## Abstract

Finding the relationship between two coordinate systems using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task. It finds applications in stereophotogrammetry and in robotics. I present here a closed-form solution to the least-squares problem for three or more points. Currently various empirical, graphical, and numerical iterative methods are in use. Derivation of the solution is simplified by use of unit quaternions to represent rotation. I emphasize a symmetry property that a solution to this problem ought to possess. The best translational offset is the difference between the centroid of the coordinates in one system and the rotated and scaled centroid of the coordinates in the other system. The best scale is equal to the ratio of the root-mean-square deviations of the coordinates in the two systems from their respective centroids. These exact results are to be preferred to approximate methods based on measurements of a few selected points. The unit quaternion representing the best rotation is the eigenvector associated with the most positive eigenvalue of a symmetric 4 × 4 matrix. The elements of this matrix are combinations of sums of products of corresponding coordinates of the points.

© 1987 Optical Society of America

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### References

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1. C. C. Slama, C. Theurer, S. W. Henrikson, eds., Manual of Photogrammetry (American Society of Photogrammetry, Falls Church, Va., 1980).
2. B. K. P. Horn, Robot Vision (MIT/McGraw-Hill, New York, 1986).
3. P. R. Wolf, Elements of Photogrammetry (McGraw Hill, New York, 1974).
4. S. K. Gosh, Theory of Stereophotogrammetry (Ohio U. Bookstores, Columbus, Ohio, 1972).
5. E. H. Thompson, “A method for the construction of orthogonal matrices,” Photogramm. Record 3, 55–59 (1958).
[Crossref]
6. G. H. Schut, “On exact linear equations for the computation of the rotational elements of absolute orientation,” Photogrammetria 16, 34–37 (1960).
7. H. L. Oswal, S. Balasubramanian, “An exact solution of absolute orientation,” Photogramm. Eng. 34, 1079–1083 (1968).
8. G. H. Schut, “Construction of orthogonal matrices and their application in analytical photogrammetry,” Photogrammetria 15, 149–162 (1959).
9. E. H. Thompson, “On exact linear solution of the problem of absolute orientation,” Photogrammetria 15, 163–179 (1959).
10. E. Salamin, “Application of quaternions to computation with rotations,” Internal Rep. (Stanford University, Stanford, California, 1979).
11. R. H. Taylor, “Planning and execution of straight line manipulator trajectories,” in Robot Motion: Planning and Control, M. Bradey, J. M. Mollerbach, T. L. Johnson, T. Lozano-Pérez, M. T. Mason, eds. (MIT, Cambridge, Mass., 1982).
12. G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw Hill, New York, 1968).
13. J. H. Stuelpnagle, “On the parameterization of the three-dimensional rotation group,”SIAM Rev. 6, 422–430 (1964).
[Crossref]
14. G. Birkhoff, S. MacLane, A Survey of Modern Algebra (Macmillan, New York, 1953).
15. P. H. Winston, B. K. P. Horn, LISP (Addison-Wesley, Reading, Mass., 1984).

#### 1968 (1)

H. L. Oswal, S. Balasubramanian, “An exact solution of absolute orientation,” Photogramm. Eng. 34, 1079–1083 (1968).

#### 1964 (1)

J. H. Stuelpnagle, “On the parameterization of the three-dimensional rotation group,”SIAM Rev. 6, 422–430 (1964).
[Crossref]

#### 1960 (1)

G. H. Schut, “On exact linear equations for the computation of the rotational elements of absolute orientation,” Photogrammetria 16, 34–37 (1960).

#### 1959 (2)

G. H. Schut, “Construction of orthogonal matrices and their application in analytical photogrammetry,” Photogrammetria 15, 149–162 (1959).

E. H. Thompson, “On exact linear solution of the problem of absolute orientation,” Photogrammetria 15, 163–179 (1959).

#### 1958 (1)

E. H. Thompson, “A method for the construction of orthogonal matrices,” Photogramm. Record 3, 55–59 (1958).
[Crossref]

#### Balasubramanian, S.

H. L. Oswal, S. Balasubramanian, “An exact solution of absolute orientation,” Photogramm. Eng. 34, 1079–1083 (1968).

#### Birkhoff, G.

G. Birkhoff, S. MacLane, A Survey of Modern Algebra (Macmillan, New York, 1953).

#### Gosh, S. K.

S. K. Gosh, Theory of Stereophotogrammetry (Ohio U. Bookstores, Columbus, Ohio, 1972).

#### Horn, B. K. P.

B. K. P. Horn, Robot Vision (MIT/McGraw-Hill, New York, 1986).

P. H. Winston, B. K. P. Horn, LISP (Addison-Wesley, Reading, Mass., 1984).

#### Korn, G. A.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw Hill, New York, 1968).

#### Korn, T. M.

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw Hill, New York, 1968).

