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 u .fat 
 
 ynaXi 
 
 C00-2 383-000 8 
 
 7 
 
 Some Matrix Results for Stable m- dimensional Rotations 
 
 by 
 C. W. Gear 
 
 April 15, 197^ 
 
 THE LIBRARY OF THE 
 
 JUL 9 1974 
 
UIUCDCS-R-7U-63U 
 
 Some Matrix Results for Stable m- dimensional Rotations 
 
 by 
 C. W. Gear 
 
 April 15, 197 1 * 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 
 
 URBANA, ILLINOIS 6l801 
 
 This report was supported in part by the Atomic Energy Commission 
 under grant US AEC AT( 11-1)2383 
 
Some Matrix Results for Stable m-dimensional Rotations 
 
 "by 
 C. W. Gear 
 
 Abstract Continuous (real-time) rotation of an object about a line 
 
 in 3-space can be done using an incremental technique. An 
 equivalent technique in 2-space has a simple modification that 
 makes it stable. The same is shown to be true in m- space. 
 
A general three dimensional rotation of angle about a line with direction 
 
 T 
 coordinates [a,b,c] through the origin is given by 
 
 x f = A(e)x 
 
 T 
 where x = [x,y,z] are the old coordinates, x' the new, and A(0) is 
 
 2 
 
 k+(l-k)a 
 
 ab(l-k)+cs 
 
 ac(l-k)-bs 
 
 ab (l-k)-cs 
 
 k+(l-k)b' 
 
 bc(l-k)+as 
 
 ac(l-k)+bs 
 
 bc(l-k)-as 
 
 k+(l-k)c' 
 
 where s = sin0 and k = cosQ. (if we wish to rotate about a line not through 
 
 the origin, translations before and after take care of it. Alternatively 
 
 we could use homogeneous coordinates.) If it is desired to rotate an object 
 
 "continuously" in 3-space (say for display purposes), we could choose a small 
 
 and compute 
 
 x = A(0)x 
 
 -n+1 — n 
 
 thus deriving successive "frames". A(0) is fairly complex, and involves sin0 
 
 and cos0. For very small 0, we could neglect 0(0 ) and replace k by 1 and s 
 
 by 0, getting 
 
 A(0) * A = 
 
 -C0 
 
 c0 
 
 -b© 
 
 b0 
 
 -a0 
 
 30 
 
 Unfortunately, at least one eigenvalue of this approximation is larger than 
 
 one in magnitude for 0^0, hence x = A x„ is unbounded for almost all x^ . 
 
 - n —0 —0 
 
 This means that x will sprial outwards as n increases, 
 -n 
 
The solution in two dimensions is well known. Instead of 
 
 1 -c0 
 
 — n+1 
 
 c0 
 
 1 
 
 x 
 
 -n 
 
 that i s 
 
 we compute 
 
 x = x - c0 y 
 
 n+1 n n 
 
 y , . = c0 x + y 
 J n+1 n °n 
 
 x = x - c0 y 
 
 n+1 n n 
 
 y , _ = c0 x , , + y 
 J n+1 n+1 °r 
 
 That is equivalent to 
 "l 0~ 
 
 -c0 1 
 
 or 
 
 ^n+1 
 
 x 
 
 -n+1 
 
 1 
 -cO 
 
 -c0 
 
 1 
 
 X 
 
 -n 
 
 _-l 
 
 ■C0 
 
 1 
 
 x = A x 
 
 -n -n 
 
 Simple computation shows that the eigenvalues of A are distinct and on the 
 
 2 2 ~N 
 
 unit circle if < c < k. Hence, for small 0, A is bounded for all N. 
 
 2 2 
 
 More important, simple calculation shows that x + y - c0 x y is invariant, 
 
 n n n n 
 
 2 2 
 
 hence all points lie on the conic x + y - c0xy = K where K is determined by 
 
 the starting point x~. 
 
