Mapping Two-Dimensional Plots to a Spherical Surface
using Elliptical Grid Mapping
Naoki Kita
Tokyo University of Agriculture and Technology
Tokyo, 184-8588, Japan

Grégoire Cliquet
L’École de design Nantes Atlantique
Nantes, 44200, France

nkita@go.tuat.ac.jp

Kazunori Miyata
Japan Advanced Institute of Science and Technology
Ishikawa, 923-1292, Japan
miyata@jaist.ac.jp

Abstract

CAVE [5] is also available for immersive display, it requiresa large physical space and there are significant costs formaintaining the system. In part for these reasons, HMDs
are gaining popularity as the devices of choice for immersive VR environments. When using HMDs, the visualization covers a wider range of the user’s field of view (FOV);
hence, such a visualization provides a more immersive experience to the user. Although geospatial data can be naturally illustrated on a spherical display, it is not straightforward to display plots defined on a 2D Cartesian coordinate
plane on a spherical display. Doing so inevitably introduces
distortions from projecting the 2D plots on a spherical surface, and such distortions restrict the potential applicationsof spherical displays of 2D data. Spherical displays still
have uses for 2D data, however; for example, we can visualize various 2D plots such as scatter plots on a sphericalsurface surrounding a user in an immersive VR environment, which enriches the user’s experience. We can also
exploit a wider FOV using spherical displays compared toplots that are displayed in a 2D window.
This paper proposes a method for mapping plots definedon a 2D Cartesian coordinate to a spherical surface. Because distortions are inevitably introduced while mapping,
we determine a mapping with minimal distortions. Our purpose is to enable displays that cover a wider range of a user’s
FOV than the 2D plots from which they are constructed bydetermining a mapping that minimizes distortion when a
2D plot is projected on a sufficiently large spherical surface. To accomplish this, we employ cube as an intermediate proxy before mapping the 2D data onto a sphere because we can determine correspondences between points ona cube and those on a sphere analytically via an ellipticalgrid mapping without parameterization. Thus, we first calculate the largest inscribed projection for a given 2D plot
inside an unfolded cube pattern. We then map the plot on

In this paper, we present a technique that maps 2D
plots to a spherical surface. We project 2D plots aslarge as possible while minimizing distortions. To accomplish this projection, we employ elliptical grid mapping, which maps points on a cube to those on a sphere.
To determine the largest projection from points on the2D plot to the cube, we find the largest projection inside
an unfolded cube pattern. The unfolded cube is thenfolded to construct a cube, and the projected 2D plot ismapped to a spherical surface. We consider all unfolding patterns of a cube and determine the optimal pattern for the projection. We apply the proposed mapping
to high-dimensional data visualizations. Since the technique maps a plot to a spherical surface surrounding theuser, and this covers a wider range of the user’s field ofview, our mapping provides an immersive experience tousers.

1. Introduction
Spherical displays are effective mediums for visualizing
geospatial data, such as meteorological data or human activities on a globe. In contrast to planar displays, sphericaldisplays can be viewed and sensically interpreted from any
direction because of their omnidirectional property. Whena user positions at the center of a sphere, he or she can visualize data on the spherical surface surrounding the user invirtual reality (VR) space, which is a kind of immersive VR
visualization [31]. In recent years, head-mounted displays(HMDs) such as Oculus Rift, HTC Vive, Sony PlayStation
VR, Samsung Gear VR, Google Cardboard, and DaydreamView have become easily available and are commonly usedin VR games and other applications using HMDs. Although1Figure 1. We propose an elliptical grid mapping technique for projecting 2D plots onto spherical surfaces that are as large as possible withminimal distortion. The images show mapping for 2D plots of dimensionality reduction results of the S-curve dataset including 2000 data
points (top) and their spherical mappings (bottom).

the cube constructed from the unfolded pattern to a sphereusing the mapping after folding the cube. Another benefit of our method is that cube mapping can be implemented
more easily than mappings using other regular polyhedra.
The decision to use cube mapping will be further discussed
in Sections 4 and 8. Without loss of generality, we focus
on scatter plots as the representative example of 2D plots inthis paper.
Our contributions can be summarized as follows:• We propose a method for mapping 2D plots (Cartesian
coordinates) to a spherical surface while introducing
minimal distortions.
• We solve the largest inscribed projection problem in an
unfolded cube pattern.
• We apply the proposed method to high-dimensional
data visualization to demonstrate the effectiveness of
our method for that application.

