Linearized static panoramic optical mirror

1. A method for designing a fabrication profile for a panoramic mirrored cone element, said cone element comprising a base and an apex, using calculating means for performing mathematical calculations, comprising the steps, starting with a plurality of predetermined input parameters for said cone element, of;

calculating a parabolic profile for said cone element;

calculating a spherical profile for said cone element, based on said parabolic profile;

calculating a composite profile for said cone element, based on said parabolic profile and said spherical profile; and

calculating an elliptical profile for said cone element, based on said composite profile; wherein

said fabrication profile comprises said elliptical profile.

2. The method of claim 1, said predetermined input parameters comprising:

a position of a detector device in a proposed application of said cone element;

a most positive vertical scene angle 1;

a most negative vertical scene angle 2;

an initial diameter of said base of said cone element;

an initial aspect ratio of said cone element, defined as a ratio of said apex of said cone to said base of said cone.

3. The method of claim 2, said step of calculating said parabolic profile for said cone element further comprising the steps of:

A) for a first approximation:

1: solving for y1, x1: y1Kp/Tan2(1)

x12(Kp*y1)1/2

2: solving for y2, x2: y2Kp/Tan2(2)

x22(Kp*y2)1/2,

where 1451/2 and 2452/2, and KpDb/2, where Db is a diameter of said base of said cone element, and

3: determining a correction factor dX for Kp:

dX(Rc*(x1x2))/(Rc1); where Rc denotes an aspect ratio of said cone element apex to said cone element base;

B) for a second approximation:

4: setting Kp(Db/2)2/(X2dX),

recalculating said equations 1, 2, and 3 using said value of Kp determined by said equation 4, and calculating the factors:

KpXdX, recalculated with said Kp of equation B)4,

KpYy1, recalculated with said Kp of equation B)4,

Ymy2y1, where Ym is a height of said cone element;

C) generating data files containing angular and associated radial positions of all points of said parabolic profile, and y and x coordinates of all points of said parabolic profile and associated tangents to an optical axis and a value of for each value of said y coordinates; and

D) using said data files generated in said step C), solving for x, Tan ()f(y), the parabolic profile equations:

X2(Kp*(KpYy))1/2KpX

Tan()(Kp/(KpYy1))1/2

902*.

4. The method of claim 3, said step of calculating said spherical profile for said cone element further comprising the steps of:

E) determining spherical constants Kc, KcY, KcX by calculating:

1: a length of a chord to a face arc:

RO(dy2dx2)1/2

2: a spherical radius: KcRO*Sin((21)/2)/2

3: spherical constants: KcYKc*Sin(1)

KcXKc*Cos(1),

where said height of said cone element dyy2y1, a width of a face arc dxx2x1, and tangents to a surface which are 2 at y2 and 1 at y1, using said y1, y2, x1, and x2 generated from said step of calculating said parabolic profile; and

F) solving the spherical profile equations:

X((Kc2(KcYy)2)KcX)1/2

T(KcYy)/((Kc2KcY)2)1/2,

where X designates a radial displacement and T designates a tangent to the surface as a function of an axial displacement from said apex of said cone.

5. The method of claim 4, said step of calculating said composite profile for said cone element further comprising the steps of:

G) using the parameters and data files generated in said steps of calculating said parabolic and spherical profiles, calculating:

1: Dr(x2x1)/(PxRPxA), radial displacement per pixel,

2: D(12)/(PxRPxA), angle displacement per pixel,

3: Dpx(n1)/D(xnx1)/Dr, pixel error, for all records in said data files, and if abs (Dpx)>a prior Dpx, storing the larger magnitude Dpx,

4: 1Dpx/(PxRPxA), linearity, where:

PxRnumber of pixels per radius in detector,

PxAPxR/Rcnumber of radial pixels subtended by said apex of said cone element, and

PxRPxAnumber of active pixels;

H) determining proportionality constants Js for said spherical profile and Jp for said parabolic profile, as:

JsDpp/(DppDps)

Jp1Js,

where DppParabolic profile Dpx and DpsSpherical profile Dpx; and

I) solving the composite profile equations:

XJp*(2*(Kp*(KpYY1))1/2KpXJs*(Kc2(KcYY1)2)1/2KcX)

TJp*(Kp/(KpYY1))1/2Js((KcYY1)/(Kc2(KcYY1)2)1/2).

