Table 1
Summary of the Subordinate Variables Discussed in Section 2
Variable Name Description Variables associated with the configuration of the final mirror t ′ Displacement from the second-to-last interface to the last mirror θ Tilt of the last mirror s Curvature of the last mirror ϕ Rotation of the last mirror Variables associated with the configuration of the object and image planes θ obj Tilt of the object ϕ obj Rotation of the object t im Displacement from the last mirror to the image plane θ im Tilt of the image ϕ im Rotation of the image
Table 2
Constraints on the Value of t ′ That Guarantee a Physically Realizable System for At Least One Value of the Angle of Incidence on the Final Mirror, θ
g
2
2 , m 2 , b I, α b I, β , and (t ′/n ′) Satisfy One and Only One Set of Inequalities in This Column Additional Constraint on t ′ Such That |m 1,max (t ′)| ≥ m (If Necessary) (
g
2
2
≥
m
2 ) No additional constraint required
[(
g
2
2
<
m
2 ) and (t ′/n ′ < b I, α < b I, β )] No additional constraint required
[(
g
2
2
<
m
2 ) and (b I, β < b I, α < t ′/n ′)] No additional constraint required
[(
g
2
2
<
m
2 ) and (b I, α < t ′/n ′ < b I, β )]
t
′
≤
n
′
(
b
I
,
β
+
b
I
,
α
A
0
A
0
+
1
)
[(
g
2
2
<
m
2 ) and (b I, α < b I, β < t ′/n ′)]
t
′
≥
n
′
(
b
I
,
β
-
b
I
,
α
A
0
1
-
A
0
)
,
A
0
<
1
t
′
≤
n
′
(
b
I
,
β
-
b
I
,
α
A
0
1
-
A
0
)
,
A
0
>
1
[(
g
2
2
<
m
2 ) and (b I, β < t ′/n ′ < b I, α )]
t
′
≥
n
′
(
b
I
,
β
+
b
I
,
α
A
0
1
+
A
0
)
[(
g
2
2
<
m
2 ) and (t ′/n ′ < b I, β < b I, α )]
t
′
≤
n
′
(
b
I
,
β
-
b
I
,
α
A
0
1
-
A
0
)
,
A
0
<
1
,
t
′
≥
n
′
(
b
I
,
β
-
b
I
,
α
A
0
1
-
A
0
)
,
A
0
>
1
Table 3
Constraints That Must Be Imposed on the Value of t ′
b I, α , b I, β , A 0 ,
g
2
2 , and m 2 Satisfy One and Only One Set of Inequalities in This Columnt ′ Must Satisfy the Corresponding Inequality in This Column(
g
2
2
>
m
2 ) t ′ > 0or [(
g
2
2
<
m
2 ) and (b I, β < b I, α < 0)]
{(
g
2
2
<
m
2 ) and [b I, α < min(0, b I, β )] and [max(0, −b I, β /b I, α ) < A 0 < 1]}
t
′
>
n
′
max
[
0
,
(
b
I
,
β
-
b
I
,
α
A
0
1
-
A
0
)
]
{(
g
2
2
<
m
2 ) and (b I, α > 0) and (b I, β < b I, α ) and [A 0 > max(0, b I, β /b I, α )]}
t
′
>
n
′
max
[
0
,
(
b
I
,
β
+
b
I
,
α
A
0
1
+
A
0
)
]
[(
g
2
2
<
m