2. Related Work
Mappings. In computer graphics, texture mapping is used
to mapping images to a 2D shape or the surface of a 3Dmodel. When mapping a texture, texture coordinates aredefined on the model. Mesh parameterization defines amapping between two surfaces with the same topologies,
and various parameterization techniques have been proposed [28]. A decaling interface was proposed by Pedersen [23], and other similar approaches have been developed including lapped textures [25], texture sprites [17],and texture mapping based on decals generated using a discrete exponential map approximation [27]. However, decals are small texture patches, and we wish to map plots
to large spherical surfaces, so these methods do not serve
our purpose. Carlos et al. proposed an area-preserving

parametrization for mapping planar rectangles to spherical
rectangles using an analytical function [30]. In our case, we
also employ an analytical function for our mapping. While
Carlos et al. map the unit rectangle to a spherical rectangle,
we map 2D plots with arbitrary boundary shapes to a spherical surface. More applicable to our goal are the mappings
among spheres, discs [8, 6, 7, 36] and environment mapping [9, 10]. We will briefly describe the mappings fromsquares to discs and cubes to spheres in Section 4. For a
more comprehensive overview, please refer to [14, 36].

Spherical Layouts. Spherical displays are used for information and scientific visualizations. They are especiallysuitable for the visualization of geospatial data because
data such as air flow, distribution of facilities, and earthquakes are already on a globe. Such phenomena are notthe only ones well-described by spherical displays. Wu
and Takatsuka proposed a method of visualizing multivariate networks using a spherical self-organizing map (GeoSOM) [34]. Kwon et al. studied spherical graph layouts in
an immersive VR environment using an HMD [13]. The results of their user study showed that participants performed
better using the spherical graph layouts than when they used
traditional 2D graph visualizations in an immersive environment, especially for more difficult tasks and larger graphs.
In addition to GeoSOM [34], direct spherical embedding
techniques have also been proposed [33, 19, 18]. Du et al.
proposed the iSphere, focus+context technique to facilitate
the exploration of large graphs [4]. They mapped a nodelink diagram onto a Riemann sphere and orthogonally projected the sphere onto a 2D plane. Although they visualizedgraphs (node-link diagrams) using spherical layouts, we canmap any 2D plot to a spherical surface. Compared to thesetechniques, our method requires 2D projection before mapping to a spherical surface; however, our method can be ap-

1 .5
1 .0
0 .5
0 .0
0 .5
1 .0
1 .5
2

1

0

1

2

(b) Largest inscribed projection(a) 2D plot(c) Spherical mapping(d) Immersive visualization

Figure 2. Overview of the proposed technique. Given a 2D plot (a), we first compute the largest inscribed projection inside an unfolded
cube pattern (b). Then, the pattern is folded, and the projected 2D plot is on the faces of a cube. Finally, the plot is mapped from the cube toa spherical surface (c). The proposed mapping allows a user with an HMD a more immersive experience when applied to data visualization
in a VR environment (d).

plied to inherently planar plots such as existing photographs
or drawings.

3. Overview
The overview of the proposed technique is shown in
Fig. 2. The input is a plot defined on a 2D Cartesian coordinate plane (Fig. 2(a)). First, we inscribe the plot inside
an unfolded cube pattern (Fig. 2(b)), which is then folded
to construct a cube. The projected plot is now on the facesof the cube. Finally, the points of the plot that are now on
the cube are mapped to points on a sphere (Fig. 2(c)). In
Fig. 2(d), we show an application of the mapping for datavisualization in an immersive VR environment. The resulting mapped plot covers a wider range of the user’s FOV andgives users an immersive experience.
In Section 4, we describe the mapping technique. Section 5 describes the largest inscribed projection problem andits solution. We demonstrate the applications in Section 6,
evaluate the mapping in Section 7, and discuss our proposed
approach in Section 8, followed by the conclusion and future work in Section 9.

