## Abstract

In this work we provide a theoretical analysis of gradient deflectometric method for 3D topography measurements of optically smooth surfaces. It is shown that the surface reconstruction problem leads to a nonlinear partial differential equation. A shape of a surface can be calculated by solution of a derived equation. An advantage of the presented method is a noncontact character and no need for a reference surface.

## 1. Introduction

Todays, it is paid a large attention to the development of methods for surface topography measurements, which find a wide application area in many parts of science, engineering, and biomedicine. The existing measurement and evaluation methods are based on different physical principles. Generally, it is possible to divide these methods into contact [1,2] and noncontact techniques [28]. One type of optical methods is based on deflectometry [7,915], which is used frequently for noncontact surface measurements.

In this work we present a deflectometric approach to solving a 3D surface reconstruction problem, which is based on measurements of a surface gradient of optically smooth surfaces. It is shown that a description of this problem leads to a nonlinear partial differential equation, from which the surface shape can be reconstructed numerically. The reconstruction process is presented on an example of 3D aspheric shape reconstruction.

## 2. Differential equation for surface reconstruction

Consider an optical surface, mathematically described by the formula $z=f(x,y)$. The principle of the gradient deflectometric method is shown in Fig. 1 , where O is the origin of a chosen coordinate system and P(x,y,z) is a point at the surface.

Fig. 1 Principle of gradient deflectometric method

Further, assume that the axis z is perpendicular to the plane ξ, which is located at the distance a from the plane xy. The ray described by its unit direction vector s, which is parallel to the z axis, intersects the plane ξ in the point C and the measured surface in the point P(x,y,z). The incidence angle ε is defined as an angle between the direction vector s and the unit vector n of the surface normal in the point P(x,y,z). After the reflection from the surface the ray intersects the plane ξ in the point Q, which is located in the distance t from the point C. The distance t = t(x,y) depends on the position of the point P(x,y,z) at the measured surface. The direction of the reflected ray is characterized by the unit vector s' and the angle ε' with respect to the unit surface normal vector n.

Corresponding to the law of reflection the vectors s, s' and n as well as the points P, C and Q lie in one plane. It holds due to the reflection law [16,17]