6. The method of claim 5, said step of calculating said elliptical profile for said cone element further comprising the steps of:

J) approximating an eccentricity e from the data produced by said steps of calculating said parabolic, spherical, and composite profiles for said cone element, and by iterative procedure, determining a value of e for a least pixel error;

K) calculating elliptical constants Ke, KeY, KeX using parameters ba(1e2)1/2 and YmParabolic y2y1, with initial conditions Km(1e2)1/2 and Kean initial, user-defined value, according to:

1: y1Ke*Tan(1)/(Km2Tan2(1))1/2

2: y2Ke*Tan(2)/(Km2Tan2(2))1/2

3: KeKe*Ym/(y1y2)

4: Recalculate equations K) 1, 2, 3

5: x1Km*(Ke2y12)1/2

6: x2Km*(Ke2y22)1/2

7: KeX(x2(Rc*x1))/(1Rc)

8: KeKe*Db/2/(x2Kex)

9: Recalculate equations K) 1, 2, 5, 6, 7

10: KeYy1

11: YaKeYy, y being a controlling variable; and

L) solving the elliptical profile equations:

XKm*(Ke2Ya2)1/2

TKm*y/(Ke2Ya2)1/2.

7. The method of claim 1, further comprising the step of fabricating said panoramic mirrored cone element using said fabrication profile.

8. The method of claim 6, further comprising the step of fabricating said panoramic mirrored cone element using said fabrication profile.

9. The method of claim 7, further comprising the step of fabricating a central cylinder running longitudinally through said panoramic mirrored cone element.

10. A panoramic mirrored cone element product, produced by a process for designing a fabrication profile for said cone element, said cone element comprising a base and an apex, said process comprising the steps, starting with a plurality of predetermined input parameters for said cone element, of:

calculating a parabolic profile for said cone element;

calculating a spherical profile for said cone element, based on said parabolic profile;

calculating a composite profile for said cone element, based on said parabolic profile and said spherical profile; and

calculating an elliptical profile for said cone element, based on said composite profile; wherein

said fabrication profile comprises said elliptical profile.

11. The product of claim 10, said predetermined input parameters comprising:

a position of a detector device in a proposed application of said cone element;

a most positive vertical scene angle 1;

a most negative vertical scene angle 2;

an initial diameter of said base of said cone element;

an initial aspect ratio of said cone element, defined as a ratio of said apex of said cone to said base of said cone.

12. The product of claim 11, said step of calculating said parabolic profile for said cone element further comprising the steps of:

A) for a first approximation:

1: solving for y1, x1: y1Kp/Tan2(1)

x12(Kp*y1)1/2

2: solving for y2, x2: y2Kp/Tan2(2)

x22(Kp*y2)1/2,

where 1451/2 and 2452/2, and KpDb/2, where Db is a diameter of said base of said cone element, and

3: determining a correction factor dX for Kp:

dX(Rc*(x1x2))/(Rc1); where Rc denotes an aspect ratio of said cone element apex to said cone element base;

B) for a second approximation:

4: setting Kp(Db/2)2/(X2dX),

recalculating said equations 1, 2, and 3 using said value of Kp determined by said equation 4, and

calculating the factors:

KpXdX, recalculated with said Kp of equation B)4,

KpYy1, recalculated with said Kp of equation B)4,

Ymy2y1, where Ym is a height of said cone element;

C) generating data files containing angular and associated radial positions of all points of said parabolic profile, and y and x coordinates of all points of said parabolic profile and associated tangents to an optical axis and a value of for each value of said y coordinates; and

D) using said data files generated in said step C), solving for x, Tan ()f(y), the parabolic profile equations:

X2(Kp*(KpYy))1/2KpX

Tan()(Kp/(KpYy1))1/2

902*.

13. The product of claim 12, said step of calculating said spherical profile for said cone element further comprising the steps of:

E) determining spherical constants Kc, KcY, KcX by calculating:

1: a length of a chord to a face arc:

RO(dy2dx2)1/2

2: a spherical radius: KcRO*Sin((21)/2)/2

3: spherical constants: KcYKc*Sin(1)

KcXKc*Cos(1),

where said height of said cone element dyy2y1, a width of a face arc dxx2x1, and tangents to a surface which are 2 at y2 and 1 at y1, using said y1, y2, x1, and x2 generated from said step of calculating said parabolic profile; and

F) solving the spherical profile equations:

X((Kc2(KcYy)2)KcX)1/2

T(KcYy)/((Kc2KcY)2)1/2,

where X designates a radial displacement and T designates a tangent to the surface as a function of an axial displacement from said apex of said cone.