4. Mapping Cube to Sphere
Mapping Square to Circle. In this section, we briefly review mapping techniques used in a 2D case. Namely, tomap a point on a square with a side of length 2 (from -1
to 1) to that on a unit circle, one uses the following assignment:
vsquare = [ x y ]> → vcircle .

(1)

The simplest way to map points on a square to those on acircle is to normalize vsquare :"vcircle =pxyx2 + y 2px2 + y 2

#>
.

(2)

Elliptical Grid Mapping. Nowell introduced a mappingthat converts horizontal and vertical edges of a square toelliptical arcs inside a circle [22]. The mapping is given by
the transformation below:
#>
" r
r
y2
x2
vcircle = x 1 −
y 1−
.
(3)
2
2
We hereafter refer to Nowell’s mapping as elliptical grid
mapping [7].
Discussion. Lambers provided comparisons of various
mapping techniques that map a square to a disk, including the elliptical grid, equal-area, and conformal mapping methods [14]. In any such comparisons, which mapping technique provides a better result to the user depends
strongly on the application area. However, since we focus
on mapping a 2D plot that is in the interior of a square,
elliptical grid mapping is the best available option regardless of the intended application of the projection [14], andtherefore, we employ the mapping. In this paper, we do
not focus on providing comprehensive comparisons of our
choice with alternative mapping techniques, although othermethods can certainly be used.
Mapping Cube to Sphere. It is straightforward to extend
the elliptical grid mapping to three dimensions, i.e., to map
points on a cube to those on a unit sphere [21], rather thanpoints on a square to those on a disk. That is, a point ona cube; vcube = [ x y z ]> is mapped onto a unit sphere
vsphere , by the assignment
 q

2
2 2
2
x 1 − y2 − z2 + y 3z
 q



vsphere =  y 1 − x22 − z22 + x23z2  .
(4)
 q

2
2
2
2
z 1 − x2 − y2 + x 3y
For mapping 2D Cartesian coordinates to a spherical surface, we employ a cube mapping-like approach. That is,Figure 4. Concave hulls for various plots.
Figure 3. Left: unfolded cubes. Right: flipped patterns. There aretwenty patterns in total (excluding the symmetrical patterns withwhite face color).

given a 2D plot, we first project the plot to the inside of an
unfolded cube pattern. The unfolded cube is then folded
and an elliptical grid mapping is applied to map points on
the cube to a spherical surface. However, there are various
ways to unfold a cube, and the unfoldings do not result in
identical projections to the sphere. In particular, there are
11 unfolding patterns shown in Fig. 3 (left). By distinguishing two sides of the patterns (Fig. 3 (right)), this gives 20
patterns in total. The shape of the 2D plot we start withdetermines how it will best fit inside an unfolded 2D cube.
To find the unfolding pattern that best inscribes the 2D data,
we solve an optimization problem, as is described in thenext section.

5. Largest Inscribed Projection
In this section, we describe the means of inscribing aplot of 2D data inside an unfolded cube pattern. Because
it is computationally expensive to deal with all data points
in the plot, we first compute the concave hull [20] and thenuse it as a proxy polygon P for the plot (Fig. 4). We empirically set k = 60 for the k-nearest neighbors used in themethod described in [20]. If a plot is given as an image(Fig. 5 (left)), we first perform boundary tracking to extract
the boundary pixels (Fig. 5 (middle)). Because the boundary may be composed of many pixels, we use the RamerDouglas-Peucker algorithm [26, 3] to simplify the polygon(Fig. 5 (right)). In the B UNNY case, although Fig. 5 (middle) contains 2559 vertices, Fig. 5 (right) contains only 72
vertices, which reduces the computational costs associatedwith checking whether the polygon is contained inside an
unfolded cube polygon during the optimization step that follows.
5.1. Problem Formulation

Formally, by denoting the set of the unfolded cube polygons as {C1 , · · · , C20 }, we would like to find the transformation function Proj(·) that projects P inside an unfolded
cube polygon Ck .
maximize ss,R,csubject to(5)
Proj(P) is inside Ck ,where we decompose the transformation function Proj(·)

Figure 5. Left:input plot as an image. Middle: boundary tracking.
Right: simplified boundary polygon.

