Spherical Mirror: A New Approach toHemispherical Dome Projection
Paul Bourke
Swinburne University
P.O.Box 218
Hawthorn
Melbourne, Vic 3122 Australia
pdb@swin.edu.auThe projector and lens are located inAbstract
Historically dome environments
Planetariums and smaller personal the center of the dome; fisheye frames
from movies or generated by real time
have been restricted to large planedomes can provide an immersive envi- interactive applications are projectedtariums and used primarily for pubronment for science education, virtual through the lens, and if created corlic education in astronomy, illustrating the positions/motion of planets,reality, and entertainment. Digital projec- rectly they look undistorted on thedome. Such solutions have the benefit
stars, and constellations. These plantion into domes, called “full dome projec- of being easy to manage and don’t
etariums have used a variety of spetion”, can be a technically challenging usually require specialised computercialised projection hardware such asstar projectors [1], laser projectors,
and expensive exercise, particularly for hardware. There are issues such as resolution and brightness, but they areand multiple edge-blended slide proinstallations with modest budgets. An al- largely a reflection of the price one isjectors. If the planetarium had theternative full dome digital projection sys- prepared to pay for the projector.
ability to present real-time digital
graphics, the graphics were limited
tem is presented here that is based upon However for small operations based
public education or researchto a small portion of the dome, typia single projector and a spherical mirror aroundbased
virtualenvironments, the costcally using a single CRT projector.
to scatter the light onto the dome sur- of a good fisheye projection system
Even though planetariums haveface. The approach offers many advan- may still be prohibitive. The alternabeen limited by the available technology, the immersive possibilities
tages over the fisheye lens alternatives, tive projection system introduced
here significantly reduces the cost of
have been obvious, mostly due toresults in a similar quality, but at a frac- dome projection while maintaining a
two characteristics of the hemisimilar quality and even offers sometion of the cost.
spherical surface: the viewers’ periinteresting advantages over fisheye
pheral vision is engaged, and prolimit the content to movies. The content isprojection.
ceedings were conducted in the dark where
not even limited to astronomy or even to
there are often no frames of reference otherscience education, but indeed to any subject
Spherical mirror projection
than the projected imagery. The former ismatter including, but not limited to, a wider
The projection system proposed here uses
responsible for the vertigo one often experirange of educational topics, immersive spaa
spherical
mirror instead of a fisheye lens toences with rapidly rotating imagery; thetial environments, virtual heritage, and even
distributelightin a wide solid angle. It canlater allows the apparent shape of the dome
pure entertainment.
be readily appreciated that a spherical mirror
to be changed and is also credited with
With the success of digital projection in
can reflect light from a rectilinear frustum
depth perception similar to stereoscopic 3D
large planetariums and the development of(produced by a commodity data projector)
effects.
formal standards [2], interest has been growover almost the whole surface of a dome (seeIn more recent times planetariums haveing in how to offer the same experience infigure 1). There are a number of options forbeen upgraded to provide full dome digital
smaller domes. These smaller domes are typithe projector/mirror placement in relation
projection, that is, a movie is seamlessly procally around 10m in diameter (found into the dome, but the geometry discussed
jected onto the dome surface at typically 30
many science centers) down to the smaller
here will consider a single projector within aframes per second. For larger planetariums5m diameter inflatable domes [3] that can besmall dome. In this case the spherical mirror
this full dome projection is achieved withinstalled almost anywhere. The differenceis placed as close as possible to the rim of themultiple projectors, most commonly CRT
between these smaller domes and the large
dome (see figure 2). A number of alternative
projectors. The projectors are carefully
planetariums is largely in the system costgeometries and environments have beenaligned and edge blended across overlapping
they can sustain. Not only do multiple proproposed in the past, for example “Enspherprojection regions. The system is driven byjector systems have a high initial cost, theyed Vision” that uses a convex mirror to promovies made up of fisheye images, these arealso have higher requirements in local experject into cylindrical environments [4] as wellusually diced into pieces and played backtise, and incur a significant cost of owneras polyhedral spaces. The author has addiusing specialised graphics hardware. Even
ship. The solution to these cost problems hastionally explored dual mirrors/projectors
more recently, high-end graphics systems
been to employ a fisheye lens attached genlocated in the middle of the dome [5] with ahave been able to project interactive grapherally to a single commodity data projector.
single edge blend across the middle. Anotherics in real-time so it is no longer necessary toIntroduction