14. The product of claim 13, said step of calculating said composite profile for said cone element further comprising the steps of:

G) using the parameters and data files generated in said steps of calculating said parabolic and spherical profiles, calculating:

1: Dr(x2x1)/(PxRPxA), radial displacement per pixel,

2: D(12)/(PxRPxA), angle displacement per pixel,

3: Dpx(n1)/D(xnx1)/Dr, pixel error, for all records in said data files, and if abs (Dpx)>a prior Dpx, storing the larger magnitude Dpx,

4: 1Dpx/(PxRPxA), linearity, where:

PxRnumber of pixels per radius in detector,

PxAPxR/Rcnumber of radial pixels subtended by said apex of said cone element, and

PxRPxAnumber of active pixels;

H) determining proportionality constants Js for said spherical profile and Jp for said parabolic profile, as:

JsDpp/(DppDps)

Jp1Js,

where DppParabolic profile Dpx and DpsSpherical profile Dpx; and

I) solving the composite profile equations:

XJp*(2*(Kp*(KpYY1))1/2KpXJs*(Kc2(KcYY1)2)1/2KcX)

TJp*(Kp/(KpyY1))1/2Js((KcYY1)/(Kc2(KcYY1)2)1/2).

15. The product of claim 14, said step of calculating said elliptical profile for said cone element further comprising the steps of:

J) approximating an eccentricity e from the data produced by said steps of calculating said parabolic, spherical, and composite profiles for said cone element, and by iterative procedure, determining a value of e for a least pixel error;

K) calculating elliptical constants Ke, KeY, KeX using parameters ba(1e2)1/2 and YmParabolic y2y1, with initial conditions Km(1e2)1/2 and Kean initial, user-defined value, according to:

1: y1Ke*Tan(1)/(Km2Tan2(1))1/2

2: y2Ke*Tan(2)/(Km2Tan2(2))1/2

3: KeKe*Ym/(y1y2)

4: Recalculate equations K) 1, 2, 3

5: x1Km*(Ke2y12)1/2

6: x2Km*(Ke2y22)1/2

7: KeX(x2(Rc*x1))/(1Rc)

8: KeKe*Db/2/(x2Kex)

9: Recalculate equations K) 1, 2, 5, 6, 7

10: KeYy1

11: YaKeYy, y being a controlling variable; and

L) solving the elliptical profile equations:

XKm*(Ke2Ya2)1/2

TKm*y/(Ke2Ya2)1/2.

16. The product of claim 10, said process further comprising the step of fabricating said panoramic mirrored cone element using said fabrication profile.

17. The product of claim 15, said process further comprising the step of fabricating said panoramic mirrored cone element using said fabrication profile.

18. The product of claim 10, further comprising a central cylinder running therethrough.

19. A computerized device used for designing a fabrication profile for a panoramic mirrored cone element, said cone element comprising a base and an apex, said computerized device comprising means, starting with a plurality of predetermined input parameters for said cone element, for:

calculating a parabolic profile for said cone element;

calculating a spherical profile for said cone element, based on said parabolic profile;

calculating a composite profile for said cone element, based on said parabolic profile and said spherical profile; and

calculating an elliptical profile for said cone element, based on said composite profile; wherein

said fabrication profile comprises said elliptical profile.

20. The computerized device of claim 19, said predetermined input parameters comprising:

a position of a detector device in a proposed application of said cone element;

a most positive vertical scene angle 1;

a most negative vertical scene angle 2;

an initial diameter of said base of said cone element;

an initial aspect ratio of said cone element, defined as a ratio of said apex of said cone to said base of said cone.

