Reflective surface producing a non-reversed, undistorted, perspective view
The invention relates to a reflective surface substantially perpendicular to a vector field described by the equation: W(x,y,z)=T(proj(x,v,z))−(x,v,z)+proj(x,v,z)−(x,v,z)∥T(proj(x,y,z))−(x,y,z)∥ ∥proj(x,y,z)−(x,y,z)∥ and a method for forming the reflective surface. The reflective surface is capable of providing a non-reversed, substantially undistorted direct reflection.
1. A non-reversing mirror comprising: a reflective surface M perpendicular to a vector field W of the following equations:
proj( x,y,z )=(1, y/x,z/x )
T (proj( x,y,z ))=( x 0 ,−αy/x,βz/x )
wherein k is the distance between the reflective surface and observer, s′ is the distance from the reflective surface M to the object plane, x 0 =−(s′−k), and α and β are magnification factors,
W
(
x
,
y
,
z
)
=
T
(
proj
(
x
,
y
,
z
)
)
-
(
x
,
y
,
z
)
T
(
proj
(
x
,
y
,
z
)
)
-
(
x
,
y
,
z
)
+
proj
(
x
,
y
,
z
)
-
(
x
,
y
,
z
)
proj
(
x
,
y
,
z
)
-
(
x
,
y
,
z
)
wherein T is a transformation from an image plane to an object plane of a non-reversed, undistorted direct reflection of an object or object plane, wherein the reflective surface M is represented by a minimizer ƒ* represented by the polynomial function ƒ(x,y,z):
f
(
x
,
y
,
z
)
=
∑
i
+
j
+
k
≤
N
a
(
i
,
j
,
k
)
x
i
y
j
z
k
where N is a fixed positive integer, there are at least three variable coefficients α(i,j,k) and a(1,0,0)=1;
wherein the reflective surface M produces a non-reversed perspective view reflection at the image plane when viewed from a perspective of an observer positioned within the field of view of said reflective surface M;
wherein the reflective surface M has an image error quantity, I e , of less than about 15% when viewed from the perspective of the observer is positioned within the field of view of the reflective surface M, and I e is calculated according to the following equation:
I
e
=
1
diameter
(
T
(
A
)
)
(
∫
A
T
(
1
,
y
,
z
)
-
T
M
(
1
,
y
,
z
)
2
ⅆ
y
ⅆ
z
)
1
2
wherein A is the image of a domain in the image plane over which the reflective surface M is a graph T is a transformation from the image plane to the object plane of a non-reversed, undistorted direct reflection of an object or object plan and T M is a non-reversing image transformation induced from the image plane to the object surface by reflecting at least one ray off reflective surface M; and
wherein the reflective surface M is convex or saddle-shaped.
2. The non-reversing mirror of claim 1 , wherein the reflective surface M has an image error quantity I e of less than about 10%.
3. The non-reversing mirror of claim 1 , wherein the reflective surface M has an image error quantity I e of less than about 5%.
4. The non-reversing mirror of claim 1 , wherein the reflective surface M has an image error quantity I e of less than about 3%.
5. The non-reversing mirror of claim 1 , wherein said reflective surface M is capable of reflecting at least a 30° field of view when viewed from the perspective of an observer.
6. The non-reversing mirror of claim 1 , wherein said reflective surface M is capable of reflecting at least a 40° field of view when viewed from a perspective of an observer.
7. The non-reversing mirror of claim 1 , wherein said reflective surface M is capable of reflecting at least a 45° field of view when viewed from a perspective of an observer.
8. The non-reversing mirror of claim 1 , wherein said reflective surface M is saddle shaped.
9. The non-reversing mirror of claim 1 , wherein said reflective surface M has a magnification of about unit magnification or less.