2 ) and (0 < b I, β < b I, α ) and (A 0 < b I, β /b I, α )]
0
<
t
′
<
n
′
(
b
I
,
β
-
b
I
,
α
A
0
1
-
A
0
)
or
t
′
>
n
′
(
b
I
,
β
+
b
I
,
α
A
0
1
+
A
0
)
{(
g
2
2
<
m
2 ) and (b I, β > 0) and (b I, α < b I, β ) and [A 0 < min(1, |b I, β /b I, α |)]}
0
<
t
′
<
n
′
(
b
I
,
β
+
b
I
,
α
A
0
1
+
A
0
)
,
or
t
′
>
n
′
(
b
I
,
β
-
b
I
,
α
A
0
1
-
A
0
)
{(
g
2
2
<
m
2 ) and [−(b I, β ) < b I, α < 0] and (1 < A 0 < |b I, β /b I, α |)}
0
<
t
′
<
n
′
(
b
I
,
β
+
b
I
,
α
A
0
1
+
A
0
) or [(
g
2
2
<
m
2 ) and (0 < b I, α < b I, β ) and (A 0 > 1)]
{(
g
2
2
<
m
2 ) and [b I, α < min(0, b I, β )] and [A 0 > max(1, |b I, β /b I, α |)]} No solution
Table 4
Constraints That Must Be Imposed on the Angle of Incidence on the Final Mirror, θ
α 0 , β 0 ,
m
¯
1
,
m
¯
2 and m Satisfy One and Only One Set of Inequalities in a Row of This Columnθ Must Satisfy the Corresponding Constraints in This Column(α 0 β 0 < 0)
[
max
(
0
,
1
-
∣
α
0
m
/
m
¯
2
∣
1
+
∣
β
0
m
/
m
¯
2
∣
)
≤
cos
2
(
θ
)
≤
1
-
∣
α
0
m
/
m
¯
1
∣
1
+
∣
β
0
m
/
m
¯
1
∣
] a
[(α 0 β 0 > 0) and (|β 0 | < |α 0 |) and (|β 0 m /
m
¯
2 | < 1)]
[
max
(
0
,
1
-
∣
α
0
m
/
m
¯
2
∣
1
-
∣
β
0
m
/
m
¯
2
∣
)
≤
cos
2
(
θ
)
≤
1
-
∣
α
0
m
/
m
¯
1
∣
1
-
∣
β
0
m
/
m
¯
1
∣
] a
[(α 0 β 0 > 0) and (|β 0 | < |α 0 |) and (|β 0 m /
m
¯
2 | > 1)]
[
0
≤
cos
2
(
θ
)
≤
1
-
∣
α
0
m
/
m
¯
1
∣
1
-
∣
β
0
m
/
m
¯
1
∣
] a
[(α 0 β 0 > 0) and (|α 0 | < |β 0 |) and (|α 0 m /
m
¯
2 | < 1)]
[
1
+
∣
α
0
m
/
m
¯
2
∣
1
+
∣
β
0
m
/
m
¯
2
∣
≤
cos
2
(
θ
)
≤
1
+
∣
α
0
m
/
m
¯
1
∣
1
+
∣
β
0
m
/
m
¯
1
∣
] b
[(α 0 β 0 > 0) and (|α 0 | < |β 0 |) and (|α 0 m /
m
¯
2 | > 1) and (|β 0 m /
m
¯
1 < 1)]
[
0
≤
cos
2
(
θ
)
≤
1
-
∣
α
0
m
/
m
¯
2
∣
1
-
∣
β
0
m
/
m
¯
2
∣
] a or
[
1
+
∣
α
0
m
/
m
¯
2
∣
1
+
∣
β
0
m
/
m
¯
2
∣
≤
cos
2
(
θ
)
≤
1
+
∣
α
0
m
/
m
¯
1
∣
1
+
∣
β
0
m
/
m
¯
1
∣
] b
[(α 0 β 0 > 0) and (|α 0 | < |β 0 |) and (|α 0 m /
m
¯
2 | > 1) and (|β 0 m /
m
¯
1 > 1)]
[
max
(
0
,
1
-
∣
α
0
m
/
m
¯
1
∣
1
-
∣
β
0
m
/
m
¯
1
∣
)
≤
cos
2
(
θ
)
≤
1
-
∣
α
0
m
/
m
¯
2
∣
1
-
∣
β
0
m
/
m
¯
2
∣
] a or
[
1
+
∣
α
0
m
/
m
¯
2
∣
1
+
∣
β
0
m
/
m
¯
2
∣
≤
cos
2
(
θ
)
≤
1
+
∣
α
0
m
/
m
¯
1
∣
1
+
∣
β
0
m
/
m
¯
1
∣
] b
a A real image is formed if and only if
β 0 is greater than zero.
b A real image is formed if and only if
β 0 is less than zero.
