Abstract

We present an analytic solution for the irradiance at a point due to a polygonal Lambertian emitter with radiant exitance that varies with position according to a polynomial of arbitrary degree. This is a basic problem that arises naturally in radiative transfer and more specifically in global illumination, a subfield of computer graphics. Our solution is closed form except for a single nonalgebraic special function known as the Clausen integral. We begin by deriving several useful formulas for high-order tensor analogs of irradiance, which are natural generalizations of the radiation pressure tensor. We apply the resulting tensor formulas to linearly varying emitters, obtaining a solution that exhibits the general structure of higher-degree cases, including the dependence on the Clausen integral. We then generalize to higher-degree polynomials with a recurrence formula that combines solutions for lower-degree polynomials; the result is a generalization of Lambert’s formula for homogeneous diffuse emitters, a well-known formula with many applications in radiative transfer and computer graphics. Similar techniques have been used previously to derive closed-form solutions for the irradiance due to homogeneous polygonal emitters with directionally varying radiance. The present work extends this previous result to include inhomogeneous emitters, which proves to be significantly more challenging to solve in closed form. We verify our theoretical results with numerical approximations and briefly discuss their potential applications.

© 2003 Optical Society of America

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  1. A. F. Emery, O. Johansson, M. Lobo, A. Abrous, “A comparative study of methods for computing the diffuse radiation viewfactors for complex structures,” ASME J. Heat Transfer 113, 413–422 (1991).
    [CrossRef]
  2. E. M. Sparrow, “A new and simpler formulation for radiative angle factors,” ASME J. Heat Transfer 85, 81–88 (1963).
    [CrossRef]
  3. J. R. Howell, A Catalog of Radiation Configuration Factors (McGraw-Hill, New York, 1982).
  4. P. Schröder, P. Hanrahan, “On the form factor between two polygons,” in SIGGRAPH 93, Computer Graphics Proceedings, Annual Conference Series (Association of Computing Machinery, New York, 1993), pp. 163–164.
  5. D. L. DiLaura, “Non-diffuse radiative transfer 3: inhomogeneous planar area sources and point receivers,” J. Illum. Eng. Soc. 26, 182–187 (1997).
    [CrossRef]
  6. J. Arvo, “Applications of irradiance tensors to the simulation of non-Lambertian phenomena,” in SIGGRAPH 95, Computer Graphics Proceedings, Annual Conference Series (Association of Computing Machinery, New York, 1995), pp. 335–342.
  7. Bui Tuong Phong, “Illumination for computer generated pictures,” Commun. ACM 18, 311–317 (1975).
    [CrossRef]
  8. Leonard Lewin, Dilogarithms and Associated Functions (MacDonald, London, 1958).
  9. G. C. Pomraning, The Equations of Radiation Hydrodynamics (Pergamon, New York, 1973).
  10. M. Spivak, Calculus on Manifolds (Benjamin/Cummings, Reading, Mass., 1965).
  11. P. Moon, The Scientific Basis of Illuminating Engineering (Dover, New York, 1961).
  12. M. Schreiber, Differential Forms: A Heuristic Introduction (Springer-Verlag, New York, 1984).
  13. M. Berger, Geometry, (Springer-Verlag, New York, 1987) Vol. 2. (Translated by M. Cole, S. Levy).
  14. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, New York, 1994).
  15. M. Chen, “Mathematical methods for image synthesis,” Ph.D. thesis (California Institute of Technology, Pasadena, Calif., 2001); http://www.cs.caltech.edu/∼chen/papers/thesis/phd_thesis.ps.gz .
  16. C. C. Grosjean, “Formulae concerning the computation ofthe Clausen integral cl2(θ),” J. Comput. Appl. Math. 11, 331–342 (1984).
    [CrossRef]
  17. J. Arvo, “Analytic methods for simulated light transport,” Ph.D. thesis (Yale University, New Haven, Conn., 1995).
  18. M. Chen, J. Arvo, “A closed-form solution for the irradiance due to linearly-varying luminaries,” in Rendering Techniques 2000, B. Péroche, H. Rushmeier, eds. (Springer-Verlag, New York, 2000), pp. 137–148.
  19. M. Chen, J. Arvo, “Simulating non-Lambertian phenomena involving linearly-varying luminaries,” in Rendering Techniques 2001, S. Gortler, K. Myszkowski, eds. (Springer-Verlag, New York, 2001), pp. 25–38.