6

Planetarian

December 2005

Figure 1. Representative rays off a projection source and reflected from a sphericalsurface. All illustrations are by the author.

the mirror and the projector or by varying
the projector zoom. While it is true thatthe whole dome surface cannot be totally
covered, it is equally not common for fisheye projection to cover the whole dome
for pixel efficiency reasons [4].
4. The system is scalable to multiple projectors and mirrors in order to achieve higher
resolution and complete dome projection.
For example, a dual mirror and projector
arrangement would give a single edge
blend across the middle of the dome [5].
5. Unlike a fisheye projector located in thecenter of the dome, the path length fromthe projector to the dome is not constant,resulting in an intensity variation. Fortunately this is straightforward to computeand correct for.
6. Unlike fisheye projection, where not all
the available pixels in the typically rectangular aspect ratio of the projector are used,
all the pixels can be used in spherical mirror projection if the image is entirely contained on the mirror. Note, however, that
not all pixels are used equally efficiently.
7. Unlike the fisheye lens solution, theimages projected need to be warped before
projection. Strictly speaking this is no different to fisheye projection; it too is awarped image, but one we are more familiar with.
8. Angular fisheye lens with good opticaldesign is in focus at all positions on thedome surface. When using a spherical mirror there is a variation in path length fromthe projector to different parts of thedome. The effect of this focusing problem
can be minimised by choosing projectors
with a good depth of focus.

Warping
For the image on the dome surface to lookcorrect and undistorted, a precisely warped
image needs to be projected. The form of thedistortion can be seen in figure 4. Figure 4a isFigure 2. Typical position of the projector (16:9), mirror, and dome in aplanetarium environment. Red lines illustrate the projected distribution of a regular grid.

installation by the author located a mirror at
the base of a vertically mounted truncateddome [6].
There are a number of comparisons one
can make between a spherical mirror reflection arrangement and a fisheye lens system:
1. It can be advantageous to locate the projection hardware away from the center ofthe dome since the center is generally thebest location for undistorted viewing. Thisis the classic problem for single-person
domes with fisheye lens projection; theviewer and fisheye lens cannot occupy thesame space.

December 2005

2. The projector and optics
have been separated, making it possible to choose
projectors based upon thecharacteristics important
for the application at hand,for example: price, brightness, resolution, or contrast
ratio. Fisheye lens can typically only fitted to a very
narrow range of projectors.
3. The coverage on the dome
can be controlled by varying the distance between

Planetarian

Figure 3. Geometry after the coordinate system has
been transformed to place the spherical mirror at
the origin and the intersection of the projected rayon the mirror/dome in the x-z plane.

7Figure 4. Warping of lines of longitude and latitude. The fisheye image in 4a consists of equally separated lines of longitude andlatitude and is a convenient test pattern for dome projection. If 4b is correctly projected and viewed from the center of the dome,
the central pole should be at the highest point in the dome, the lines of longitude and latitude should all appear vertical and horizontal respectively, and the line of 0 latitude should encircle the horizon of the dome. Note the intensity is varied so it fades gradually to back at the rear of the dome.