#### MacLane, S.

G. Birkhoff, S. MacLane, A Survey of Modern Algebra (Macmillan, New York, 1953).

#### Oswal, H. L.

H. L. Oswal, S. Balasubramanian, “An exact solution of absolute orientation,” Photogramm. Eng. 34, 1079–1083 (1968).

#### Salamin, E.

E. Salamin, “Application of quaternions to computation with rotations,” Internal Rep. (Stanford University, Stanford, California, 1979).

#### Schut, G. H.

G. H. Schut, “On exact linear equations for the computation of the rotational elements of absolute orientation,” Photogrammetria 16, 34–37 (1960).

G. H. Schut, “Construction of orthogonal matrices and their application in analytical photogrammetry,” Photogrammetria 15, 149–162 (1959).

#### Stuelpnagle, J. H.

J. H. Stuelpnagle, “On the parameterization of the three-dimensional rotation group,”SIAM Rev. 6, 422–430 (1964).
[Crossref]

#### Taylor, R. H.

R. H. Taylor, “Planning and execution of straight line manipulator trajectories,” in Robot Motion: Planning and Control, M. Bradey, J. M. Mollerbach, T. L. Johnson, T. Lozano-Pérez, M. T. Mason, eds. (MIT, Cambridge, Mass., 1982).

#### Thompson, E. H.

E. H. Thompson, “On exact linear solution of the problem of absolute orientation,” Photogrammetria 15, 163–179 (1959).

E. H. Thompson, “A method for the construction of orthogonal matrices,” Photogramm. Record 3, 55–59 (1958).
[Crossref]

#### Winston, P. H.

P. H. Winston, B. K. P. Horn, LISP (Addison-Wesley, Reading, Mass., 1984).

#### Wolf, P. R.

P. R. Wolf, Elements of Photogrammetry (McGraw Hill, New York, 1974).

#### Photogramm. Eng. (1)

H. L. Oswal, S. Balasubramanian, “An exact solution of absolute orientation,” Photogramm. Eng. 34, 1079–1083 (1968).

#### Photogramm. Record (1)

E. H. Thompson, “A method for the construction of orthogonal matrices,” Photogramm. Record 3, 55–59 (1958).
[Crossref]

#### Photogrammetria (3)

G. H. Schut, “On exact linear equations for the computation of the rotational elements of absolute orientation,” Photogrammetria 16, 34–37 (1960).

G. H. Schut, “Construction of orthogonal matrices and their application in analytical photogrammetry,” Photogrammetria 15, 149–162 (1959).

E. H. Thompson, “On exact linear solution of the problem of absolute orientation,” Photogrammetria 15, 163–179 (1959).

#### SIAM Rev. (1)

J. H. Stuelpnagle, “On the parameterization of the three-dimensional rotation group,”SIAM Rev. 6, 422–430 (1964).
[Crossref]

#### Other (9)

G. Birkhoff, S. MacLane, A Survey of Modern Algebra (Macmillan, New York, 1953).

P. H. Winston, B. K. P. Horn, LISP (Addison-Wesley, Reading, Mass., 1984).

E. Salamin, “Application of quaternions to computation with rotations,” Internal Rep. (Stanford University, Stanford, California, 1979).

R. H. Taylor, “Planning and execution of straight line manipulator trajectories,” in Robot Motion: Planning and Control, M. Bradey, J. M. Mollerbach, T. L. Johnson, T. Lozano-Pérez, M. T. Mason, eds. (MIT, Cambridge, Mass., 1982).

G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw Hill, New York, 1968).

C. C. Slama, C. Theurer, S. W. Henrikson, eds., Manual of Photogrammetry (American Society of Photogrammetry, Falls Church, Va., 1980).

B. K. P. Horn, Robot Vision (MIT/McGraw-Hill, New York, 1986).

P. R. Wolf, Elements of Photogrammetry (McGraw Hill, New York, 1974).

S. K. Gosh, Theory of Stereophotogrammetry (Ohio U. Bookstores, Columbus, Ohio, 1972).

### Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

### Figures (6)

Fig. 1

The coordinates of a number of points is measured in two different coordinate systems. The transformation between the two systems is to be found.