Table 5
Constraints That Ensure a Real, Nonanamorphic Image with Magnification, m
b I, α , b I, β , A 0 , m 2 ,
g
1
2
,
g
2
2 , and det2 (
G ) Satisfy One and Only One Set of Inequalities in This Columnt ′ and θ Must Satisfy the Corresponding Inequalities in This Column[b I, β < min(0, b I, α )]
t
′
>
max
(
0
,
n
′
b
I
,
α
)
and
1
+
∣
α
0
m
/
m
¯
2
∣
1
+
∣
β
0
m
/
m
¯
2
∣
≤
cos
2
(
θ
)
≤
1
+
∣
α
0
m
/
m
¯
1
∣
1
+
∣
β
0
m
/
m
¯
1
∣ or [(0 < b I, β < b I, α ) and (A 0 > b I, β /b I, α ) and (
g
2
2
<
m
2 )]
[(b I, α < 0 < b I, β ) and (
g
2
2
>
m
2 )]
max
(
0
,
n
′
b
I
,
α
)
<
t
′
<
{
n
′
b
I
,
β
,
g
2
2
>
m
2
n
′
(
b
I
,
β
+
b
I
,
α
A
0
1
+
A
0
)
g
2
2
<
m
2
and
max
(
0
,
1
-
∣
α
0
m
/
m
¯
2
∣
1
+
∣
β
0
m
/
m
¯
2
∣
)
≤
cos
2
(
θ
)
≤
1
-
∣
α
0
m
/
m
¯
1
∣
1
+
∣
β
0
m
/
m
¯
1
∣ or {(0 < b I, α < b I, β ) and (A 0 < b I, β /b I, α ) and [
m
2
<
det
2
(
G
)
/
g
1
2 ] and (
g
2
2
>
m
2 )} or [(b I, α < 0 < b I, β ) and (A 0 < |b I, β /b I, α |) and (
g
2
2
<
m
2 )]
[(0 < b I, β < b I, α ) and (
g
2
2
>
m
2 )]
0
<
t
′
<
{
n
′
b
I
,
β
,
g
2
2
>
m
2
n
′
(
b
I
,
β
-
b
I
,
α
A
0
1
-
A
0
)
,
g
2
2
<
m
2
and
1
-
∣
α
0
m
/
m
¯
1
∣
1
-
∣
β
0
m
/
m
¯
1
∣
≥
cos
2
(
θ
)
≥
{
0
,
|
β
0
m
m
¯
2
|
>
1
max
(
0
,
1
-
∣
α
0
m
/
m
¯
2
∣
1
-
∣
β
0
m
/
m
¯
2
∣
)
,
|
β
0
m
m
¯
2
|
<
1
or
t
′
>
n
′
b
I
,
α
and
1
+
∣
α
0
m
/
m
¯
2
∣
1
+
∣
β
0
m
/
m
¯
2
∣
≤
cos
2
(
θ
)
≤
1
+
∣
α
0
m
/
m
¯
1
∣
1
+
∣
β
0
m
/
m
¯
1
∣ or [(0 < b I, β < b I, α ) and (A 0 < b I, β /b I, α ) and (
g
2
2
<
m
2 )]
{(0 < b I, α < b I, β ) and (A 0 > b I, β /b I, α ) and [
m
2
<
det
2
(
G
)
/
g
1
2 ] and (
g
2
2
>
m
2 )}
0
<
t
′
<
n
′
(
b
I
,
β
-
b
I
,
α
A
0
1
-
A
0
)
and
1
-
∣
α
0
m
/
m
¯
2
∣
1
-
∣
β
0
m
/
m
¯
2
∣
≥
cos
2
(
θ
)
≥
{
0
,
|
β
0
m
m
¯
1
|
<
1
max
(
0
,
1
-
∣
α
0
m
/
m
¯
1
∣
1
-
∣
β
0
m
/
m
¯
1
∣
)
,
|
β
0
m
m
¯
1
|
>
1
or
n
′
b
I
,
α