1997

D. L. DiLaura, “Non-diffuse radiative transfer 3: inhomogeneous planar area sources and point receivers,” J. Illum. Eng. Soc. 26, 182–187 (1997).
[CrossRef]

1991

A. F. Emery, O. Johansson, M. Lobo, A. Abrous, “A comparative study of methods for computing the diffuse radiation viewfactors for complex structures,” ASME J. Heat Transfer 113, 413–422 (1991).
[CrossRef]

1984

C. C. Grosjean, “Formulae concerning the computation ofthe Clausen integral cl2(θ),” J. Comput. Appl. Math. 11, 331–342 (1984).
[CrossRef]

1975

Bui Tuong Phong, “Illumination for computer generated pictures,” Commun. ACM 18, 311–317 (1975).
[CrossRef]

1963

E. M. Sparrow, “A new and simpler formulation for radiative angle factors,” ASME J. Heat Transfer 85, 81–88 (1963).
[CrossRef]

Abrous, A.

A. F. Emery, O. Johansson, M. Lobo, A. Abrous, “A comparative study of methods for computing the diffuse radiation viewfactors for complex structures,” ASME J. Heat Transfer 113, 413–422 (1991).
[CrossRef]

Arvo, J.

J. Arvo, “Applications of irradiance tensors to the simulation of non-Lambertian phenomena,” in SIGGRAPH 95, Computer Graphics Proceedings, Annual Conference Series (Association of Computing Machinery, New York, 1995), pp. 335–342.

J. Arvo, “Analytic methods for simulated light transport,” Ph.D. thesis (Yale University, New Haven, Conn., 1995).

M. Chen, J. Arvo, “A closed-form solution for the irradiance due to linearly-varying luminaries,” in Rendering Techniques 2000, B. Péroche, H. Rushmeier, eds. (Springer-Verlag, New York, 2000), pp. 137–148.

M. Chen, J. Arvo, “Simulating non-Lambertian phenomena involving linearly-varying luminaries,” in Rendering Techniques 2001, S. Gortler, K. Myszkowski, eds. (Springer-Verlag, New York, 2001), pp. 25–38.

Berger, M.

M. Berger, Geometry, (Springer-Verlag, New York, 1987) Vol. 2. (Translated by M. Cole, S. Levy).

Chen, M.

M. Chen, J. Arvo, “Simulating non-Lambertian phenomena involving linearly-varying luminaries,” in Rendering Techniques 2001, S. Gortler, K. Myszkowski, eds. (Springer-Verlag, New York, 2001), pp. 25–38.

M. Chen, J. Arvo, “A closed-form solution for the irradiance due to linearly-varying luminaries,” in Rendering Techniques 2000, B. Péroche, H. Rushmeier, eds. (Springer-Verlag, New York, 2000), pp. 137–148.

DiLaura, D. L.

D. L. DiLaura, “Non-diffuse radiative transfer 3: inhomogeneous planar area sources and point receivers,” J. Illum. Eng. Soc. 26, 182–187 (1997).
[CrossRef]

Emery, A. F.

A. F. Emery, O. Johansson, M. Lobo, A. Abrous, “A comparative study of methods for computing the diffuse radiation viewfactors for complex structures,” ASME J. Heat Transfer 113, 413–422 (1991).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, New York, 1994).

Grosjean, C. C.

C. C. Grosjean, “Formulae concerning the computation ofthe Clausen integral cl2(θ),” J. Comput. Appl. Math. 11, 331–342 (1984).
[CrossRef]

Hanrahan, P.

P. Schröder, P. Hanrahan, “On the form factor between two polygons,” in SIGGRAPH 93, Computer Graphics Proceedings, Annual Conference Series (Association of Computing Machinery, New York, 1993), pp. 163–164.

Howell, J. R.

J. R. Howell, A Catalog of Radiation Configuration Factors (McGraw-Hill, New York, 1982).

Johansson, O.

A. F. Emery, O. Johansson, M. Lobo, A. Abrous, “A comparative study of methods for computing the diffuse radiation viewfactors for complex structures,” ASME J. Heat Transfer 113, 413–422 (1991).
[CrossRef]

Lewin, Leonard

Leonard Lewin, Dilogarithms and Associated Functions (MacDonald, London, 1958).

Lobo, M.

A. F. Emery, O. Johansson, M. Lobo, A. Abrous, “A comparative study of methods for computing the diffuse radiation viewfactors for complex structures,” ASME J. Heat Transfer 113, 413–422 (1991).
[CrossRef]

Moon, P.