Fig. 2

Three points define a triad. Such a triad can be constructed by using the left measurements. A second triad is then constructed from the right measurements. The required coordinate transformation can be estimated by finding the transformation that maps one triad into the other. This method does not use the information about each of the three points equally.

Fig. 3

When the two sets of measurements each lie in a plane, the rotation decomposes conveniently into rotation about the line of intersection of the two planes and rotation about a normal of one of the planes. (In this figure the two coordinate systems have been aligned and superimposed.)

Fig. 4

The second rotation, when both sets of measurements are coplanar, is about the normal of one of the planes. The rotation that minimizes the sum of errors is found easily in this special case since there is only one degree of freedom.

Fig. 5

Three different optimal scale factors can be obtained by choosing three different forms for the error term. In general, a symmetric form for the error is to be preferred.

Fig. 6

The rotation of a vector r to a vector r′ can be understood in terms of the quantities appearing in this diagram. Rodrigues’s formula follows.

### Equations (169)

$x l = r l , 2 - r l , 1 .$
$x ^ l = x l / ‖ x l ‖$
$y l ( r l , 3 - r l , 1 ) - [ ( r l , 3 - r l , 1 ) · x ^ l ] x ^ l$
$y ^ l = y l / ‖ y l ‖$
$z ^ l = x ^ l × y ^ l$
$M = ∣ x ^ l y ^ l z ^ l ∣ , M r = ∣ x ^ r y ^ r z ^ r ∣ .$
$M l T r l$
$r r = M r M l T r l .$
$R = M r M l T .$
${ r l , i } and { r r , i } ,$
$r r = s R ( r l ) + r 0$
$‖ R ( r l ) ‖ 2 = ‖ r l ‖ 2 ,$
$e i = r r , i - s R ( r l , i ) - r 0 .$
$∑ i = 1 n ‖ e i ‖ 2 .$
$r ¯ l = 1 n ∑ i = 1 n r l , i , r ¯ r = 1 n ∑ i = 1 n r r , i .$
$r ′ l , i = r l , i - r ¯ l , r ′ r , i = r r , i - r ¯ r .$
$∑ i = 1 n r ′ l , i = 0 , ∑ i = 1 n r ′ r , i = 0 .$
$e i = r ′ r , i - s R ( r ′ l , i ) - r ′ 0 ,$
$r ′ 0 = r 0 - r ¯ r + s R ( r ¯ l ) .$
$∑ i = 1 n ‖ r ′ r , i - s R ( r ′ l , i ) - r ′ 0 ‖ 2$
$∑ i = 1 n ‖ r ′ r , i - s R ( r ′ l , i ) ‖ 2 - 2 r ′ 0 · ∑ i = 1 n [ r ′ r , i - s R ( r ′ l , i ) ] + n ‖ r ′ 0 ‖ 2 .$
$r 0 = r ¯ r - s R ( r ¯ l ) .$
$e i = r ′ r , i - s R ( r ′ l , i )$