$s′=s−2n(sn)$
and
$cosε=−sn, ε′=ε.$
We obtain for the normal vector [17]
$n=s−s′2[1−(ss′)].$
Corresponding to Fig. 1 we can write
$tPC¯=ta−z=tan2ε=2tanε1−tan2ε.$
Moreover, we denote the derivatives of the function $z=f(x,y)$ as
$p=∂z∂x, q=∂z∂y.$
Then, it is well known that the unit normal vector to the surface $z=f(x,y)$ can be expressed as [18,19]
$n=[−pN, −qN, 1N],$
where $N=1+p2+q2$. Assuming that the direction vector s of the incident ray is identical with the z axis direction, then
$s=(0,0,−1).$
It holds
$cosε=−sn=1/N.$
Using the following relationship [18,19]
$tanε=±1−cos2εcosε=±p2+q2,$
we obtain from Eqs. (4) and (6)
$tan2ε=±2p2+q21−p2−q2.$
By substitution of previous formula into Eq. (4) we have
$K=ta−z=±2p2+q21−p2−q2.$
Equation (11) can be also written in the form
$K(1−p2−q2)=±2p2+q2.$
By squaring the previous equation we obtain
$(1−p2−q2)2−4K2(p2+q2)=0.$
After simplification of Eq. (13) we can write the final equation
$(p2+q2)2−2b(p2+q2)+1=0,$
where we denoted
$b=(1+2/K2).$
The solution of Eq. (14) has the following form
$p2+q2=b±b2−1.$
Equation (16) can be generally written in the form
$(∂z∂x)2+(∂z∂y)2=F(x,y,z),$
where
$F(x,y,z)=b±b2−1=1+2/K2±21+1/K2/K.$
The nonlinear partial differential equation of the first order (17) is a general solution of the problem of the reconstruction of the surface $z=f(x,y)$ using the deflectometric method. Due to the fact, that the general solution of Eq. (17) cannot be found in an explicit form, Eq. (17) must be solved numerically. Several methods exists how to solve this nonlinear differential equation [1820]. An universal method is to write the equation of surface as a sum of certain basis functions and then to find the coefficients in the sum in order to satisfy the differential Eq. (17) as well as possible. To find a solution $z=f(x,y)$ of Eq. (17), the unknown function $z=f(x,y)$ can be expressed in the form of two-dimensional polynomials, for example
$z=f(x,y)=∑m=0M∑n=0Ncmnxmyn.$
By substitution of derivatives (5) of the chosen function $z=f(x,y)$ into differential Eq. (17), we obtain the optimization problem instead of the solution of the partial differential equation. The measured area of the surface can be discretized into a regular grid of sampling points $(xi,yk)$ and the value $t(xi,yk)$ is measured using the detector ($i=1,2,...,I$, $k=1,2,...,K$). It is also possible to discretize the measured area by an irregular grid. By substitution of values $(xi,yk)$ and $t(xi,yk)$ into Eq. (5) and Eq. (19) we obtain values $pik=p(xi,yk)$, $qik=q(xi,yk)$, and $zik=z(xi,yk)$. These values of derivatives are substituted into the following function
$M=(∂z∂x)2+(∂z∂y)2−F(x,y,z).$
We obtain values $Mik(c00,c10,c01,c20,c11,c02,.....)$ and we can create a merit functions Φ for the formulation of the optimization problem in the form:
$Φ=∑i=1I∑k=1K[Mik(c00,c10,c01,c20,c11,c02,.....)] 2.$
The goal is to find such values of unknown coefficients $cmn$, in order the function (21) was minimized. Different optimization techniques can be used of the mentioned optimization problem [21,22]. Knowing coefficients $cmn$ one can calculate coordinates $(xi,yk,zik)$ of the measured surface for each measurement point using Eq. (19). We transformed the problem of solving the partial differential equation to the optimization problem. Generally, the convergence of optimization techniques is dependent on initial values of coefficients $cmn$. In the case of testing spherical and aspheric surfaces, which are fabricated, for example, in optics industry, one knows the nominal shape of the test surface very well and good initial parameters for the optimization procedures can be easily found, which ensure a very good convergence.

## 3. Example and simulation

As an example we present a numerical simulation of the described reconstruction method for the case of an aspheric surface.

Figure 2 presents a principal scheme of a possible technical realization of the sensor head for the measurement of distance t. The ray with the direction vector s propagates through the semitransparent mirror M and reflects at the measured surface $S(x,y,z)=0$ in the point P. The reflected ray, which is characterized by the direction vector s′, is reflected by the mirror M and intersects the plane of the two dimensional array CCD sensor in the point Q. The position of the point Q is evaluated and the distance t of the point Q from the center of the sensor C can be determined (see Fig. 1). During the measurement the surface is scanned and the distance t(x,y) is measured for different scanned points at the surface. Consider now the case of the shape reconstruction of a rotationally symmetric aspheric surface, given by the following formula [5]

$z=f(x,y)=c(x2+y2)1+1−c2(x2+y2)+∑s=1SAs+1(x2+y2)s+1,$
where $c=1/R$ is the surface vertex curvature, R is the vertex radius, and $Ai$ are aspheric coefficients. Using Eq. (5) and Eq. (22) we obtain values of the surface gradient. As an example we have chosen an aspheric surface has the vertex radius $R=−133.2126 mm$ (vertex curvature $c=−7,5067974⋅10-3$), diameter $D=20 mm$and aspheric coefficients $A2=2,6934596⋅10-7$, $A3=-4,0803344⋅10-10$. As the starting point for the optimization we choose a flat surface with the following coefficients: $c=0$, $A2=A3=A4=...=0$. We obtained by the solution of the Eq. (17) using the proposed method for the approximation coefficients: $c=−7,5067974⋅10-3$, $A2=2,6934596⋅10-7$, $A3=-4,0803344⋅10-10$. Figure 3 presents an approximation error of the shape using the coefficients obtained from the optimization procedure Broyden-Fletcher-Goldfarb-Shanno (BFGS) method [21,22]. We can see that the approximation error is very small, in the order of 10−13 mm and the proposed mathematical technique for the solution of Eq. (17) is working very well. It is evident from the previous example that a very good surface shape approximation can be found from an initial optimization point (approximation coefficients), which is really different from the optimal solution. We performed the analysis on several examples of spherical and aspheric surfaces with similar results, which prove the robustness of the described method. The proposed measurement method and the approach to find the solution of the partial differential Eq. (17), which describes theoretically the problem of surface reconstruction, give very good results and can be principally used for measurements of flat, spherical and aspheric surfaces.