P. Moon, The Scientific Basis of Illuminating Engineering (Dover, New York, 1961).

Phong, Bui Tuong

Bui Tuong Phong, “Illumination for computer generated pictures,” Commun. ACM 18, 311–317 (1975).
[CrossRef]

Pomraning, G. C.

G. C. Pomraning, The Equations of Radiation Hydrodynamics (Pergamon, New York, 1973).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, New York, 1994).

Schreiber, M.

M. Schreiber, Differential Forms: A Heuristic Introduction (Springer-Verlag, New York, 1984).

Schröder, P.

P. Schröder, P. Hanrahan, “On the form factor between two polygons,” in SIGGRAPH 93, Computer Graphics Proceedings, Annual Conference Series (Association of Computing Machinery, New York, 1993), pp. 163–164.

Sparrow, E. M.

E. M. Sparrow, “A new and simpler formulation for radiative angle factors,” ASME J. Heat Transfer 85, 81–88 (1963).
[CrossRef]

Spivak, M.

M. Spivak, Calculus on Manifolds (Benjamin/Cummings, Reading, Mass., 1965).

ASME J. Heat Transfer

A. F. Emery, O. Johansson, M. Lobo, A. Abrous, “A comparative study of methods for computing the diffuse radiation viewfactors for complex structures,” ASME J. Heat Transfer 113, 413–422 (1991).
[CrossRef]

E. M. Sparrow, “A new and simpler formulation for radiative angle factors,” ASME J. Heat Transfer 85, 81–88 (1963).
[CrossRef]

Commun. ACM

Bui Tuong Phong, “Illumination for computer generated pictures,” Commun. ACM 18, 311–317 (1975).
[CrossRef]

J. Comput. Appl. Math.

C. C. Grosjean, “Formulae concerning the computation ofthe Clausen integral cl2(θ),” J. Comput. Appl. Math. 11, 331–342 (1984).
[CrossRef]

J. Illum. Eng. Soc.

D. L. DiLaura, “Non-diffuse radiative transfer 3: inhomogeneous planar area sources and point receivers,” J. Illum. Eng. Soc. 26, 182–187 (1997).
[CrossRef]

Other

J. Arvo, “Applications of irradiance tensors to the simulation of non-Lambertian phenomena,” in SIGGRAPH 95, Computer Graphics Proceedings, Annual Conference Series (Association of Computing Machinery, New York, 1995), pp. 335–342.

J. Arvo, “Analytic methods for simulated light transport,” Ph.D. thesis (Yale University, New Haven, Conn., 1995).

M. Chen, J. Arvo, “A closed-form solution for the irradiance due to linearly-varying luminaries,” in Rendering Techniques 2000, B. Péroche, H. Rushmeier, eds. (Springer-Verlag, New York, 2000), pp. 137–148.

M. Chen, J. Arvo, “Simulating non-Lambertian phenomena involving linearly-varying luminaries,” in Rendering Techniques 2001, S. Gortler, K. Myszkowski, eds. (Springer-Verlag, New York, 2001), pp. 25–38.

J. R. Howell, A Catalog of Radiation Configuration Factors (McGraw-Hill, New York, 1982).

P. Schröder, P. Hanrahan, “On the form factor between two polygons,” in SIGGRAPH 93, Computer Graphics Proceedings, Annual Conference Series (Association of Computing Machinery, New York, 1993), pp. 163–164.

Leonard Lewin, Dilogarithms and Associated Functions (MacDonald, London, 1958).

G. C. Pomraning, The Equations of Radiation Hydrodynamics (Pergamon, New York, 1973).

M. Spivak, Calculus on Manifolds (Benjamin/Cummings, Reading, Mass., 1965).

P. Moon, The Scientific Basis of Illuminating Engineering (Dover, New York, 1961).

M. Schreiber, Differential Forms: A Heuristic Introduction (Springer-Verlag, New York, 1984).

M. Berger, Geometry, (Springer-Verlag, New York, 1987) Vol. 2. (Translated by M. Cole, S. Levy).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, New York, 1994).

M. Chen, “Mathematical methods for image synthesis,” Ph.D. thesis (California Institute of Technology, Pasadena, Calif., 2001); http://www.cs.caltech.edu/∼chen/papers/thesis/phd_thesis.ps.gz .

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Figures (7)

Fig. 1
Fig. 1

Computing the irradiance at the origin due to a polygonal emitter by integrating over its spherical projection on the hemisphere above the origin.