$∑ i = 1 n ‖ r ′ r , i - s R ( r ′ l , i ) ‖ 2 .$
$‖ R ( r ′ l , i ) ‖ 2 = ‖ r ′ l , i ‖ 2 ,$
$∑ i = 1 n ‖ r ′ r , i ‖ 2 - 2 s ∑ i = 1 n r ′ r , i · R ( r ′ l , i ) + s 2 ∑ i = 1 n ‖ r ′ l , i ‖ 2 ,$
$S r - 2 s D + s 2 S l ,$
$( s S l - D / S l ) 2 + ( S r S l - D 2 ) / S l .$
$s = ∑ i = 1 n r ′ r , i · R ( r ′ l , i ) / ∑ i = 1 n ‖ r ′ l , i ‖ 2 .$
$r r = s R ( r l ) + r 0 ,$
$r l = s ¯ R ¯ ( r r ) + r ¯ 0 ,$
$s ¯ = 1 / s , r ¯ 0 = - 1 s R - 1 ( r 0 ) , R ¯ = R - 1 .$
$s ¯ = ∑ i = 1 n r ′ l , i · R ¯ ( r ′ r , i ) / ∑ i = 1 n ‖ r ′ r , i ‖ 2 ,$
$e i = 1 s r ′ r , i - s R ( r ′ l , i ) .$
$1 s ∑ i = 1 n ‖ r ′ r , i ‖ 2 - 2 ∑ i = 1 n r ′ r , i · R ( r ′ l , i ) + s ∑ i = 1 n ‖ r ′ l , i ‖ 2$
$1 s S r - 2 D + s S l .$
$( s S l - 1 s S r ) 2 + 2 ( S l S r - D ) .$
$s = ( ∑ i = 1 n ‖ r ′ r , i ‖ 2 / ∑ i = 1 n ‖ r ′ l , i ‖ 2 ) 1 / 2 .$
$∑ i = 1 n r ′ r , i · R ( r ′ l , i )$
$∑ i = 1 n r ′ r , i · R ( r ′ l , i )$
$p ∘ = q 0 + i q x + j q y + k q z ,$
$i 2 = - 1 , j 2 = - 1 , k 2 = - 1 ; i j = k , j k = i , k i = j ;$
$j i = - k , k j = - i , i k = - j .$
$r ∘ = r 0 + i r x + j r y + k r z ,$
$r ∘ q ∘ = ( r 0 q 0 - r x q x - r y q y - r z q z ) + i ( r 0 q x + r x q 0 + r y q z - r z q y ) + j ( r 0 q y - r x q z + r y q 0 + r z q x ) + k ( r 0 q z + r x q y - r y q x + r z q 0 ) .$
$r ∘ q ∘ = [ r 0 - r x - r y - r z r x r 0 - r z r y r y r z r 0 - r x r z - r y r x r 0 ] q ∘ = IR q ∘$
$q ∘ r ∘ = [ r 0 - r x - r y - r z r x r 0 r z - r y r y - r z r 0 r x r z r y - r x r 0 ] q ∘ = IR ¯ q ∘ .$
$r 0 2 + r x 2 + r y 2 + r z 2 ,$
$p ∘ · q ∘ = p 0 q 0 + p x q x + p y q y + p z q z .$
$‖ q ∘ ‖ 2 = q ∘ · q ∘ .$
$q ∘ * = q 0 - i q x - j q y - k q z .$
$q ∘ q ∘ * = ( q 0 2 + q x 2 + q y 2 + q z 2 ) = q ∘ · q ∘ .$
$q ∘ - 1 = ( 1 / q ∘ · q ∘ ) q ∘ * .$
$( q ∘ p ∘ ) · ( q ∘ r ∘ ) = ( Q p ∘ ) · ( Q r ∘ ) = ( Q p ∘ ) T ( Q r ∘ )$
$( Q p ∘ ) T ( Q r ∘ ) = p ∘ T Q T Q r ∘ = p ∘ T ( q ∘ · q ∘ ) I r ∘ .$
$( q ∘ p ∘ ) · ( q ∘ r ∘ ) = ( q ∘ · q ∘ ) ( p ∘ · r ∘ ) ,$
$( p ∘ q ∘ ) · ( p ∘ q ∘ ) = ( p ∘ · p ∘ ) ( q ∘ · q ∘ ) ;$
$( p ∘ q ∘ ) · r ∘ = p ∘ · ( r ∘ q ∘ * ) ,$
$r ∘ = 0 + i x + j y + k z .$
$IR T = - IR and IR ¯ T = - IR ¯$
$( q ∘ p ∘ ) · ( q ∘ r ∘ ) = p ∘ · r ∘ ,$
$r ′ ∘ = q ∘ r ∘ q * ∘ ,$
$q ∘ r ∘ q * ∘ = ( Q r ∘ ) q * ∘ = Q ¯ T ( Q r ∘ ) = ( Q ¯ T Q ) r ∘ ,$
