can consider the room as a monogon. In the second case, the room has two mirrors that intersect at $\frac{\pi}{k}$ at both ends where $k>1$. These are diangular orbifolds $*kk$, which are the section of the unit sphere that are between two longitudes that are $\frac{\pi}{k}$ apart. Note that $*11$ is the same as $*$ since the corner angles are $\pi$. In fact, every corner with an angle $\pi$ can be removed from the list of corners. Thus, in the orbifold notation, we require every number to be at least $2$ when there are at least two walls.

For the triangular spherical orbifolds, i.e., three mirrors, there are two sub-types. The first sub-type has the form $*22k$ where $k>1$. Figure 13 (a) shows one such orbifold ($*222$). This type of orbifolds can be obtained by taking half of the orbifold $*kk$ in the northern hemisphere and adding a mirror on the equator. The second sub-type has the form $*23k$ where $k=3,4,5$. Notice that when $k=6$ we have $*236$, a Euclidean orbifold. From the discussion, we can see there are more spherical orbifolds than Euclidean orbifolds.

There are bad orbifolds, namely, $*k$ where $k>1$ and $*k_{1}k_{2}$ where $k_{2}>k_{1}>1$. Note that neither type of the bad orbifolds can be realized because it is not physically possible to have one great circle intersecting itself at an angle not equal to $\pi$, nor is it possible to have two different great circles that intersect at different angles where they meet. In our system, we do not construct bad orbifolds.

The rest of polygonal kaleidoscopic orbifolds are hyperbolic, and there are no bad hyperbolic orbifolds. There are three cases: (1) three mirrors, (2) four mirrors, and (3) five or more mirrors. An orbifold is hyperbolic if its has five or more mirrors (e.g. Figure 13 (c)). In addition, all quadrangular orbifolds except $*2222$ are hyperbolic (e.g. Figure 13 (b)). Finally, triangular hyperbolic orbifolds include six sub-types: (1) $*23k$ where $k>6$, (2) $*24k$ where $k>4$, (3) $*2k_{2}k_{3}$ where $k_{3}\geq k_{2}>4$, (4) $*33k$ where $k>3$, (5) $*3k_{2}k_{3}$ where $k_{3}\geq k_{2}>3$, and (6) $*k_{1}k_{2}k_{3}$ where $k_{3}\geq k_{2}\geq k_{1}>3$. Notice the three cases, each of which corresponds to a Euclidean orbifold that serves as the border between the set of spherical and the set of hyperbolic orbifolds, namely, $*236$ for type (1), $*244$ for type (2), and $*333$ for type (4).

Our enumeration of all two-dimensional kaleidoscopic orbifolds based on the combination of the cardinality of the underlying polygon and the type of its universal cover is shown in Table 1. We provide the computations behind our enumeration in Appendix A.

Our orbifold visualization system consists of two components: a design panel and the display (Figure 2). We employ the Irrlicht game engine [1], which provides an effective balance between interactivity and functionality.

In the design panel, the user can specify the type of the scene by entering its orbifold notation in the form of a number $N$ for the number of walls in the scene and a list of $N$ numbers, $k_{1},k_{2},...,k_{N}$. Here, $k_{i}$ indicates that the angle of the $i$-th corner is $\frac{\pi}{k_{i}}$.

The default value of $N$ is five, and five evenly spaced nodes are displayed on the disk inside the design panel (Figure 2), each of which has a default value of two, i.e. $*22222$. The user can change the value of each node, which can be a non-integer in order to create non-orbifold scenes (Figure 1 (c)). The user can also change $N$, which results in a room with more or fewer walls. The default value for each node in the new setting is again two. Recall that there are two cases that are not physically realizable: (1) a circular room with a single mirror ($N=1$) that self-intersects at an angle not equal to $\pi$ ($*k$ where $k>1$), and (2) a room with two mirrors whose two intersection angles are mutually distinct ($*k_{1}k_{2}$ where $k_{2}>k_{1}$). Thus, we disallow these cases from occurring during the design phase. For example, when $N=1$, the value of the only node is set to one and cannot be changed. Similarly, when $N=2$, if the user changes the value of one node, the value of the other node is automatically updated to match it.