Fig. 2
Fig. 2

(a) The solid angle of a spherical triangle is easily obtained from Girard’s formula. (b) The solid angle of nonconvex polygons can be computed by generalizing Girard’s formula.

Fig. 3
Fig. 3

A floor is illuminated by a square emitter with a linear radiant exitance distribution specified by three points p 1 , p 2 , and p 3 and their associate radiant excitant values shown beside. h is the distance between the floor and the light source. (a) distribution B ( x ) = y + z / 2 , (b) distribution B ( x ) = ( x - y + z ) / 2 .

Fig. 4
Fig. 4

Comparison of our closed-form solutions with Monte Carlo simulations by varying the number of stratified samples. (a) B ( x ) = y + z / 2 , (b) B ( x ) = ( x - y + z ) / 2 .

Fig. 5
Fig. 5

Comparison of our closed-form solutions with finite-element approximations by varying subdivision levels. Bottom plots show the relative errors of finite-element solutions. (a) B ( x ) = y + z / 2 , (b) B ( x ) = ( x - y + z ) / 2 .

Fig. 6
Fig. 6

Quadratic radiant exitance distribution B ( x ) = y 2 . (a) Comparison of our closed-form solutions with finite-element approximations. (b) Comparison of our closed-form solutions with Monte Carlo simulations.

Fig. 7
Fig. 7

Quadratic radiant exitance distribution B ( x ) = x 2 + y 2 + z 2 . (a) Comparison of our closed-form solutions with finite-element approximations using 8, 50, and 128 patches. (b) Comparison of our closed-form solutions with Monte Carlo simulations using 49, 100, and 400 stratified samples.

Equations (143)