$Q ¯ T Q = [ q ∘ · q ∘ 0 0 0 0 ( q 0 2 + q x 2 - q y 2 - q z 2 ) 2 ( q x q y - q 0 q z ) 2 ( q x q z + q 0 q y ) 0 2 ( q y q x + q 0 q z ) ( q 0 2 - q x 2 + q y 2 - q z 2 ) 2 ( q y q z - q 0 q x ) 0 2 ( q z q x - q 0 q y ) 2 ( q z q y + q 0 q x ) ( q 0 2 - q x 2 - q y 2 + q z 2 ) ] .$
$r ′ = R r .$
$( - q ∘ ) r ∘ ( - q * ∘ ) = q ∘ r ∘ q * ∘$
$q ∘ = cos θ 2 + sin θ 2 ( i ω x + j ω y + k ω z ) .$
$r ″ ∘ = p ∘ r ′ ∘ p * ∘ = p ∘ ( q ∘ r ∘ q * ∘ ) p * ∘ .$
$r ″ ∘ = ( p ∘ q ∘ ) r ∘ ( p ∘ q ∘ ) * .$
$∑ i = 1 n ( q ∘ r ′ l , i q * ∘ ∘ ) · r ′ r , i ∘ .$
$∑ i = 1 n ( q ∘ r ′ l , i ∘ ) · ( r ′ r , i q ∘ ∘ ) .$
$q ∘ r ′ l , i ∘ = [ 0 - x ′ l , i - y ′ l , i - z ′ l , i x ′ l , i 0 z ′ l , i - y ′ l , i y ′ l , i z ′ l , i 0 x ′ l , i z ′ l , i y ′ l , i - x ′ l , i 0 ] q ∘ = IR ¯ l , i q ∘ ,$
$r ′ r , i ∘ q ∘ = [ 0 - x ′ r , i - y ′ r , i - z ′ r , i x ′ r , i 0 - z ′ r , i y ′ r , i y ′ r , i z ′ r , i 0 - x ′ r , i z ′ r , i - y ′ r , i x ′ r , i 0 ] q ∘ = IR r , i q ∘ .$
$∑ i = 1 n ( IR ¯ l , i q ∘ ) · ( IR r , i q ∘ )$
$∑ i = 1 n q T IR ¯ l , i T IR r , i q ∘ ∘ ;$
$q T ∘ ( ∑ i = 1 n IR ¯ l , i T IR r , i ) q ∘$
$q T ∘ ( ∑ i = 1 n N i ) q ∘ = q T ∘ N q ∘ ,$
$M = ∑ i = 1 n r ′ l , i r ′ r , i T$
$M = [ S x x S x y S x z S y z S y y S y z S z x S z y S z z ] ,$
$S x x = ∑ i = 1 n x ′ l , i x ′ r , i , S x y = ∑ i = 1 n x ′ l , i y ′ r , i ,$
$N = [ ( S x x + S y y + S z z ) S y z - S z y S z x - S x z S x y - S y x S y z - S z y ( S x x - S y y - S z z ) S x y + S y x S z x + S x z S z x - S x z S x y + S y x ( - S x x + S y y - S z z ) S y z + S z y S x y - S y x S z x + S x z S y z + S z y ( - S x x - S y y + S z z ) ] .$
$q T ∘ N q ∘$
$det ( N - λ I ) = 0 ,$
$[ N - λ m I ] e m ∘ = 0.$
$N = [ a e h j e b f i h f c g j i g d ] ,$
$det ( N - λ I ) = 0$
$λ 4 + c 3 λ 3 + c 2 λ 2 + c 1 λ + c 0 = 0 ,$
$c 3 = a + b + c + d , c 2 = ( a c - h 2 ) + ( b c - f 2 ) + ( a d - j 2 ) + ( b d - i 2 ) + ( c d + g 2 ) + ( a b - e 2 ) , c 1 = [ - b ( c d - g 2 ) + f ( d f - g i ) - i ( f g - c i ) - a ( c d - g 2 ) + h ( d h - g j ) - j ( g h - c j ) - a ( b d - i 2 ) + e ( d e - i j ) - j ( e i - b j ) - a ( b c - f 2 ) + e ( c e - f h ) - h ( e f - b h ) ] , c 0 = ( a b - e 2 ) ( c d - g 2 ) + ( e h - a f ) ( f d - g i ) + ( a i - e j ) ( f g - c i ) + ( e f - b h ) ( h d - g j ) + ( b j - e i ) ( h g - c j ) + ( h i - f j ) 2 .$