10. A method for producing a non-reversing mirror comprising generating a non-reversed perspective view reflective surface M comprising the steps of:
a) inputting data comprising an image surface, a domain of an image surface, an object surface, a non-reversing undistorted direct correspondence T and coordinates of an eye of an observer;
b) computing a vector field W(x,y,z) as an algebraic expression from the following three equations;
proj( x,y,z )=(1, y/x,z/x )
T (proj( x,y,z ))=( x 0 ,−αy/x,βz/x )
wherein k is the distance between the reflective surface and observer, s′ is the distance from the reflective surface M to the object plane, x 0 =−(s′−k), and α and β are magnification factors,
W
(
x
,
y
,
z
)
=
T
(
proj
(
x
,
y
,
z
)
)
-
(
x
,
y
,
z
)
T
(
proj
(
x
,
y
,
z
)
)
-
(
x
,
y
,
z
)
+
proj
(
x
,
y
,
z
)
-
(
x
,
y
,
z
)
proj
(
x
,
y
,
z
)
-
(
x
,
y
,
z
)
;
d) representing ƒ as a combination of basis functions of said reflective surface M with unknown coefficients;
e) solving for said unknown coefficients by minimizing an integral of the following equation, over the volume V of the cross-product of vector field W(x,y,z) and a gradient vector field ∇ƒ:
Cost
(
f
)
=
∫
∫
∫
V
∇
f
×
(
W
/
W
)
2
ⅆ
x
ⅆ
y
ⅆ
z
=
f
*
(
x
,
y
,
z
)
,
wherein ∇ƒ is the gradient of the function represented by select basis functions; and
f) calculating a resulting minimizer ƒ* that represents the reflective surface M as a solutions to an equation ƒ*(x,y,z)=C, where C=ƒ*(a,b,c) for a chosen point (a,b,c) in V wherein minimizer ƒ* is represented by the polynomial function ƒ(x,y,z):
f
(
x
,
y
,
z
)
=
∑
i
+
j
+
k
≤
N
a
(
i
,
j
,
k
)
x
i
y
j
z
k
where N is a fixed positive integer, there are at least three variable coefficients α(i,j,k) and α 1,0,0 =1; and
g) producing a non-reversing mirror using the calculated minimizer ƒ*.
11. The method of claim 10 , wherein the reflective surface M reflects a field of view of at least 30° when viewed from a perspective of an observer.
12. The method of claim 10 , wherein the reflective surface M reflects a field of view of at least 40° when viewed from a perspective of an observer.
13. The method of claim 10 , wherein the reflective surface M reflects a field of view of at least 45° when viewed from a perspective of an observer.
14. The method of claim 10 , wherein the reflective surface M is saddle shaped.
15. The method of claim 10 , wherein the reflective surface M has an image error quantity, I e , of less than about 15%, wherein I e is calculated according to the following equation:
I
e
=
1
diameter
(
T
(
A
)
)
(
∫
A
T
(
1
,
y
,
z
)
-
T
M
(
1
,
y
,
z
)
2
ⅆ
y
ⅆ
z
)
1
2
wherein A is the image of a domain in the image plane over which the reflective surface M is a graph and T M is the non-reversing reflection transformation induced from the image plane to the object surface by reflective surface M.
16. The method of claim 10 , wherein the reflective surface M has an image error quantity I e of less than about 10%, and I e is calculated according to the following equation:
I
e
=
1
diameter
(
T
(
A
)
)
(
∫
A
T
(
1
,
y
,
z
)
-
T
M
(
1
,
y
,
z
)
2
ⅆ
y
ⅆ
z
)
1
/
2
wherein A is the image of a domain in the image plane over which the reflective surface M is a graph T is a transformation from the image plane to the object plane of a non-reversed, undistorted direct reflection of an object or object plane and T M is a non-reversing image transformation induced from the image plane to the object surface by reflecting at least one ray off reflective surface M.
17. The method of claim 16 , wherein the reflective surface M has an image error quantity I e of less than about 5%.
18. The method of claim 10 , wherein said magnification factors α and β are about k+s′ or less.