<
t
′
<
n
′
b
I
,
β
and
max
(
0
,
1
-
∣
α
0
m
/
m
¯
2
∣
1
+
∣
β
0
m
∣
m
¯
2
)
≤
cos
2
(
θ
)
≤
1
-
∣
α
0
m
/
m
¯
1
∣
1
+
∣
β
0
m
/
m
¯
1
∣
{(0 < b I, α < b I, β ) and [
m
2
<
det
2
(
G
)
/
g
1
2 ] and (
g
2
2
>
m
2 )}
0
<
t
′
<
n
′
b
I
,
α
and
1
-
∣
α
0
m
/
m
¯
2
∣
1
-
∣
β
0
m
/
m
¯
2
∣
≥
cos
2
(
θ
)
≥
{
0
,
|
β
0
m
m
¯
1
|
<
1
max
(
0
,
1
-
∣
α
0
m
/
m
¯
1
∣
1
-
∣
β
0
m
/
m
¯
1
∣
)
,
|
β
0
m
m
¯
1
|
>
1
or
n
′
b
I
,
α
<
t
′
<
{
n
′
b
I
,
β
,
g
2
2
>
m
2
n
′
(
b
I
,
β
+
b
I
,
α
A
0
1
+
A
0
)
,
g
2
2
<
m
2
and
max
(
0
,
1
-
∣
α
0
m
/
m
¯
2
∣
1
+
∣
β
0
m
/
m
¯
2
∣
)
≤
cos
2
(
θ
)
≤
1
-
∣
α
0
m
/
m
¯
1
∣
1
+
β
0
m
/
m
¯
1
∣ or [(0 < b I, α < b I, β ) and (
g
2
2
<
m
2 )]
(b I, α < b I, β < 0) No real image or [(b I, α < 0 < b I, β ) and (A 0 > |b I, β /b I, α |) and (
g
2
2
<
m
2 )]
Table 6
Constraints That Must Be Imposed on the Angle of Incidence on the Final Mirror, θ
α
0
11
,
β
0
11
,
m
¯
1
11 , and
m
¯ Satisfy One and Only One Set of Inequalities in a Row of This Column θ Must Satisfy the Corresponding Equality in This Column(
α
0
11
β
0
11
<
0 )
(
cos
2
(
θ
)
=
1
-
∣
α
0
11
m
/
m
¯
1
11
∣
1
+
∣
β
0
11
m
/
m
¯
1
11
∣
) a
[(
α
0
11
β
0
11
>
0 ) and (
∣
β
0
11
∣
<
∣
α
0
11
∣ )]
(
cos
2
(
θ
)
=
1
-
∣
α
0
11
m
/
m
¯
1
11
∣
1
-
∣
β
0
11
m
/
m
¯
1
11
∣
) a
[(
α
0
11
β
0
11
>
0 ) and (
∣
α
0
11
∣
<
β
0
11
∣ ) and (
∣
α
0
11
m
/
m
¯
1
11
∣
<
1 )]
(
cos
2
(
θ
)
=
1
+
∣
α
0
11
m
/
m
¯
1
11
∣
1
+
∣
β
0
11
m
/
m
¯
1
11
∣
) a
[(
α
0
11
β
0
11
>
0 ) and (
∣
α
0
11
∣
<
∣
β
0
11
∣ ) and (
∣
α
0
11
m
/
m
¯
1
11
∣
>
1 )]
(
cos
2
(
θ
)
=
1
-
∣
α
0
11
m
/
m
¯
1
11
∣
1
-
∣
β
0
11
m
/
m
¯
1
11
∣
) a or
(
cos
2
(
θ
)
=
1
+
∣
α
0
11
m
/
m
¯
1
11
∣
1
+
∣
β
0
11
m
/
m
¯
1
11
∣
) b
a A real image is formed if and only if
β
0
11 is greater than zero.
b A real image is formed if and only if
β
0
11 is less than zero.