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E = Ω L ( u ) b ,   u d σ ( u ) ,
L ( u ) = 1 π   B h u w ,   u ,
E = 1 π A B h u w ,   u b ,   u d σ ( u ) .
B ( x ) = [ w 1 w 2 w 3 ] [ p 1 p 2 p 3 ] - 1 x = s a ,   x ,
E 1 ( A ,   a ,   b ,   w ) = sh π A a ,   u b ,   u w ,   u d σ ( u ) .
B ( x ) = x p y q z r = x ,   e 1 p x ,   e 2 q x ,   e 3 r ,
E n ( A ,   p ,   q ,   r ,   b ,   w )
= h n π A e 1 ,   u p e 2 ,   u q e 3 ,   u r b ,   u w ,   u n d σ ( u ) ,
τ n ; q ( A ,   v 1 , ,   v n ;   w ) A v 1 ,   u v n ,   u w ,   u q d σ ( u ) ,
{ v 1 , , v n + 1 } = { e 1 , , e 1 p , e 2 , , e 2 q , e 3 , , e 3 r , b } .
T n ; q ( A ,   w ) A u u w ,   u q d σ ( u ) ,
τ n ; q ( A ,   v 1 , ,   v n ;   w ) = T I n ; q ( A ,   w ) ( v 1 v n ) I ,
Ψ 1 c S 2 uu T L ( u ) d σ ( u ) ,
A ω = A d ω ,
I \ k = ( i 1 , ,   i k - 1 ,   i k + 1 , ,   i n ) ,
I j = ( i 1 ,   i 2 , ,   i n , j ) .
T I n ; q ( A ,   w )
= 1 q - 1 k = 1 n w I k T I \ k n - 1 ; q - 1 - ( n - q + 3 ) T I n ; q - 2
- A u I n w ,   n w ,   u q - 1 d s ] ,
T Ij n ; 1 ( A ,   w )
= w j T I n - 1 + 1 n   ( δ jm - w j w m ) × k = 1 n - 1 δ mI k T I \ k n - 2 ; 1 - A u I n - 1 n m w , u d s ,
T Ij n ( A ) = 1 n + 1 k = 1 n - 1 δ jI k T I \ k n - 2 ( A ) - A u I n - 1 n j d s ,
T 0 ; 1 ( A ,   w ) = A ln w ,   u 1 - w ,   u 2   w ,   n d s ,
τ n ; q ( A ,   v J ; w )
= 1 q - 1 k = 1 n w ,   v k τ n - 1 ; q - 1 ( A ,   v J \ k ;   w ) - ( n - q + 3 ) τ n ; q - 2 ( A ,   v J ;   w ) - A v 1 ,   u v n ,   u w ,   u q - 1   w ,   n d s ,
τ n ; 1 ( A ,   v J ;   w )
= w ,   v n τ n - 1 ( A ,   v J \ n ) + v n T ( I - ww T ) n × k = 1 n - 1 v k τ n - 2 ; 1 ( A ,   v J \ { k , n } ;   w ) - A v 1 ,   u v n - 1 ,   u w ,   u n d s ,
τ n ( A ,   v J ) = 1 n + 1 k = 1 n - 1 v n ,   v k τ n - 2 ( A ,   v J \ { k , n } ) - A v 1 ,   u v n - 1 ,   u v n ,   n d s ,
σ ( Δ ABC ) = α + β + γ - π ,
σ ( A ) = i = 1 n θ i - ( n - 2 ) π ,
C * = ζ ln w ,   u 1 - w ,   u 2 d s ,
C m , l ( v 1 , ,   v m ) = ζ v 1 ,   u v m ,   u w ,   u l d s .
u ( θ ) = s cos   θ + t sin   θ ,
a i = v i ,   s , b i = v i ,   t , c i = a i 2 + b i 2 ,
a = w ,   s ,   b = w ,   t , c = a 2 + b 2 ,
( cos   ϕ i ,   sin   ϕ i ) = ( a i / c i ,   b i / c i ) ,
( cos   ϕ ,   sin   ϕ ) = ( a / c ,   b / c ) ,
C * = - ϕ Θ - ϕ ln ( c   cos   θ ) 1 - c 2 cos 2   θ d θ = Λ ( c ,   Θ - ϕ ) - Λ ( c ,   - ϕ ) ,
Λ ( α ,   β ) 0 β ln ( α   cos   θ ) 1 - ( α   cos   θ ) 2 d θ
Cl 2 ( x ) - 0 x ln 2   sin θ 2 d θ
Υ ( μ ,   ν ) 0 μ ( 1 + ν 2 sec 2   θ ) d θ ,
Λ ( α ,   β ) = - 1 2 1 - α 2   Υ tan - 1 tan   β 1 - α 2 ,   1 - α 2 α .
Λ ( α ,   β ) = 1 4 1 - α 2   [ 2 ( η - μ ) ln   γ + 2 Cl 2 ( 2 μ ) - Cl 2 ( 4 μ - 2 η ) - Cl 2 ( 2 η ) ] ,
μ ( α ,   β ) = tan - 1 tan   β 1 - α 2 ,
γ ( α ) = 1 - 1 - α 2 α 2 ,
η ( α ,   β ) = tan - 1 sin ( 2 μ ) γ + cos ( 2 μ ) .