Fig. 2 Principal scheme of measuring sensor.

Fig. 3 Approximation error of shape of reconstructed aspheric surface

We also performed an analysis of the influence of random errors in coordinates x,y using a computer simulation, where we considered the same uncertainty in both coordinates. The root-mean-square (RMS) of approximation errors for the given aspheric surface example is 0.1 μm for the uncertainty 3 μm, 0.4 μm for the uncertainty 10 μm, and 0.75 μm for the uncertainty 20 μm. These results prove the robustness of the proposed method.

## 4. Conclusion

The previous theoretical analysis had shown that a general solution of the 3D shape reconstruction of the optical surface is given by a nonlinear partial differential equation of the first order, which is a completely original mathematical approach in surface topography. The shape of the measured surface can be numerically calculated from the derived equation. We performed a numerical simulation of the presented method, which confirmed a robustness of the reconstruction method and a good possibility for measurements of spherical and aspheric specular surfaces. To our knowledge such a mathematical method of reconstruction of the surface shape has not been published yet.

## Acknowledgment

This work has been supported by the Ministry of Industry and Trade of the Czech Republic the grant FR−TI2/074.

1. J. A. Bosch, Coordinate Measuring Machines and Systems (CRC Press, 1995).

2. D. J. Whitehouse, Handbook of Surface and Nanometrology (Institute of Physics Publishing, 2003).

3. T. Yoshizawa, Handbook of Optical Metrology: Principles and Applications (CRC Press, 2009).

4. F. M. Santoyo, Handbook of Optical Metrology (CRC Press, 2008).

5. D. Malacara, Optical Shop Testing (John Wiley & Sons, 2007).

6. J. Geng, “Structured-light 3D surface imaging: a tutorial,” Adv. Opt. Photon. 3(2), 128–160 (2011). [CrossRef]

7. R. Leach, Optical Measurement of Surface Topography (Springer, 2011).

8. A. Mikš, J. Novák, and P. Novák, “Theory of chromatic sensor for topography measurements,” Proc. SPIE 6609, 66090U (2007).

9. T. Bothe, W. S. Li, C. von Kopylow, and W. Juptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004). [CrossRef]

10. M. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004). [CrossRef]

11. Y. Tang, X. Su, Y. Liu, and H. Jing, “3D shape measurement of the aspheric mirror by advanced phase measuring deflectometry,” Opt. Express 16(19), 15090–15096 (2008). [CrossRef]   [PubMed]

12. M. Rosete-Aguilar and R. Díaz-Uribe, “Profile testing of spherical surfaces by laser deflectometry,” Appl. Opt. 32(25), 4690–4697 (1993). [CrossRef]   [PubMed]

13. W. D. van Amstel, S. M. Bäumer, and J. L. Horijon, “Optical figure testing by scanning deflectometry,” Proc. SPIE 3739, 283–290 (1999). [CrossRef]

14. I. Weingärtner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE 3782, 306–317 (1999). [CrossRef]

15. M. Schulz, G. Ehret, and A. Fitzenreiter, “Scanning deflectometric form measurement avoiding path-dependent angle measurement errors,” J. Eur. Opt. Soc. Rap. Publ. 5, 10026 (2010).