Table 7
Constraints That Ensure a Real, Nonanamorphic Image with Magnification m of the Object at Infinity
b
I
,
α
11
,
b
I
,
β
11 , A 0 ,
g
2
2 , and m 2 Satisfy One and Only One Set of Inequalities in a Row of This Column t ′ and θ Must Satisfy the Corresponding Constraints in This Column
b
I
,
β
11
<
min
(
0
,
b
I
,
α
11
)
t
′
>
max
(
0
,
n
′
b
I
,
α
11
)
and
cos
2
(
θ
)
=
1
+
∣
α
0
11
m
/
m
¯
1
11
∣
1
+
∣
β
0
11
m
/
m
¯
1
11
∣ or [(
0
<
b
I
,
β
11
<
b
I
,
α
11 ) and (
A
0
>
b
I
,
β
11
/
b
I
,
α
11 ) and (
g
2
2
<
m
2 )]
[(
b
I
,
β
11
>
0 ) and (
b
I
,
α
11
<
b
I
,
α
11 ) and (
g
2
2
>
m
2 )]
max
(
0
,
n
′
b
I
,
α
11
)
<
t
′
<
{
n
′
b
I
,
β
11
,
g
2
2
>
m
2
n
′
(
b
I
,
β
11
+
b
I
,
α
11
A
0
1
+
A
0
)
,
g
2
2
<
m
2
and
cos
2
(
θ
)
=
1
-
∣
α
0
11
m
/
m
¯
1
11
∣
1
+
∣
β
0
11
m
/
m
¯
1
11
∣ or [(
b
I
,
β
11
>
0 ) and (
b
I
,
α
11
<
b
I
,
β
11 ) and (
A
0
<
∣
b
I
,
β
11
/
b
I
,
α
11
∣ ) and (
g
2
2
<
m
2 )]
[(
0
<
b
I
,
β
11
<
b
I
,
α
11 ) and (
g
2
2
>
m
2 )]
0
<
t
′
<
{
n
′
b
I
,
β
11
,
g
2
2
>
m
2
n
′
(
b
I
,
β
11
-
b
I
,
α
11
A
0
1
-
A
0
)
,
g
2
2
<
m
2
and
cos
2
(
θ
)
=
1
-
∣
α
0
11
m
/
m
¯
1
11
∣
1
-
∣
β
0
11
m
/
m
¯
1
11
∣
or
t
′
>
n
′
b
I
,
α
11
and
cos
2
(
θ
)
=
1
+
∣
α
0
11
m
/
m
¯
1
11
∣
1
+
∣
β
0
11
m
/
m
¯
1
11
∣ or [(
0
<
b
I
,
β
11
<
b
I
,
α
11 ) and (
A
0
<
b
I
,
β
11
/
b
I
,
α
11 ) and (
g
2
2
<
m
2 )]
[(
0
<
b
I
,
α
11
<
b
I
,
β
11 ) and (
A
0
>
b
I
,
β
11
/
b
I
,
α
11 ) and (
g
2
2
<
m
2 )]
0
<
t
′
<
n
′
(
b
I
,
β
11
-
b
I
,
α
11
A
0
1
-
A
0
)
and
cos
2
(
θ
)
=
1
-
∣
α
0
11
m
/
m
¯
1
11
∣
1
-
∣
β
0
11
m
/
m
¯
1
11
∣
or
b
I
,
α
11
<
t
′
<
n
′
(
b
I
,
β
11
+
b
I
,
α
11
A
0
1
+
A
0
)
and
cos
2
(
θ
)
=
1
-
∣
α
0
11
m
/
m
¯
1
11
∣
1
+
∣
β
0
11
m
/
m
¯
1
11
∣
(
b
I
,
α
11
<
b
I
,
β
11
<
0 ) No real image or [(
b
I
,
α
11
<
0
<
b
I
,
β
11 ) and (
A
0
>
∣
b
I
,
β
11
/
b
I
,
α
11
∣ ) and (
g
2
2
<
m
2 )]