to a scaling factor s ∈ R, rotation matrix R ∈ SO(2), anddisplaced centroid of P, c ∈ R2 .
5.2. Optimization

We employ particle swarm optimization (PSO) [24] tosolve Equation (5), which solves a problem using a population of candidate solutions that are represented as particles. The candidates have positions and velocities, andmove around the high-dimensional search space to determine the optimal solution. The positions and velocities are
updated for each iteration by calculating the following equations:vi (t + 1) ← wvi (t) + c1 r1 (xpbest − xi (t)) + c2 r2 (xgbest − xi (t)),
xi (t + 1) ← xi (t) + vi (t),(6)

where the position of the i-th particle at time t is represented
as xi ∈ Rn and the velocity is represented as vi ∈ Rn .
The variables r1 and r2 ∈ [0, 1) are random values drawn
uniformly from [0, 1), and w, c1 , and c2 are the control parameters. The position xpbest is the current best positionof the particle, and xgbest is the current best position of allparticles. The best positions are determined by calculating
an objective function. Using these local and global parameters, the particles communicate with one another and explore possible better solutions. We automatically determine
parameters w, c1 , and c2 by employing a constriction coefficient [2]. In our case, x = [s, R(θ), c] and the objectivefunction is defined as follows:(s
Proj(P) is inside Ck
Objective(x) =(7)−∞ otherwise.
Jacobson employed a hybrid PSO by combining PSO withbinary search to solve a problem similar to ours [12]. In
our case, however, we found empirically that the hybrid
PSO with binary search is computationally more expensive
than PSO without binary search because it checks whether
Proj(P) is inside Ck several times. Without the hybridiza-Table 1. Timings. #particle = 500, #iteration = 500.

Hull Polygon#verticesAvg. Time

41
0.796 s300.510 s33
0.510 s600.875 sFigure 6. Largest inscribed projections for Fig 4.

Figure 8. Optimization statistics for B UNNY (Fig. 5).
Figure 7. Developments of the mean objective values andstandard deviations over iterations. #particle = 500,
#iteration = 500.

tion, we can obtain approximately optimal results withinhundreds of iterations (200 − 500) in most cases.
5.3. Optimization Results and Discussions

Figure 6 shows the largest inscribed projections foundby our method for each of the polygons in Fig. 4. Depending on the shape of the hull polygons, the most appropriate
unfolded cube pattern is chosen for the projection.
Optimization Developments. We measure the objective
value of each unfolded cube pattern and shows the developments in the mean objective values and standard deviations over iterations in Fig. 7. We set PSO parameters#particle = 500 and #iteration = 500. The
curve for each plot shows how fast the optimization converges. We can observe that the blue plot converges fasterthan the others, and that the red and orange plots converge
especially slowly. It seems that the convergence depends onthe convexity of the concave hull. Here, the convexity of apolygon can be measured using the ratio of the area of theconcave hull and its convex hull. The standard deviations
in each plot show the difficulty in finding the optimal values among the unfolded cube patterns because the optimalvalues are different for each pattern. It also seems that variations in the standard deviation depend on the convexity ofthe polygon as well as the convergence speed.
Here, we have to note that, the above findings are inferences based on the experiments. We have to conduct more
dedicated experiments for drawing any conclusions.
Timing. We implement the optimization method in C++,and measure the computation time using an Intel Core i5@ 2.9 GHz personal computer with 16 GB RAM. Table 1
shows the average time required for the optimization per
unfolded cube pattern for each polygon in Fig. 4. It seems
that the optimization time depends linearly on the numberof vertices of the polygon. Figure 8 shows the optimization
statistics for the 2D B UNNY plot in Fig. 5. We compare theoriginal polygon extracted using boundary tracking (Original) and its simplified polygon (Simplified). As we cansee, although the obtained optimal scaling values and thedevelopments of the objective values are almost the same,the original polygon required approximately 30 times moretime to optimize than the simplified one.
Comparison between Patterns. Figure 9 shows the results of optimization for 2D L UCY polygons. In this example, we inscribe a L UCY polygon inside an unfolded cube
polygon. Although the L UCY polygons are projected withmaximum scaling in both cases, the right one occupies approximately twice as much area in the unfolded cube as theleft one. In the case of immersive spherical display applications where a user stands at the center of a surrounding
spherical display, the right projection gives a more immer-

sional scaling (MDS) [16], Laplacian Eigenmaps [1], andt-SNE [32]. We then apply our technique to 2D plots, andthe resulting spherical plots are shown in Fig. 1 (bottom).