$c 2 = - 2 ( S x x 2 + S x y 2 + S x z 2 + S y x 2 + S y y 2 + S y z 2 + S z x 2 + S z y 2 + S z z 2 ) ,$
$c 1 = 8 ( S x x S y z S z y + S y y S z x S x z + S z z S x y S y x ) - 8 ( S x x S y y S z z + S y z S z x S x y + S z y S y x S x z ) .$
$r ′ r , i · n r = 0$
$( x ′ r , i , y ′ r , i , z ′ r , i ) T · n r = 0.$
$( S x x , S x y , S x z ) T · n r = 0 , ( S y x , S y y , S y z ) T · n r = 0 , ( S z x , S z y , S z z ) T · n r = 0 ,$
$M n r = 0 .$
$λ 4 + c 2 λ 2 + c 0 = 0.$
$μ 2 + c 2 μ + c 0 = 0 ,$
$μ = ± ½ ( c 2 2 - 4 c 0 ) 1 / 2 - c 2 .$
$λ = ± μ + , λ = ± μ - .$
$λ m = [ ½ ( c 2 2 - 4 c 0 ) 1 / 2 - c 2 ] 1 / 2 .$
$n l = r ′ l 2 × r ′ l 1 , n r = r ′ r 2 × r ′ r 1 .$
$a = n l × n r .$
$cos ϕ = n ^ l · n ^ r , sin ϕ = ‖ n ^ l × n ^ r ‖ .$
$q a ∘ = cos ϕ 2 + sin ϕ 2 ( i a x + j a y + k a z ) ,$
$∑ i = 1 n ‖ r ′ r , i - r ″ l , i ‖ 2 .$
$r ′ r , i r ″ l , i cos α i = r ′ r , i · r ″ l , i$
$r ′ r , i r ″ l , i sin α i = ( r ′ r , i × r ″ l , i ) · n ^ r ,$
$r ′ r , i = ‖ r ′ r , i ‖ , r ″ l , i = ‖ r ″ l , i ‖ = ‖ r ′ l , i ‖ .$
$‖ r ′ r , i × r ″ l , i ‖$
$( r ′ r , i ) 2 + ( r ″ l , i ) 2 - 2 r ′ r , i r ″ l , i cos α i .$
$∑ i = 1 n r ′ r , i r ″ l , i cos ( α i - θ )$
$C cos θ + S sin θ ,$
$C = ∑ i = 1 n r ′ r , i r ″ l , i cos α i = ∑ i = 1 n ( r ′ r , i · r ″ l , i )$
$S = ∑ i = 1 n r ′ r , i r ″ l , i sin α i = ( ∑ i = 1 n r ′ r , i × r ″ l , i ) · n ^ r .$
$C sin θ = S cos θ$
$sin θ = ± S S 2 + C 2 , cos θ = ± C S 2 + C 2 .$
$q P ∘ = cos θ 2 + sin θ 2 ( i n x + j n y + k n z )$
$q ∘ = q p ∘ q a ∘ or R = R p R a .$
$cos θ / 2 = [ ( 1 + cos θ ) / 2 ] 1 / 2 ,$
$sin θ / 2 = sin θ / [ 2 ( 1 + cos θ ) ] 1 / 2 .$
$E r l = 2 ( c - s a ) 2 + 2 ( d - s b ) 2 ,$
$s = s r l = a c + b d a 2 + b 2 .$
$E = 2 ( a - s ¯ c ) 2 + 2 ( b - s ¯ d ) 2 ,$
$s ¯ = a c + b d c 2 + d 2 .$
$s = s l r = c 2 + d 2 a c + b d .$
$E sy = 2 ( 1 s c - s a ) 2 + 2 ( 1 s d - s c ) 2 ,$
$S = S sy = ( c 2 + d 2 a 2 + b 2 ) 1 / 2 .$
$∑ i = 1 n w i ‖ e i ‖ 2 ,$
$r ¯ l = ∑ i = 1 n w i r l , i / ∑ i = 1 n w i$
$r ¯ r = ∑ i = 1 n w i r r , i / ∑ i = 1 n w i .$
$S = ( ∑ i = 1 n w i ‖ r ′ r , i ‖ 2 / ∑ i = 1 n w i ‖ r ′ l , i ‖ 2 ) 1 / 2 .$