w ,   u = a   cos   θ + b   sin   θ = c   cos ( θ - ϕ ) .
C m , l = c 1 c m c l - ϕ Θ - ϕ i = 1 m cos ( θ + ϕ - ϕ i ) cos l   θ d θ = c 1 c m c l - ϕ Θ - ϕ i = 1 m ( α i cos   θ + β i sin   θ ) cos l   θ d θ = c 1 c m β 1 β m c l - ϕ Θ - ϕ P m ( tan   θ ) cos m - l θ   d θ ,
G ( r ,   s ,   x ,   y ) x y sin r   θ   cos s   θ   d θ ,
C m , l = c 1 c m β 1 β m c l × i = 0 m p i G ( i ,   m - l - i ,   - ϕ ,   Θ - ϕ ) .
G ( r ,   s ,   x ,   y )
= 1 r + s   [ sin r + 1   y   cos s - 1   y - sin r + 1   x   cos s - 1   x + ( s - 1 ) G ( r ,   s - 2 ,   x ,   y ) ] ,
G ( r ,   s ,   x ,   y )
= 1 s + 1   [ sin r + 1   x   cos s + 1   x - sin r + 1   y   cos s + 1   y + ( r + s + 2 ) G ( r ,   s + 2 ,   x ,   y ) ] ,
G ( r ,   s ,   x ,   y )
= 1 r + s   [ sin r - 1   x   cos s + 1   x - sin r - 1   y   cos s + 1   y + ( r - 1 ) G ( r - 2 ,   s ,   x ,   y ) ] ,
G ( 0 ,   - 1 ,   x ,   y ) = ln tan ( π / 4 + y / 2 ) tan ( π / 4 + x / 2 ) ,
G ( 1 ,   0 ,   x ,   y ) = cos   x - cos   y ,
G ( 1 ,   - 1 ,   x ,   y ) = ln cos   x cos   y ,
G ( 0 ,   0 ,   x ,   y ) = y - x .
E 1 ( A ,   a ,   b ,   w ) = sh π   τ 2 ; 1 ( A ,   a ,   b ;   w ) = sh π T ij 2 ; 1 ( A ,   w ) a i b j .
T ij 2 , 1 = w j T i 1 + 1 2   ( δ jm - w j w m ) × δ im T 0 , 1 - A u i n m w ,   u d s = - 1 2 A δ ik w j - ( δ jm - w j w m ) × δ im w k η - δ km u i w ,   u n k d s ,
η ( w ,   u ) ln w ,   u 1 - w ,   u 2 .
M ijk 3 ( w ,   u ) = δ ik w j + ( δ jm - w j w m ) × δ km u i w ,   u - δ im w k η ,
E 1 ( A ,   a ,   b ,   w ) = - l 2 π A M ijk 3 a i b j n k d s ,
E 1 ( A ,   a ,   b ,   w ) = - l 2 π i = 1 k { a ,   n b ,   w Θ + b T ( I - ww T ) [ C 1 , 1 ( a ) n - C * w ,   n a ] } ,
C 1 , 1 ( a ) = ζ i a ,   u w ,   u d s .
C 1 , 1 ( a ) = c 1 c cos ( ϕ - ϕ 1 ) Θ + sin ( ϕ - ϕ 1 ) ln cos ( Θ - ϕ ) cos   ϕ ,
B ( x ) = x p y q z r = x ,   e 1 p x ,   e 2 q x ,   e 3 r ,
V ( e 1 , , e 1 p , e 2 , , e 2 q , e 3 , , e 3 r ) ,
E 0 ( A ,   b ) = c π A b ,   u d σ ( u ) = c π T j 1 ; 0 ( A ,   w ) b j ,
E 0 ( A ,   b ) = - l 2 π A δ jk b j n k d s ,
E 1 ( A ,   a ,   b ,   w ) = - l 2 π A M ijk 3 ( w ,   u ) a i b j n k d s .
E n ( A ,   V ,   b ,   w ) = - l 2 π A M Ijk n + 2 ( w ,   u ) V I b j n k d s ,
T Ij n + 1 ; n ( A ,   w ) = - 1 2 A M Ijk n + 2 ( w ,   u ) n k d s
M Ijk n + 2 = 1 n - 1 i = 1 n [ w I i M ( I \ i ) jk n + 1 - δ jI i M I \ i k n ] + ( n - 1 ) w j M Ik n + 1 + 2 w k u Ij n + 1 w ,   u n - 1 - 2 δ jk u I n w ,   u n - 2 ,
M ijk 3 = δ ik w j + ( δ jm - w j w m ) δ km u i w ,   u - δ im w k η ,
M jk 2 = δ jk .
E n ( A ,   V ,   b ,   w ) = h n π   τ n + 1 ; n ( A ,   V ,   b ,   w ) = h n π T Ij n + 1 ; n ( A ,   w ) V I b j ,
E n ( A ,   V ,   b ,   w ) = - l 2 π i = 1 m ζ i M Ijk n + 2 V I b j d s n k .
B ( x ) = x 2 + y 2 + z 2 = x ,   x .
E 2 = h 2 π A b ,   u w ,   u 2 d σ ( u ) = h 2 π   τ 1 ; 2 ( A ,   b ;   w ) ,