Figure 9. Largest inscribed projections for L UCY polygons.

Figure 10. The largest inscribed projections for a single concave
hull (left) and multiple concave hulls (right).

sive projection because the projection covers a wider FOVof the user.
Approximating with Multiple Concave Hulls. In thecase of a plot composed of clusters that are far apart fromone another, a single concave hull does not approximate theplot well (Fig. 10 (left)). In such cases, we first computek-means clustering, then construct concave hulls for eachcluster, and finally, calculate the largest inscribed projection
while maintaining the center of the plot in order to retain therelative positional relations (Fig. 10 (right)). In the current
implementation, we manually select k depending on input
plots. As can be seen in Fig. 10, the projection with multiple concave hulls occupies a larger area than that of thesingle hull.
5.4. Mapping Cube Patterns to a Spherical Surface

Because we have 20 unfolded cube patterns, we define
mappings for each pattern to a spherical surface. That is, wedefine the LEFT (−X), RIGHT (+X), UP (+Y ), DOWN(−Y ), FRONT (+Z), and BACK (−Z) directions for eachface of an unfolded cube pattern. Then, a coordinate defined on the face is mapped to a corresponding part of thespherical surface using elliptical grid mapping. Figure 11
shows the mapping results of Fig. 6.

6. Applications
6.1. Visualizing High-Dimensional Data

S-curve. High-dimensional data are difficult to understand. Therefore, dimensionality reduction techniques areused to project the data to two or three dimensions. We
apply our proposed technique to the visualization of highdimensional data. We use the S-curve dataset, where theS-curve is a 2D manifold embedded in 3D space. We generate 2000 sample points, and then apply dimensionality reduction techniques to project the data to 2D space. Figure 1 (top) shows the results of Isomap [29], multidimen-

MNIST. We also visualize the MNIST dataset [15].
MNIST consists of images of handwritten digits. For the visualization, we randomly sample 2500 images and apply tSNE to project 784-dimensional feature vectors to 2D space(Fig. 12(a)). Here, we empirically determine the number ofsamples i.e., 2500 samples since it is adequate for demonstrating our visualization. We then apply our technique tomap the 2D plot to a spherical surface. To compare the 2D
planar plot and the spherical plot, we visualize the plots in
an immersive VR environment (Fig. 12(b-c)), where a user
is positioned at the center of a unit sphere. Figure 12(b)shows the 2D planar plot. The digit images are used as the
texture of a plane representing the data points. The centroidof the plot is at the point of contact between the plane andthe sphere. Although we can adjust the perceived distance
between the user and the plot and the scaling, it is difficult
to locate data points in the surroundings. In contrast, because our spherical mapping maps the data on a unit sphere,
the distance between the user and any data point is constant(Fig. 12(c)). Figure 12(c) shows planes constructed to represent the data point billboards, namely those in which all
digit images always face the user. These help a user to correctly interpret the size of the digits, which is important if
the data points are encoded in variably sized objects. Although we can walk among visualized data in a VR Cartesian 3D space, data visualization in a Cartesian 3D spacecan occlude data behind another data because of depth. In
contrast, our spherical display visualizes data on a spherethat holds images at a constant distant from the user, andtherefore, cannot occlude data due to depth.

7. Evaluation
7.1. Comparison of Different Mapping Techniques

Qualitative Comparison. We compare different mapping techniques shown in Fig. 13, and find that cylindrical mapping stretches and distorts the mesh/texture, especially in regions close to the poles (Fig. 13(b)). We
also observe that normalization-based mapping distorts themesh/textures (arms and legs in Fig. 13(c)). In contrast, although some distortions are observed after our mapping, it
preserves the mesh better than other techniques.
User Study. We conduct a user study to perceptuallycompare the mapping techniques. We develop an immersive VR data visualization demo for displaying S-curve datavisualized by t-SNE (Fig. 1(d)) using three mapping techniques: cylindrical, normalization-based (norm-based), andFigure 11. Results of spherical mapping for the projections listed in Fig. 6.