16. M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

17. A. Mikš and P. Novák, “Determination of unit normal vectors of aspherical surfaces given unit directional vectors of incoming and outgoing rays: comment,” J. Opt. Soc. Am. A 29(7), 1356–1357, discussion 1358 (2012). [CrossRef]   [PubMed]

18. K. Rektorys, Survey of Applicable Mathematics (M.I.T. Press, 1969).

19. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Courier Dover Publications, 2000).

20. E.Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen (B.G.Teubner GmbH, 1977).

21. L. E. Scales, Introduction to Non-linear Optimization (Springer-Verlag, 1985).

22. M. Aoki, Introduction to Optimization Techniques: Fundamentals and Applications of Nonlinear Programming (Maxmillian, 1971).

### References

• View by:
• |
• |
• |

1. J. A. Bosch, Coordinate Measuring Machines and Systems (CRC Press, 1995).
2. D. J. Whitehouse, Handbook of Surface and Nanometrology (Institute of Physics Publishing, 2003).
3. T. Yoshizawa, Handbook of Optical Metrology: Principles and Applications (CRC Press, 2009).
4. F. M. Santoyo, Handbook of Optical Metrology (CRC Press, 2008).
5. D. Malacara, Optical Shop Testing (John Wiley & Sons, 2007).
6. J. Geng, “Structured-light 3D surface imaging: a tutorial,” Adv. Opt. Photon.3(2), 128–160 (2011).
[CrossRef]
7. R. Leach, Optical Measurement of Surface Topography (Springer, 2011).
8. A. Mikš, J. Novák, and P. Novák, “Theory of chromatic sensor for topography measurements,” Proc. SPIE6609, 66090U (2007).
9. T. Bothe, W. S. Li, C. von Kopylow, and W. Juptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE5457, 411–422 (2004).
[CrossRef]
10. M. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE5457, 366–376 (2004).
[CrossRef]
11. Y. Tang, X. Su, Y. Liu, and H. Jing, “3D shape measurement of the aspheric mirror by advanced phase measuring deflectometry,” Opt. Express16(19), 15090–15096 (2008).
[CrossRef] [PubMed]
12. M. Rosete-Aguilar and R. Díaz-Uribe, “Profile testing of spherical surfaces by laser deflectometry,” Appl. Opt.32(25), 4690–4697 (1993).
[CrossRef] [PubMed]
13. W. D. van Amstel, S. M. Bäumer, and J. L. Horijon, “Optical figure testing by scanning deflectometry,” Proc. SPIE3739, 283–290 (1999).
[CrossRef]
14. I. Weingärtner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE3782, 306–317 (1999).
[CrossRef]
15. M. Schulz, G. Ehret, and A. Fitzenreiter, “Scanning deflectometric form measurement avoiding path-dependent angle measurement errors,” J. Eur. Opt. Soc. Rap. Publ.5, 10026 (2010).
16. M. Herzberger, Modern Geometrical Optics (Interscience, 1958).
17. A. Mikš and P. Novák, “Determination of unit normal vectors of aspherical surfaces given unit directional vectors of incoming and outgoing rays: comment,” J. Opt. Soc. Am. A29(7), 1356–1357, discussion 1358 (2012).
[CrossRef] [PubMed]
18. K. Rektorys, Survey of Applicable Mathematics (M.I.T. Press, 1969).
19. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Courier Dover Publications, 2000).
20. E.Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen (B.G.Teubner GmbH, 1977).
21. L. E. Scales, Introduction to Non-linear Optimization (Springer-Verlag, 1985).
22. M. Aoki, Introduction to Optimization Techniques: Fundamentals and Applications of Nonlinear Programming (Maxmillian, 1971).

#### 2010 (1)

M. Schulz, G. Ehret, and A. Fitzenreiter, “Scanning deflectometric form measurement avoiding path-dependent angle measurement errors,” J. Eur. Opt. Soc. Rap. Publ.5, 10026 (2010).