Given the orbifold notation, our system instantaneously generates an empty room (a right polygonal prism) whose floor and ceiling are congruent to the orbifold and whose walls are the sides of the prism. In our system, it is possible to have multiple mirrors on a wall as shown in Figure Interactive Design and Optics-Based Visualization of Arbitrary Non-Euclidean Kaleidoscopic Orbifolds. The user can also change the height of the room, the color and attenuation of a mirror, and the textures for the ceiling and the floor. Objects can be added to the scene, whose locations, orientations, sizes, base colors, and material properties (e.g. marble, glass) can be modified from their default values as needed. Light sources can also be added to the scene, with control over their locations, shapes, and optical properties. Unwanted objects and light sources can be removed from the scene.

All of the above scene design operations are interactively rendered in order to support the What-You-See-Is-What-You-Get (WYSIWYG) paradigm, and all of the examples included in the paper and accompanying video were created using our design system.

At the core of our system is the ability to create a room given any arbitrary two-dimensional kaleidoscopic orbifold and to correctly deform an object in the scene when it moves. In addition, in our design system, mirrors are not explicitly generated. Instead, we emulate the mirror effects by creating copies of the original room which together approximate the universal cover of the orbifold. We provide detail on each of these topics next.

Our mirror-based visual metaphor and interactive orbifold design algorithm are motivated by a number of tasks for understanding the concept of orbifolds. Additional results illustrating concepts and properties related to orbifolds can be found in Appendix B. Below we list four tasks and show the results of using our approach to address them.

Recognizing Non-Orbifolds: One of the most fundamental tasks for orbifold visualization is to decide whether a given mirror scene is an orbifold. Non-orbifolds can be difficult to conceptualize, especially when one or more angles at the corners violates $\frac{\pi}{k}$ for $k\in\mathbb{N}^{+}$. The resulting rendered images and/or videos can deliver a sense of whether the room is a kaleidoscopic orbifold. When the room is a kaleidoscopic orbifold, the viewer should be able to move around the scene and see a consistent larger scene (the universal cover) that is seamlessly tiled by copies of the original room (the orbifold). On the other hand, if the viewer sees conflicting results, such as the double-headed dragon in Figure 1 (c), it is an indication of a non-orbifold. In this particular example, the room has a corner with an angle of $\frac{\pi}{k}$ where $k=\frac{2}{3}$, which is not an integer.

Sensing the Orbifold: An orbifold is a topological space that stands on its own. Using our mirrored scenes, we can produce visualizations that emphasize different aspects of an orbifold. For example, for the scene shown in Figure 3, we can adjust the attenuation factors for the mirrors to emphasize the orbifold itself (Figure 3 (a)) by setting high attenuation for both mirrors, or the translational cover (Figure 3 (b)) by setting high attenuation for one mirror and low attenuation for the other mirror, or the universal cover (Figure 3 (c)) by setting low attenuation for both mirrors. Notice that in all of these cases, our mirror-based visualization provides a clear sense of the orbifold, the room.

Determining the Orbifold Notation: Once an intuitive sense of the orbifold is established and the structure of the reflection is observed, it becomes a relatively straightforward task to determine the type of the orbifold in terms of its orbifold notation. Given the power of 3D graphics and animation, we produce either a panorama of the scene or an animation in which an avatar walks along the walls (see the accompanying video). As the avatar approaches the $i$-th corner, so do its nearest reflections. There appear to be $k_{i}$ pairs of avatars approaching the corner. Then, $k_{i}$ pairs of the same avatars leave the corner for the next one. This helps the user identify $k_{i}$ for the $i$-th corner. By walking around the room one time, each corner is visited. This can help the user write down the orbifold notation.

Identifying the Type of the Universal Cover: With the same panorama and walkaround animations, the user can gain insight into the type of the universal cover. Both the spherical space and the hyperbolic space can bring unfamiliar experience to someone new to the concept. For example, inside the spherical space, objects do not always appear smaller when they are further away. Specifically, when the viewer is at the south pole, objects near the north pole appear much wider than the same object at the equator (Figure 13 (a)). On the other hand, objects farther away always look smaller in the hyperbolic space. However, with the seemingly same distance to the viewers, an object can look much smaller in the hyperbolic space than in the Euclidean space (Figure 13 (b-c)). As the objects move around the scene such as the avatar in Figure 13 and the accompanying video, the way the reflections of the objects deform in non-Euclidean spaces is rather different from that in the Euclidean space. For example, a reflection of the avatar may suddenly grow much bigger and then shrink quickly in the spherical space. With a relatively simple scene (the orbifold), the user can gain insight into the entire universal cover through mirror reflections.