E 2 = h 2 π   τ 0 ; 1 ( A ,   w ) ,
( q - 1 ) T I n ; q - k = 1 n w I k T I \ k n - 1 ; q - 1 + ( n - q + 3 ) T I n ; q - 2
= - A u I n w ,   n w ,   u q - 1 d s .
A u I n w ,   n w ,   u q - 1 d s = A r q - 1 w ,   r q - 1 r I n r n kpl w k r p d r l r 2 = A D Il n + 1 d r l ,
D Il , m = kpl w k r m 1 w ,   r q - 1 r p r I n r n - q + 3 + 1 w ,   r q - 1 r m r p r I n r n - q + 3 ,
r m 1 w ,   r q - 1 = - ( q - 1 ) w m w ,   r q ,
r m r p r I n r n - q + 3 = δ pm r I n + r p r I , m n r n - q + 3 - ( n - q + 3 )   r m r p r I n r n - q + 5 .
A D Il d r l = A D Il , m d r m d r 1 = A D Il , m d r m d r l - d r l d r m 2 = A qml D Il , m qst d r s d r t 2 ,
qml kpl w k r m 1 w ,   r q - 1   r p r I n r n - q + 3
= ( q - 1 )   r I n w ,   r q r n - q r q r 3 - ( q - 1 )   w q r I n w ,   r q - 1 r n - q + 3 ,
qml kpl w k 1 w ,   r q - 1 r m r p r I n r n - q + 3
= ( q - 1 )   w q r I n w ,   r q - 1 r n - q + 3 + ( n - q + 3 ) × r I n w ,   r q - 2 r n - q + 2 - w m r I , m n w ,   r q - 1 r n - q r q r 3 ,
r m r I , m n = r m k = 1 n δ mI k r I \ k n - 1 = k = 1 n r I k r I \ k n - 1 = n r I n .
A D Il d r l = - A ( q - 1 )   r I n w ,   r q r n - q - w m r I , m n w , r q - 1 r n - q + ( n - q + 3 )   r I n w ,   r q - 2 r n - q + 2 d ω ,
d ω - qst r q d r s d r t 2 r 3 .
A D Il n + 1 d r l = - ( q - 1 ) T I n ; q - ( n - q + 3 ) T I n ; q - 2 + k = 1 n w I k T I \ k n - 1 ; q - 1 ,
A u I n - 1 n m w ,   u d s = A D Iml d r l = A qsl D Iml , s qht d r h d r t 2 ,
D Iml , s = mpl r s 1 w ,   r r p r I n - 1 r n + 1 w ,   r r s r p r I n - 1 r n = mpl ( A 1 + A 2 ) ,
A 1 = - w s r p r I n - 1 w ,   r 2 r n ,
A 2 = δ ps r I n - 1 + r p r I , s n - 1 w ,   r r n - n   r s r p r I n - 1 w ,   r r n + 2 .
qsl mpl A 1 = r I n - 1 w ,   r 2 r n   ( δ qm w ,   r - w m r q ) ,
qsl mpl A 2 = 1 w ,   r δ qm r I n - 1 r n - r q r I , m n - 1 r n + n   r q r m r I n - 1 r n + 2 .
A D Iml d r l = - A w m r I n - 1 w ,   r 2 r n - 3 - k = 1 n - 1 δ mI k r I \ k n - 2 w ,   r r n - 3 + n   r m r I n - 1 w ,   r r n - 1 d ω .
1 n + 1 A u I n - 1 n j d s = A D Ijl d r l
= A kml D Ijl , m kst d r s d r t 2 ,
D Ijl , m = jpl δ pm r I n - 1 + r p r I , m n - 1 ( n + 1 ) r n + 1 - r m r p r 1 n - 1 r n + 3 .
kml D Ijl , m = r k r I , j n - 1 ( n + 1 ) r n + 1 - r j r k r I n - 1 r n + 3 .
A ln w ,   u 1 - w ,   u 2   w ,   n d s = A D l d r l = Λ qml D l , m qst d r s d r t 2 ,
D l = kpl w k ln w ,   u 1 - w ,   u 2     r p r 2 = kpl w k η   r p r 2 ,
D l , m = kpl w k η   r m r p r 2 + η m r p r 2 .
r m r p r 2 = δ pm r 2 - 2 r p r m r 4 ,
η m = η r m = r 2 - w ,   r 2 + 2 w ,   r 2 ln w ,   u ( r 2 - w ,   r 2 ) 2 × r 2 w m w ,   r - r m .
qml kpl w k η   r m r p r 2 = r q r 3   [ 2 η w ,   u ] ,
qml kpl w k η m r p r 2 = - r q r 3 1 w ,   u + 2 η w ,   u .
