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Without loss of generality we can assume that the effective viewpoint
of the catadioptric system lies at the origin of a
cartesian coordinate system. Suppose that the effective pinhole is
located at the point . Then, again without loss of generality,
we can assume that the zaxis lies in the direction
. Moreover, since perspective projection
is rotationally symmetric about any line through , the
mirror can be assumed to be a surface of revolution about the zaxis
. Therefore, we work in the 2D cartesian frame
where is a unit vector orthogonal to , and try to find the
2dimensional profile of the mirror z(r) = z(x,y) where
. Finally, if the distance from to
is denoted by the parameter c, we have
and . See
Figure 1 for an illustration^{} of the coordinate frame.
Figure 1:
The geometry used to derive the fixed viewpoint
constraint equation. The viewpoint is located
at the
origin of a 2D coordinate frame , and the pinhole of the camera is located at a distance c from
along the zaxis .If a ray of light, which was about to pass through
, is reflected at the mirror point (r,z), the angle
between the ray of light and is
. If the ray is then reflected and passes
through the pinhole , the angle it makes with
is , and the angle
it makes with is . Finally, if
is the angle between the normal to the mirror at (r,z) and
, then by the fact that the angle of incidence equals
the angle of reflection, we have the constraint that

We begin the translation of the fixed viewpoint constraint into
symbols by denoting the angle between an incoming ray from a world
point and the raxis by . Suppose that this ray intersects
the mirror at the point (z,r). Then, since we assume that it also
passes through the origin we have the relationship:
 
(1) 
If we denote the angle between the reflected ray and
the (negative) raxis by , we also have:
 
(2) 
since the reflected ray must pass through the pinhole .Next, if is the angle between the zaxis and the normal to
the mirror at the point (r,z), we have:
 
(3) 
Our final geometric relationship is due to the fact that we can assume
the mirror to be specular. This means that the angle of incidence
must equal the angle of reflection. So, if is the
angle between the reflected ray and the zaxis, we have
and
.(See Figure 1 for an illustration of this constraint.)
Eliminating from these two expressions and rearranging
gives:
 
(4) 
Then, taking the tangent of both sides and using the standard
rules for expanding the tangent of a sum:
 
(5) 
we have:
 
(6) 
Substituting from Equations (1), (2), and (3)
yields the fixed viewpoint constraint equation:
 
(7) 
which when rearranged is seen to be a quadratic firstorder ordinary
differential equation:
 
(8) 
Next: General Solution of the
Up: The Fixed Viewpoint Constraint
Previous: The Fixed Viewpoint Constraint
Simon Baker
1/22/1998