$M = ∑ i = 1 n w i r ′ l , i r ′ r , i T ,$
$S x x = ∑ i = 1 n w i x ′ l , i x ′ r , i ,$
$N e i ∘ = λ i e i ∘ for i = 1 , 2 , … , 4.$
$q ∘ = α 1 e 1 ∘ + α 2 e 2 ∘ + α 3 e 3 ∘ + α 4 e 4 ∘ .$
$q ∘ · q ∘ = α 1 2 + α 2 2 + α 3 2 + α 4 2 .$
$N q ∘ = α 1 λ 1 e 1 ∘ + α 2 λ 2 e 2 ∘ + α 3 λ 3 e 3 ∘ + α 4 λ 4 e 4 ∘$
$q T ∘ N q ∘ = q ∘ · ( N q ∘ ) = α 1 2 λ 1 + α 2 2 λ 2 + α 3 2 λ 3 + α 4 2 λ 4 .$
$λ 1 ≥ λ 2 ≥ λ 3 ≥ λ 4 .$
$q T ∘ N q ∘ ≤ α 1 2 λ 1 + α 2 2 λ 1 + α 3 2 λ 1 + α 4 2 λ 1 = λ 1$
$( N - λ m I ) e m ∘ = 0.$
$N λ = N - λ I ;$
$det ( N λ ) N λ - 1 = ( N ¯ λ ) T ,$
$det ( N λ ) = ∏ j = 1 n ( λ j - λ ) .$
$q ∘ = ∑ i = 1 n λ i e i ∘ ,$
$N λ - 1 q ∘ = ∑ i = 1 n α i ( λ i - λ ) e i ∘ .$
$∑ i = 1 n ∏ j ≠ 1 n ( λ j - λ ) α i e i ∘ = ( N ¯ λ ) T q ∘ .$
$∏ j ≠ m n ( λ j - λ m ) α m e m ∘ = ( N ¯ λ m ) T q ∘ .$
$q ∘ = q 0 + i q x + j q y + k q z$
$q ∘ = q + q ,$
$p = r s - r · s , p = r s + s r + r × s ,$
$r ∘ = r + r , s ∘ = s + s , p ∘ = p + p .$
$p = - r · s , p = r × s .$
$r ′ ∘ = q ∘ r ∘ q * ∘ , s ′ ∘ = q ∘ s ∘ q * ∘ , p ′ ∘ = q ∘ p ∘ q * ∘ .$
$r ′ ∘ s ′ ∘ = ( q ∘ r ∘ q * ∘ ) ( q ∘ s ∘ q * ∘ ) = ( q ∘ r ∘ ) ( q * ∘ q ∘ ) ( s ∘ q * ∘ ) = q ∘ ( r ∘ s ∘ ) q * ∘ ,$
$- r ′ · s ′ = - r · s .$
$r ′ = cos θ r + sin θ ω ^ + r + ( 1 - cos θ ) ( ω ^ · x ) ω ^ .$
$r ′ ∘ = q ∘ r ∘ q * ∘ ,$
$r ∘ = 0 + r , r ′ ∘ = 0 + r ′ ,$
$q ∘ = cos ( θ / 2 ) + sin ( θ / 2 ) ω ^ .$
$r ′ = ( q 2 - q · q ) r + 2 q q × r + 2 ( q · r ) q .$
$q = cos ( θ / 2 ) , q = sin ( θ / 2 ) ω ^$
$2 sin ( θ / 2 ) cos ( θ / 2 ) = sin θ , cos 2 ( θ / 2 ) - sin 2 ( θ / 2 ) = cos θ ,$
$ω ^ = q / ‖ q ‖$
$sin θ = 2 q ‖ q ‖ , cos θ = ( q 2 - ‖ q ‖ 2 ) .$
$q ∘ = q 0 + i q x + j q y + k q z$
$R = [ ( q 0 2 + q x 2 - q y 2 - q z 2 ) 2 ( q x q y - q 0 q z ) 2 ( q x q z + q 0 q y ) 2 ( q y q x + q 0 q z ) ( q 0 2 - q x 2 + q y 2 - q z 2 ) 2 ( q y q z - q 0 q x ) 2 ( q z q x - q 0 q y ) 2 ( q z q y + q 0 q z ) ( q 0 2 - q x 2 - q y 2 + q z 2 ) ] .$
$1 + r 11 + r 22 + r 33 = 4 q 0 2 , 1 + r 11 - r 22 - r 33 = 4 q x 2 , 1 - r 11 + r 22 - r 33 = 4 q y 2 , 1 - r 11 - r 22 + r 33 = 4 q z 2 .$
$r 32 - r 23 = 4 q 0 q x , r 13 - r 31 = 4 q 0 q y , r 21 - r 12 = 4 q 0 q z .$
$r 21 + r 12 = 4 q x q y , r 32 + r 23 = 4 q y q z , r 13 + r 31 = 4 q z q x .$