 The equivalent three dimensional modification is to write 
 
 x^ n =x -c0y +b0z 
 n+1 n n n 
 
 y,=c0x^+y -a0z 
 n+1 n+1 n n 
 
 or 
 
 n+1 n+1 J n+1 n 
 
where 
 
 
 
 -c9 
 
 b© 
 
 
 
 
 
 -a© 
 
 
 
 
 
 
 
 u = 
 
 Hence x = Ax where A = [i+U ] [I+U]. A has all unit magnitude eigen- 
 -n+1 — n 
 
 values if U is small, as shown "by 
 Theorem 
 
 If A = [I + E T ] _1 [I + E] 
 
 for any m x m matrix E then 
 
 T T 
 
 (i) The quadratic form x Q x is invariant where Q = 21 + E + E 
 
 (ii) The eigenvalues of A are on the unit circle and correspond 
 
 to linear elementary divisors if |E| is small enough. 
 
 Proof 
 
 By simple computation 
 T 
 
 T ~T 
 Q x ._ = x A Q Ax 
 -n+1 -n+1 -n n 
 
 = x^[l+E T ][l+E] X [(I+E)+(I+E T )] [I+E T ] 1 [I+E] x^ 
 
 = x T [(l+E) + (I+E T )] x 
 -n -n 
 
 T 
 = x Q x 
 
 -n -n 
 
 as claimed in part (i). If | |e|J is small, the eigenvalues of Q can be 
 
 bounded close to +1 by the Gerschgorin circle theorem. Q is symetric, hence 
 
 T 
 it eigenvalues are real and positive. Hence Q is positive definite and x Q x = 
 
 |x| | is a norm. If an eigenvalue of A is greater than one in magnitude, or 
 
 of magnitude one and corresponds to a non-linear elementary divisor, there 
 
 exists an eigenvector or principal vector x such that x = A x grows unboundedly 
 
But this is not possible since |x^||_ = |x||q is bounded. Similarly, 
 
 if there exists an eigenvalue of A inside the unit circle, its eigenvector 
 
 ~N i i i i iiii 
 
 x is such that x^ = A x ->- as n + °°. But | |v | [ = | |x| | in contradiction. 
 
 Q.E.D. 
 
 Returning to the 3-space problem, we note that x remains on the conic 
 
 T T i i i i 
 
 surface x Q x = x^ Q x which is an ellipsoid if |U| is small enough. 
 
 2 2 2 2 
 
 (To be precise, if < [a +b +c - abc0]0 < h.) This is not quite 
 
 enough, as the "locus" of x^ (actually, a set of points) could wander all over 
 
 the conic surface. However, in this case, one of the eigenvalues of A is 1. 
 
 (Since A is real, at least one of its three eigenvalues is real, that is, ±1 
 
 When U =0, the eigenvalues are all 1, and the eigenvalues are continuous 
 
 functions of U, hence one eigenvalue is 1 for all small ||u||.) Suppose the 
 
 T 
 left eigenvector for the unit eigenvalue is l_ . Then 
 
 T T T T 
 
 i x .. = I Ax = I x = I x 
 — -n+1 — -n — -n — -n) 
 
 T 
 That is, the plane l_ x_ = K is another invariant subspace for the transformation 
 
 T T 
 
 Hence, the intersection of x Q x_ = K, and £_ x_ = K is an invariant, and 
 
 computed points x_^ move around this ellipse. 
 
 Note These invariant planes are perpendicular to the vector %_ 9 so, in a sense, 
 
 T 
 _£ is the axis of rotation. l_ = [a, b-ac , c] is a small pertubation 
 
 T 
 of the original axis [a, b, c] . However, in another sense, the line 
 
 v, where v_ is the right unit eigenvector of A, is the axis of rotation, 
 
 because, if x = v, then 
 -n — 
 
 x,.,=Ax =Av = v=x 
 n+1 -n — — -n 
 
 T 
 so points on y_ are not moved, v is [a, b, c] . Hence the approximate 
 
 rotation leaves the desired axis unchanged, but skews the planes of 
 
 rotation a small amount . 
 
