E IGENMIRRORS AND THE PASSENGER S IDE M IRROR P ROBLEM
R. A NDREW H ICKS , R ONALD K. P ERLINE , AND S ARAH G OTWALS R ODY
D EPARTMENT OF M ATHEMATICS , D REXEL U NIVERSITY
1. I NTRODUCTION
3. R OTATIONAL S YMMETRY
5. C ONSTRUCTING AN A PPROXIMATION
The standard passenger side mirror on a motor vehicle has a limited field of view which results in a
blind spot. Other mirrors, such as spherical mirrors,
reduce the blind spot but distort the image. Our
technique, the method of eigenmirrors, allows us to
construct a mirror using finite differences. The resulting mirror has a wider field of view than a standard passenger side mirror, but less distortion than
a spherical mirror.
Rotational symmetry reduces the problem to a twodimensional system. Given any mirror, we can find
a virtual surface that exactly solves the problem.
Since want the images seen by both viewpoints to
be the same, we can calculate the corresponding surface by tracing rays from both viewpoints and finding their intersection.
To construct the passenger side
mirror we set up a parametrized
vector field. We want the driver
viewpoint and the virtual viewpoint to see the same image, so
the rays of light emitting at the
same angle from both viewpoints
must intersect at the eigensurface.
2. E IGENMIRRORS
The technique uses a virtual target eigensurface to
construct the mirror.
• One viewpoint looks at the reflection of the
target eigensurface in the eigenmirror.
• The other viewpoint looks directly at the target surface.
• We want the images seen by both viewpoints
to be the same.
Figure 1: Side view of the mirror and surface
Viewpoint Looking Down
ϴ
Figure 2: Eigenmirror
Figure 3: Eigensurface
We can force the two rays to intersect if the eigenmirror is curved
so that the ray reflects off of it
in toward the path of the light
ray emitting from the virtual view
point. We create a parametrized
vector field W(x, y, z, t) that is the
vector normal to the mirror that
would have the correct reflection.
We calculate W by adding the
normalized incoming and outgoing vectors at a specific point of
the eigenmirror. The parameter
t is the distance from the vir-
tual viewpoint to the intersection
point.
W(x, y, z, t) = kINk + kOUTk
where
IN = h−x, −y, −zi
αty
αtx
OUT =h−1− tz
−x,
−y,−1−
R
R
R −z i
p
and R = x2 + y 2 + z 2
Here, α is the constant of magnification. As we increase α, the mirror shows a wider field of view
and eliminates the blind spot, although it does unfortunately introduce a small amount of distortion. As we vary t, we should be
able to find a surface normal to
the vector field. While an exact
solution may exist, we have not
yet found an explicit formula for a
mirror. Instead, we discretize the
problem, finding a mesh of points
on the mirror. We build the eigenmirror by first finding an initial
curve normal to the vector field
for a fixed function of t.
Then we use the finite difference
method to extend the initial curve
over the grid of points, moving in
the direction of the tangent vector
at each step. At every new point,
we use Newton’s method to adjust the parameter t to minimize
the dot product of W and the tangent vector. This process forces
the mirror to be more perpendicular to W.
Mirror
4. A PPLICATION
ϴ
Viewpoint Looking Up
Surface
Target Point
In all of our constructions we will make use of ray
tracing, following individual light rays. We see objects by having light rays reflect off of surfaces and
enter our eyes, but the calculations are simplified
if we reverse the light rays and assume they are
emitted from the viewpoints and end at the surface.
Our work also uses the relationship
n̂ = kincoming rayk + koutgoing rayk
to calculate the normal vectors to the eigenmirrors
and the path of light rays reflected by the mirrors.
6. R ESULTS
To apply the method of eigenmirrors to the passenger side mirror problem:
• Let the passenger side mirror play the role of
the eigenmirror.
• The virtual eigensurface will be a distant surface behind the car.
• If the eigensurface is relatively flat and sufficiently far from the viewpoint, the eigenmirror will be a good approximation of a passenger side mirror.
• This setup is not rotationally symmetric, so we
have to generate the eigenmirror and eigensurface simultaneously.
Figure 4: Flat mirror
Figure 5: Spherical mirror
Figure 6: Eigenmirror
7. R EFERENCES
[1] The sky and sand portion of the images were generated with open source code by Alexander Alexandrov.
[2] The code for the Nissan Micra car was written by Rene Bui, Creative Commons License 2.0, 2006.
[3] R. Hicks and R. Perline. "Blind-spot problem for motor vehicles," Applied Optics 44, 3893-3897. 2005.
[4] S. Larsson and V. Thomée. Partial Differential Equations with Numerical Methods. Springer, 2009.