#### 2007 (1)

A. Mikš, J. Novák, and P. Novák, “Theory of chromatic sensor for topography measurements,” Proc. SPIE6609, 66090U (2007).

#### 2004 (2)

T. Bothe, W. S. Li, C. von Kopylow, and W. Juptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE5457, 411–422 (2004).
[CrossRef]

M. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE5457, 366–376 (2004).
[CrossRef]

#### 1999 (2)

W. D. van Amstel, S. M. Bäumer, and J. L. Horijon, “Optical figure testing by scanning deflectometry,” Proc. SPIE3739, 283–290 (1999).
[CrossRef]

I. Weingärtner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE3782, 306–317 (1999).
[CrossRef]

#### Bäumer, S. M.

W. D. van Amstel, S. M. Bäumer, and J. L. Horijon, “Optical figure testing by scanning deflectometry,” Proc. SPIE3739, 283–290 (1999).
[CrossRef]

#### Bothe, T.

T. Bothe, W. S. Li, C. von Kopylow, and W. Juptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE5457, 411–422 (2004).
[CrossRef]

#### Ehret, G.

M. Schulz, G. Ehret, and A. Fitzenreiter, “Scanning deflectometric form measurement avoiding path-dependent angle measurement errors,” J. Eur. Opt. Soc. Rap. Publ.5, 10026 (2010).

#### Elster, C.

I. Weingärtner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE3782, 306–317 (1999).
[CrossRef]

#### Fitzenreiter, A.

M. Schulz, G. Ehret, and A. Fitzenreiter, “Scanning deflectometric form measurement avoiding path-dependent angle measurement errors,” J. Eur. Opt. Soc. Rap. Publ.5, 10026 (2010).

#### Häusler, G.

M. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE5457, 366–376 (2004).
[CrossRef]

#### Horijon, J. L.

W. D. van Amstel, S. M. Bäumer, and J. L. Horijon, “Optical figure testing by scanning deflectometry,” Proc. SPIE3739, 283–290 (1999).
[CrossRef]

#### Juptner, W.

T. Bothe, W. S. Li, C. von Kopylow, and W. Juptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE5457, 411–422 (2004).
[CrossRef]

#### Kaminski, J.

M. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE5457, 366–376 (2004).
[CrossRef]

#### Knauer, M.

M. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE5457, 366–376 (2004).
[CrossRef]

#### Li, W. S.

T. Bothe, W. S. Li, C. von Kopylow, and W. Juptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE5457, 411–422 (2004).
[CrossRef]

#### Mikš, A.

A. Mikš, J. Novák, and P. Novák, “Theory of chromatic sensor for topography measurements,” Proc. SPIE6609, 66090U (2007).

#### Novák, J.

A. Mikš, J. Novák, and P. Novák, “Theory of chromatic sensor for topography measurements,” Proc. SPIE6609, 66090U (2007).

#### Novák, P.

A. Mikš, J. Novák, and P. Novák, “Theory of chromatic sensor for topography measurements,” Proc. SPIE6609, 66090U (2007).

#### Schulz, M.

M. Schulz, G. Ehret, and A. Fitzenreiter, “Scanning deflectometric form measurement avoiding path-dependent angle measurement errors,” J. Eur. Opt. Soc. Rap. Publ.5, 10026 (2010).

I. Weingärtner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE3782, 306–317 (1999).
[CrossRef]

#### van Amstel, W. D.

W. D. van Amstel, S. M. Bäumer, and J. L. Horijon, “Optical figure testing by scanning deflectometry,” Proc. SPIE3739, 283–290 (1999).
[CrossRef]

#### von Kopylow, C.

T. Bothe, W. S. Li, C. von Kopylow, and W. Juptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE5457, 411–422 (2004).
[CrossRef]

#### Weingärtner, I.

I. Weingärtner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE3782, 306–317 (1999).
[CrossRef]

#### J. Eur. Opt. Soc. Rap. Publ. (1)

M. Schulz, G. Ehret, and A. Fitzenreiter, “Scanning deflectometric form measurement avoiding path-dependent angle measurement errors,” J. Eur. Opt. Soc. Rap. Publ.5, 10026 (2010).