21. The computerized device of claim 20, said means for calculating said parabolic profile for said cone element further comprising means for:

A) for a first approximation:

1: solving for y1, x1: y1Kp/Tan2(1)

x12(Kp*y1)1/2

2: solving for y2, x2: y2Kp/Tan2(2)

x22(Kp*y2)1/2,

where 1451/2 and 2452/2, and KpDb/2, where Db is a diameter of said base of said cone element, and

3: determining a correction factor dX for Kp:

dX(Rc*(x1x2))/(Rc1); where Rc denotes an aspect ratio of said cone element apex to said cone element base;

B) for a second approximation:

4: setting Kp(Db/2)2/(X2dX),

recalculating said equations 1, 2, and 3 using said value of Kp determined by said equation 4, and

calculating the factors:

KpXdX, recalculated with said Kp of equation B)4,

KpYy1, recalculated with said Kp of equation B)4,

Ymy2y1, where Ym is a height of said cone element;

C) generating data files containing angular and associated radial positions of all points of said parabolic profile, and y and x coordinates of all points of said parabolic profile and associated tangents to an optical axis and a value of for each value of said y coordinates; and

D) using said data files generated in said step C), solving for x, Tan ()f(y), the parabolic profile equations:

X2(Kp*(KpYy))1/2KpX

Tan()(Kp/(KpYy1))1/2

902*.

22. The computerized device of claim 21, said means for calculating said spherical profile for said cone element further comprising means for:

E) determining spherical constants Kc, KcY, KcX by calculating:

1: a length of a chord to a face arc:

RO(dy2dx2)1/2

2: a spherical radius: KcRO*Sin((21)/2)/2

3: spherical constants: KcYKc*Sin(1)

KcXKc*Cos(1),

where said height of said cone element dyy2y1, a width of a face arc dxx2x1, and tangents to a surface which are 2 at y2 and 1 at y1, using said y1, y2, x1, and x2 generated from said means for calculating said parabolic profile; and

F) solving the spherical profile equations:

X((Kc2(KcYy)2)KcX)1/2

T(KcYy)/((Kc2KcY)2)1/2,

where X designates a radial displacement and T designates a tangent to the surface as a function of an axial displacement from said apex of said cone.

23. The computerized device of claim 22, said means for calculating said composite profile for said cone element further comprising means for:

G) using the parameters and data files generated in said steps of calculating said parabolic and spherical profiles, calculating:

1: Dr(x2x1)/(PxRPxA), radial displacement per pixel,

2: D(12)/(PxRPxA), angle displacement per pixel,

3: Dpx(n1)/D(xnx1)/Dr, pixel error, for all records in said data files, and if abs (Dpx)>a prior Dpx, storing the larger magnitude Dpx,

4: 1Dpx/(PxRPxA), linearity, where:

PxRnumber of pixels per radius in detector,

PxAPxR/Rcnumber of radial pixels subtended by said apex of said cone element, and

PxRPxAnumber of active pixels;

H) determining proportionality constants Js for said spherical profile and Jp for said parabolic profile, as:

JsDpp/(DppDps)

Jp1Js,

where DppParabolic profile Dpx and DpsSpherical profile Dpx; and

I) solving the composite profile equations:

XJp*(2*(Kp*(KpYY1))1/2KpXJs*(Kc2(KcYY1)2)1/2KcX)

TJp*(Kp/(KpYY1))1/2Js((KcYY1)/(Kc2(KcYY1)2)1/2).

24. The computerized device of claim 23, said means for calculating said elliptical profile for said cone element further comprising means for:

J) approximating an eccentricity e from the data produced by said steps of calculating said parabolic, spherical, and composite profiles for said cone element, and by iterative procedure, determining a value of e for a least pixel error;

K) calculating elliptical constants Ke, KeY, KeX using parameters ba(1e2)1/2 and YmParabolic y2y1, with initial conditions Km(1e2)1/2 and Kean initial, user-defined value, according to:

1: y1Ke*Tan(1)/(Km2Tan2(1))1/2

2: y2Ke*Tan(2)/(Km2Tan2(2))1/2

3: KeKe*Ym/(y1y2)

4: Recalculate equations K) 1, 2, 3

5: x1Km*(Ke2y12)1/2

6: x2Km*(Ke2y22)1/2

7: KeX(x2(Rc*x1))/(1Rc)

8: KeKe*Db/2/(x2Kex)

9: Recalculate equations K) 1, 2, 5, 6, 7

10: KeYy1

11: YaKeYy, y being a controlling variable; and

L) solving the elliptical profile equations:

XKm*(Ke2Ya2)1/2

TKm*y/(Ke2Ya2)1/2.

25. The computerized device of claim 19, further comprising means for fabricating said panoramic mirrored cone element using said fabrication profile.

26. The computerized device of claim 24, further comprising means for fabricating said panoramic mirrored cone element using said fabrication profile.

27. The computerized device of claim 25, further comprising means for fabricating a central cylinder running longitudinally through said panoramic mirrored cone element.