(a) 2D plot(b) 2D plot in VR environment(c) Our spherical mapping in VR environment

Texture Mapping

Mesh Deformation

Figure 12. Visualization of the MNIST dataset in a VR environment. (a) Visualization of 2500 digit images by t-SNE. (b) Viewing of (a)in a VR environment. (c) Viewing of (a) mapped on a spherical surface by the proposed method.

(a) Original(b) Cylindrical(c) Normalization-based(d) Ours

Figure 13. Comparison of mapping techniques.

the mapping that is the subject of this paper. We ask theparticipants to answer the following questions:
Q1

Which mapping is more immersive?

Q2 Which mapping preserves the inter-element distancesand neighbors well?
In the demos, we display the original 2D plot and the spherical plots generated by the three mapping techniques. A participant can switch the plots at any time to compare them.
We first explain each participant the ways in which they canCylindrical
Norm-based
Ours

Q1
Q2
0%

50%

100%

Figure 14. User study results.

Table 2. Area variance comparison of different mapping techniques.
Area Variance
Polygon

Cylindrical

Norm-based

Ours

B UNNY

9.28 × 10−7
2.37 × 10−8
1.63 × 10−2
6.26 × 10−6

4.08 × 10−7
2.83 × 10−8
1.72 × 10−7
5.39 × 10−6

1.77 × 10−8
1.45 × 10−91.60 × 10−8
3.45 × 10−7

A RMADILLOL UCY
G INGERBREAD M AN

interact with the plots. Each user is then asked to sit in frontof a desk wearing an Oculus Rift DK2 HMD. Because thespherical layouts do not require positional tracking, we only
use the head tracking. We include 20 participants (twelve
males and eight females) with normal color vision in theuser study. The age of the participants ranges from 22 to
54. The mean age is 24.65 and the median is 22 years old(standard deviation: 7.15). Most of them are graduate students at a design school.
The results of the user survey are shown in Fig. 14. Mostof the participants answer that cylindrical mapping is themost immersive in Q1. We believe this is because the mapping result of cylindrical mapping covers almost the fullrange of the hemisphere, whereas the norm-based map andour mapping cover narrower FOVs. In Q2, our mapping
technique outperforms other mapping techniques by best
preserving the characteristics of the original 2D plot. Thisis an important property for a mapping technique when it
is applied to data visualization applications as it helps to
avoid misleading users and generating invalid interpretations. Although the cylindrical mapping technique heavily distorts the original plot, norm-based mapping and ours
cause less distortion and better preserve the inter-element
distance and neighbors. This makes it more difficult todistinguish between the two mapping techniques. Nevertheless, according to the result of Q2, the participants regard ours to be more distance-preserving than the normbased mapping technique, which shows that ours outperforms norm-based mapping in the aspects measured.

Quantitative Comparison. Rather than relying exclusively on user reports, we compare the distortions quantitatively. We measure the area variances of the triangles ofthe 2D polygonal mesh of B UNNY, A RMADILLO, L UCY,
and G INGERBREAD M AN. The area of a triangle on aunit sphere is equal to the spherical excess of the triangle. The original 2D meshes are generated using capacityconstrained Delaunay triangulation [35] to ensure that thetriangles of the meshes have a minimal capacity variance.
The result is shown in Table 2. Cylindrical mapping has
a larger area variance than the others, as is especially pronounced in the L UCY example. Our mapping has approximately ten times less area variance than normalizationbased mapping.

Figure 15. Area variance and objective value for each cube pattern.
x-labels represent cube pattern indices.

7.2. Comparison between Unfolded Cube Patterns

We also compare the area variances (less is better) andobjective values (larger is better) of the proposed mapping
for each cube pattern (Fig. 15). Since B UNNY is more convex than the others, the graph is different from those of theothers. In contrast, the graphs of A RMADILLO, L UCY, andG INGERBREAD M AN show that the area variances and objective values vary depending on the cube pattern indices.
In all these cases, the graphs show that there is a tradeoff
between area variance and objective value. Although we
select the one with the largest objective value in our current
implementation, we can select the one with the least area
variance or the maximum balance for other applications, as
appropriate.