Once the scene has been constructed, we can either display it interactively during the design stage (Figure 2) or send it to Mitsuba [19] for high-quality off-line rendering. For the latter, we make use of a ray-tracing type of approach with Mitsuba. Recall that our non-Euclidean room is modelled by a subset in $\mathbb{D}\times[0,h]$ where $\mathbb{D}$ is the unit disk and $h$ is the height of the room. Thus, any geodesic $\gamma$ in the space satisfies that its horizontal projection is a geodesic in the spherical or hyperbolic space. Consequently, $\gamma$ takes the form of a 3D spiral. To generate the correct rendering of such scenes, we modify the ray-triangle intersection algorithm in Mitsuba from line-plane intersection to spiral-plane intersection.

Given a point $p_{0}$ in the room and a tangent direction $v$, the spiral emanating from $p_{0}$ in the direction of $v$ is the tensor product of horizontal projection and the vertical projection of the spiral. The horizontal projection is a circle (can also be a line) in the floor that starts with the projection of $p_{0}$ onto the floor and travels in the direction of $v_{H}$, the projection of $v$ onto the floor. The vertical projection of the spiral is a line that starts at $p$ that travels in the direction of $v_{Z}$, the $z$-component of $v$. This leads to a parametric form of $\gamma$ which we use to find the intersection of the spiral $\gamma$ with a given triangle on an object in the scene. When there are multiple intersections with the triangle, we find the one that has the smallest positive $t$ value, which represents the closest intersection from the reference point $p$.

During the design stage, we make use of scan conversion for interactive feedback. Recall that in this case, we also render the universal cover since we do not model mirror reflections during design. For each vertex on an object in the scene or a virtual copy, we project it to the image plane by using the same spiral-triangle intersection as in the case of Mitsuba rendering. We then perform barycentric interpolation to find the footprint of any triangle in the image plane by interpolating their vertices’ locations. Given enough mesh resolution, the error in this interpolation-based approach is relatively small since all of its vertices are projected correctly using the spiral-triangle intersection.

Our interactive design system is evaluated on a computer featuring an i7-8700K @3.70GHz CPU and an NVIDIA GeForce GTX 2080 GPU. For an orbifold $*22222$ with 26,022 primitives at an $800\times 600$ resolution, our system can render its universal cover approximated by $166$ copies of the original room at $11$ frames per second. When the number of copies is reduced to $61$, the frame rate is increased to $30$ frames per second.

For off-line rendering, we make modifications to Mitsuba $0.6$. The rendering was performed on a Linux Cluster, comprised of machines equipped with $12$-core $2.67$ GHz processors and $48$ GB SDRAM, to generate high-resolution images and videos based on the scenes that we created. Due to the increased computational cost from the spiral-triangle intersections, the rendering is considerably slower for non-Euclidean orbifolds than Euclidean orbifolds. As an example, for the single frame of the spherical orbifold $*222$ that contains $12,572$ primitives (shown in Figure 13 (a)), it required $7.69$ minutes when rendered at a resolution of $800\times 600$.

In this paper, we propose the use of mirror reflections as a visual metaphor for orbifolds and provide a system in which the user can interactively design any two-dimensional kaleidoscopic orbifold. At the core of our interactive design system is the ability to determine the configuration of the room (locations of the corners, shapes of the walls) given the orbifold notation provided by the user. Our system can handle not only Euclidean orbifolds, but also spherical orbifolds and hyperbolic orbifolds. In addition, we provide an enumeration of two-dimensional kaleidoscopic orbifolds based on the combination of the cardinality of the underlying polygon and the type of the universal cover. As part of the design system, we also enable the interactive construction of the universal cover of the orbifold as well as movement of the objects in the scene. The user can also generate high-quality photorealistic renderings of the scene, panorama, and animations with Mitsuba, which we have modified to account for the geodesics in spherical and hyperbolic geometry along which light travels.

Making rendering more efficient with the off-line rendering with Mitsuba is important, and we plan to investigate efficient spatial hashing data structures for the non-Euclidean spaces. In addition, the quality of the meshes used to represent the floor and the ceiling can impact the rendering speed, and we plan to explore optimal meshing structures for our purpose.

For future directions, we wish to expand our rendering system to arbitrary three-dimensional orbifolds. In addition, not all orbifolds are kaleidoscopic, and we would like to incorporate the visualization for non-kaleidoscopic orbifolds, such as the ones involving gliding reflections. Finally, we wish to explore the visualization of non-orientable orbifolds, whose universal cover is a non-orientable surface such as the Projective plane and the Klein bottle.