a regular polar grid appropriate for fishThe alternative projection system dome, and to implement edge-blending
eye projection and figure 4b is thefor multiple projector configurations.
introduced here significantly re- Figure 6 shows a fisheye image applied as
warped version that will look correct on
the surface of the dome. Figure 5 shows
duces the cost of dome projection a texture onto a regular mesh using
the projector and mirror arrangement
while maintaining a similar quality OpenGL. Similarly, a standard approachwith a warped polar mesh on the comto creating fisheye images in interactive
and even offers some interesting OpenGL applications is to render four
puter display and the resulting image on
advantages over fisheye projection.
the mirror.
faces of a cube and form the fisheye
Creating correctly warped images
image by applying those as textures on aprojector is located at P 1 , the mirror is ofgiven a particular projector, mirror, andmesh with precisely specified texture coordiradius r, and the position on the dome is P2.
dome arrangement requires finding thenates. Figure 7a shows the mesh onto which
The path length from the projector to thepoint on the projector frustum for any point
four cubic map textures are applied to form
mirror is L1, the path length from the domeon the dome. The problem is three-dimenthe correctly warped fisheye; figure 7b shows
to the mirror is L2, these are given as a funcsional but can be turned into a simpler twoa resulting screen dump from a real timetion of ø below
dimensional problem by firstly translating
driving simulator.
the geometry so the spherical mirror is at theIt should be noted that while the discusL12 = (P1x – r cos(ø))2 + (r sin(ø))2origin and then rotating the geometry sosion here has concentrated on hemispherical
L22 = (P2x – r cos(ø))2 + (P2z – r sin(ø))2that the point on the mirror, dome, and prodomes, it can also be employed in any situajector lies in a single plane. In figure 3, thetion where extremely wide angle projection
Fermat’s principle states thatis required. In particular, it could be used tolight travels by the shortest
wrap the output from a single projector into
route, so ø can be found by
a rectangular room, achieving an undistortminimising the total light path
ed result similarly requires the calculation oflength from the projector tothe correct warping function.
the position on the dome,Conclusion
namely minimising (L 1 2 +An alternative dome projection system
L22)1/2.
has been designed and demonstrated to be
Once a relationship can besuitable for small planetarium domes. The
made between positions in themathematics required and practical issues
projection plane and the dome,
involved in warping fisheye images as a prea regular mesh can be createdprocessing stage and in real-time have been
where each node is represented
developed and tested. By comparison to theby normalised frustum coordimore conventional fisheye solutions, thenates (x, y), fisheye image texspherical mirror solution suffers from noture coordinates (u, v), and an
serious disadvantages and offers some advanintensity value. The intensity
tages at a significantly lower cost. Future
value can be used for compenwork includes creating an optimal mirror
sating for the brightness variasurface rather than using a spherical surface.
tion due to the range of light
Such an optimal surface will use all pixels inpath lengths, to softly fade theFigure 5. Projector and mirror in development
configuration, the projected image on the laptop
the rectangular image plane and attempt toimage towards the back of thescreen and mirror surface is a warped polar grid.

8

Planetarian

December 2005

Figure 6. Fisheye image shown in 6a is applied as an OpenGL textured mesh in 6b. Each mesh node in 6b is represented by (x,y) coordinate in normalized projection plane coordinates, a (u,v) texture coordinate that relates to the fisheye image, and an intensity
value that can compensate for the variable light path length.

Figure 7. 7a shows the warped appearance of the four texture regions and mesh outline as used by real-time OpenGL applications.
The textures are derived from 4 virtual cameras each with the face of a cube as the projection plane. 7b is a single frame from areal-time driving simulator using the warped texture meshes in 7a.

distribute them equally on the hemisphere.

References
1. The History of the Planetarium, Chartrand,M.R., Planetarian , Vol 2, #3, September
1973.
2. Proceedings of the IPS 2004 Special Session
Fulldome Standard Summit. Valencia,December 2005

Spain, July 7/8, 2004.
3. Inflatable planetarium domes, Starlab Inc.
[http://www.starlab.com]
4. Ensphered Vision, Hashimoto, W., Iwata,H., Ensphered Vision: Spherical immersive
display using convex mirror. Transactionsof the Virtual Reality Society of Japan, 4(3) 497-486.

Planetarian

65 Bourke, December 2004. [http://astronomy.swin.edu.au/~pbourke/projection/dualdome]
6. Spherical mirror projection for an uprightdome. Paul Bourke, May 2005. [http://astronomy.swin.edu.au/~pbourke/projection/uprightdome]
C9