8. Discussion
We have proposed a two-step approach for mapping 2D
planar plots to a spherical surface. First, we project a 2D
plot to an unfolded cube pattern. Then, the projected plot ismapped to a spherical surface using elliptical grid mapping.
We could have used another regular polyhedra (e.g., octahedron or icosahedron) as an alternative for a cube, and sucha choice has the potential to reduce distortions when thepolyhedral is mapped to a spherical surface. However, we
employ cube since we can find a correspondence between apoint on a cube to that on a sphere analytically via ellipticalgrid mapping. We do not have to pre-compute parameterizations to define a mapping from a 2D domain to a unit

sphere. In addition, the number of comparisons needed for
optimization of the initial projection into the unfolded polyhedral increases significantly when we move from the cube
to another polyhedral: while a cube has 11 essentially different unfoldings, a dodecahedron and an icosahedron have43380 essentially different unfoldings [11]. If we can determine the most appropriate unfolding pattern and define amapping between the polyhedra to a unit sphere, it would
be a better alternative to the proposed approach, but it iscomputationally very expensive. On the other side, since anoctahedron has 11 essentially different unfoldings, it might
be a reasonable alternative to our choice (cube).
As we discussed in Section 4, we compare the three selected mapping techniques, but this is not a comprehensive
comparison between our method and every possible wellsuited mapping technique, as such choices depend on thearea of application. As such, it is difficult to provide a general solution. Although we focus on mapping a 2D plot toa spherical surface as large as possible while minimizing
distortions, trying to use as much of the sphere as possible seems to be unhelpful since our FOV is limited to 120◦in the horizontal direction with less in the vertical direction.
Rather than replacing 2D planar displays with spherical displays for data visualization and exploration, we aim to combine a 2D window and a spherical layout for data exploration. For example, we can use a 2D window to overview
the data, and then dive into an immersive visualization using
the proposed spherical mapping.

9. Conclusion and Future Work
This paper proposes a mapping technique that maps 2D
plots to a spherical surface. To minimize distortions, weemploy elliptical grid mapping to map points on a cube
to a spherical surface. To fill as much of a user’s FOV as
possible, we solve the largest inscribed projection problem
for a polygon approximating a given plot and an unfolded
cube pattern. We apply the proposed technique to a highdimensional data visualization problem.
Constructing a conformal map between a sphere and a
cube does not seems to be simple, however, if such mapping
would be constructed, it can derive a better result than thatin the proposed method in terms of distortion, and is moresuited for immersive data visualization applications.
Limitation and Future Work. As reported in Section 5.3, solving the largest inscribed projection problem
is not a computationally demanding task, but may not besuited for realtime mapping. In that case, direct mapping
can be better regarding the computational performance.
In addition, although we quantitatively evaluate distortions in Section 7, the distortions are noticeable if we apply the mapping technique for textures (Fig. 13 (bottom)).

Figure 16. Since the red edges of the unfolded cube pattern shownin the left figure are merged when constructing a cube, the mapping places both ends of the plot close (see the area enclosed by
the red rectangle in the right figure).

Therefore, further optimization is necessary when one is applying the technique described here to such applications.
Our method also may not preserve neighbors for points
close to the edges of the cube because we first project a
2D plot to an unfolded cube pattern, fold it to construct a
cube, and then map points on the cube to a spherical surface. For example, in Fig. 16 and Fig. 1(a), while both endsof the plot are far apart in the 2D plot, they are close together
in the spherical plots. Therefore, we must take care not tomislead viewers if applying the technique for data visualization. To address such problems, we can scale the plotslightly smaller to avoid edges. Because mapping distortions are inevitable, we would like to further optimize themapping, and conduct a task-based user study to measure
the performance of our method in applications with sensitivity to this feature.

Acknowledgements
We thank all reviewers for their helpful comments. This
work was supported by JSPS KAKENHI grant number
19K24338.

Appendix
2D Polygonal Meshes used for Quantitative Evaluation

Figure 17. Input 2D polygonal meshes for Table 2 and Fig. 15.

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