#### Proc. SPIE (5)

W. D. van Amstel, S. M. Bäumer, and J. L. Horijon, “Optical figure testing by scanning deflectometry,” Proc. SPIE3739, 283–290 (1999).
[CrossRef]

I. Weingärtner, M. Schulz, and C. Elster, “Novel scanning technique for ultra-precise measurement of topography,” Proc. SPIE3782, 306–317 (1999).
[CrossRef]

A. Mikš, J. Novák, and P. Novák, “Theory of chromatic sensor for topography measurements,” Proc. SPIE6609, 66090U (2007).

T. Bothe, W. S. Li, C. von Kopylow, and W. Juptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE5457, 411–422 (2004).
[CrossRef]

M. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE5457, 366–376 (2004).
[CrossRef]

#### Other (12)

R. Leach, Optical Measurement of Surface Topography (Springer, 2011).

J. A. Bosch, Coordinate Measuring Machines and Systems (CRC Press, 1995).

D. J. Whitehouse, Handbook of Surface and Nanometrology (Institute of Physics Publishing, 2003).

T. Yoshizawa, Handbook of Optical Metrology: Principles and Applications (CRC Press, 2009).

F. M. Santoyo, Handbook of Optical Metrology (CRC Press, 2008).

D. Malacara, Optical Shop Testing (John Wiley & Sons, 2007).

M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

K. Rektorys, Survey of Applicable Mathematics (M.I.T. Press, 1969).

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Courier Dover Publications, 2000).

E.Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen (B.G.Teubner GmbH, 1977).

L. E. Scales, Introduction to Non-linear Optimization (Springer-Verlag, 1985).

M. Aoki, Introduction to Optimization Techniques: Fundamentals and Applications of Nonlinear Programming (Maxmillian, 1971).

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### Figures (3)

Fig. 1

Fig. 2

Principal scheme of measuring sensor.

Fig. 3

Approximation error of shape of reconstructed aspheric surface

### Equations (22)

$s ′ =s−2n(sn)$
$cosε=−sn, ε ′ =ε.$
$n= s− s ′ 2[ 1−(s s ′ ) ] .$
$t PC ¯ = t a−z =tan2ε= 2tanε 1− tan 2 ε .$
$p= ∂z ∂x , q= ∂z ∂y .$
$n=[ − p N , − q N , 1 N ],$
$s=(0,0,−1).$
$cosε=−sn=1/N.$
$tanε=± 1− cos 2 ε cosε =± p 2 + q 2 ,$
$tan2ε=± 2 p 2 + q 2 1− p 2 − q 2 .$
$K= t a−z =± 2 p 2 + q 2 1− p 2 − q 2 .$
$K(1− p 2 − q 2 )=±2 p 2 + q 2 .$
$(1− p 2 − q 2 ) 2 − 4 K 2 ( p 2 + q 2 )=0.$
$( p 2 + q 2 ) 2 −2b( p 2 + q 2 )+1=0,$
$b=(1+2/ K 2 ).$
$p 2 + q 2 =b± b 2 −1 .$
$( ∂z ∂x ) 2 + ( ∂z ∂y ) 2 =F(x,y,z),$
$F(x,y,z)=b± b 2 −1 =1+2/ K 2 ±2 1+1/ K 2 /K.$
$z=f(x,y)= ∑ m=0 M ∑ n=0 N c mn x m y n .$
$M= ( ∂z ∂x ) 2 + ( ∂z ∂y ) 2 −F(x,y,z).$
$Φ= ∑ i=1 I ∑ k=1 K [ M ik ( c 00 , c 10 , c 01 , c 20 , c 11 , c 02 ,.....) ] 2 .$
$z=f(x,y)= c( x 2 + y 2 ) 1+ 1− c 2 ( x 2 + y 2 ) + ∑ s=1 S A s+1 ( x 2 + y 2 ) s+1 ,$