Λ ( α ,   β ) = - 1 2 1 - α 2   Υ tan - 1 tan   β 1 - α 2 ,   1 - α 2 α ,
t = tan - 1 tan   θ 1 - α 2 , d θ = 1 - α 2 1 - α 2 sin 2   t d t ,
cos 2   θ = cos 2   t 1 - α 2 sin 2   t .
Λ ( α ,   β )
= 1 2 0 β ln ( α 2 cos 2   θ ) 1 - α 2 cos 2   θ d θ = - 1 2 1 - α 2 × 0 tan - 1 ( tan   β / 1 - α 2 ) ln 1 - α 2 sin 2   t α 2 cos 2   t d t = - 1 2 1 - α 2 × 0 tan - 1 ( tan   β / 1 - α 2 ) ln 1 + 1 - α 2 α 2 sec 2   t d t .
Υ ( μ ,   ν ) = - 1 2 Cl 2 ( 4 μ ) - Cl 2 ( 4 μ - 2 η ) - Cl 2 ( 2 η ) + 2 Cl 2 ( π - 2 μ ) - 2 μ   ln ( 4 ν 2 + 2 ) sin 2 a 2 + η   ln tan 2 a 2 ,
a = sin - 1 1 2 ν 2 + 1 ,
η = tan - 1 sin ( 2 μ ) tan   ( a / 2 ) + cos ( 2 μ ) .
tan a 2 = γ , η = tan - 1 sin ( 2 μ ) γ + cos ( 2 μ ) ,
sin 2 a 2 = γ 4 ν 2 + 2 ,
Υ ( μ ,   ν ) = - 1 2   [ Cl 2 ( 4 μ ) - Cl 2 ( 4 μ - 2 η ) - Cl 2 ( 2 η ) + 2 Cl 2 ( π - 2 μ ) + 2 ( η - μ ) ln   γ ] ,
Cl 2 ( 2 θ ) = 2 [ Cl 2 ( θ ) - Cl 2 ( π - θ ) ] ,
Υ ( μ ,   ν ) = - [ 2 Cl 2 ( 2 μ ) - Cl 2 ( 4 μ - 2 η ) - Cl 2 ( 2 η ) + 2 ( η - μ ) ln   γ ] / 2 .
T Ij n ; q = 1 n - q + 1 k = 1 n - 1 δ jI k T I \ k n - 2 ; q - q w j T I n - 1 ; q + 1 - A u I n - 1 n j w ,   u q d s ,
A u I n - 1 n j w ,   u q d s = A D Ijl d r l = A qml D Ijl , m qst d r s d r t 2 ,
D Ijl , m = jpl r m 1 w ,   r q r p r I n - 1 r n - q + 1 + 1 w ,   r q r m r p r I n - 1 r n - q + 1 = jpl ( A 1 + A 2 ) .
A 1 = - q   w m r p r I n - 1 w ,   r q + 1 r n - q + 1 ,
A 2 = 1 w ,   r q δ pm r I n - 1 + r p r I , m n - 1 r n - q + 1 - ( n - q + 1 )   r m r p r I n - 1 r n - q + 3 .
qml jpl A 1 = q w j r I n - 1 w ,   r q + 1 r n - q - 2 r q r 3 - q δ qj r I n - 1 w ,   r q r n - q + 1 ,
qml jpl A 2 = q δ qj r I n - 1 w ,   r q + ( n - q + 1 )   r j r I n - 1 w ,   r q r n - q - r I , j n - 1 w ,   r q r n - q - 2 r q r 3 .
A D Ijl d r l = A - q w j r I n - 1 w ,   r q + 1 r n - q - 2 - ( n - q + 1 )   r I n - 1 r j w ,   r q r n - q + k = 1 n - 1 δ jI k r I \ k n - 2 w ,   r q r n - q - 2 d ω = - q w j T I n - 1 ; q + 1 - ( n - q + 1 ) T Ij n ; q + k = 1 n - 1 δ jI k T I \ k n - 2 ; q .
T j 1 ; 0 = - 1 2 A n j d s = - 1 2 A δ jk n k d s .
T Rj k + 1 ; k = - 1 2 A M Rjk k + 2 n k d s
T Ij n + 1 ; n = 1 n - 1 i = 1 n w I i T ( I \ i ) j n ; n - 1 + w j T I n ; n - 1 - 4 T Ij n + 1 ; n - 2 - A u Ij n + 1 w ,   n w ,   u n - 1 d s .
4 T Ij n + 1 ; n - 2 = i = 1 n δ jI i T I \ i n - 1 ; n - 2 - ( n - 2 ) w j T I n ; n - 1 - A u I n n j w ,   u n - 2 d s .
T Ij n + 1 ; n = 1 n - 1 i = 1 n w I i T ( I \ i ) j n ; n - 1 + ( n - 1 ) w j T I n ; n - 1 - i = 1 n δ jI i T I \ i n - 1 ; n - 2 + A δ jk u I n w ,   u n - 2 - w k u Ij n + 1 w ,   u n - 1 n k d s .
T Ij n + 1 ; n = - 1 2 A 1 n - 1 i = 1 n w I i M ( I \ i ) jk n + 1 + ( n - 1 ) w j M Ik n + 1 - i = 1 n δ jI i M ( I \ i ) k n + 2 w k u Ij n + 1 w ,   u n - 1 - 2 δ jk u I n w ,   u n - 2 n k d s .

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