In an m- dimensional space, we must first define what we mean by a rotation. 
 A general unitary transformation whose determinant is +1 is often called a 
 rotation. This has m(m-l)/2 "degrees of freedom". These could be taken to be 
 the angles of rotation in the plane of each axis pair. In three dimensions, any 
 combination of rotations is equivalent to a single rotation in some plane, but 
 that is not true in higher dimensions. This means that we can continuously 
 rotate an object in four or more dimensions, and never get back to the original 
 position. Consequently, for display manipulation purposes, we are interested 
 Ln the class of rotations in a two dimensional subspace (plane) which leave all 
 points in an m-2 dimensional subspace invariant. We will characterize these by 
 the unitary matrix A(e) = MR(o)M where 
 
 cosQ -sino 
 
 sing cosq 
 
 R(e) = 
 
 m-2 
 
 in d where M is a unitary matrix of dimension m. To compute successive "frames" 
 'or display, we would like to replace cosO by 1 and sine by in A(e) and then 
 ise the trick used earlier to "stabilize" the method. We will show that the 
 '.odified method leaves the same points invariant as does the original rotation, 
 o that it is a reasonable approximation. Also, the rotation will be in a plane 
 hich is an 0(6) perturbation to the original plane. 
 For small 0, let 
 
 R = 
 
 -0 
 1 
 
 m-2 
 
Note that the eigenvalues of R include m-2 equal to one, and that these have 
 
 m-2 distinct right and left eigenvectors v., i=3,...,m. These are also eigenvalues 
 
 and eigenvectors of R(3). 
 
 T 
 Note that the right and left eigenvectors of A(0) and A = MRM corresponding 
 
 to the eigenvalues of one are Mv. , i-3,...,m. R can be expressed as (I + E - E ) 
 
 where | |e| | =0(0). Then A = I + MEM T - ME M T . Let MEM T = D + L + U where L 
 
 is strictly lower triangular, D is diagonal and U is strictly upper triangular. 
 
 Define E = E - M T (p + L + L )M. Then we see that R = I + E - E T where 
 
 l l I l T A T 
 
 ||E| I = 0(0) and E = U - L = U, a strictly upper triangular matrix. Hence 
 
 T . . 
 
 A = I + U - U , and the stabilized method is given by 
 
 x . = A x 
 
 -n+1 -n 
 
 where A = (I + U T ) _1 (I + U) 
 
 We will show below that A has m-2 eigenvalues equal to one with eigenvectors 
 equal to Mv. (the same as for A and A(0)). This means that A has the same 
 invariant subspace as A(0). We will also show that the left eigenvectors of A are 
 
 of the form v.M + d where Idll =0(0). 
 
 —i — — 
 
 T 
 Lemma If _z is a unit eigenvector of A = (i + U-U ) then it is a unit right 
 
 eigenvector of A = (I + U T ) _1 (i+U), A = (I + U)" 1 (I + U T ) , A = (i - U T ) _1 (I - U) 
 
 and A = (I - U)" 1 (I - U T ). 
 
 Proof 
 
 Hence 
 
 z = Az=(l + U-U)z. 
 
 (I + U T )z = (I + U)z, 
 
 and (I - U)z = (I - U T )z, 
 
T -1 
 The results follow by multiplying the first equation by (I + U ) or 
 
 1 —1 T —1 
 
 (I + U) , and the second equation by (I - U) or (I - U ) . Q.E.D. 
 
 Thus A has at least m-2 unit right eigenvectors when A does and hence leaves all 
 
 points in the subspaced spanned by these invariant. 
 
 T 
 Lemma If z is a unit eigenvector of A = (i + U-U ), a unit left eigenvector 
 
 ~ m _2_ m m m m mm 
 
 of A = (I + U ) (I + U) is l_ = z (I + U ) = z_ (I + U) = z_ + d where 
 \\i\ I = 0(| |U| | ) = 0(0). 
 
 Proof 
 
 Note that _z T ( I + U - U ) = z_ T . Hence _z T U = z_ T U T . Thus 
 
 £ T A = z_ T (l + U T ) (I + U T ) _1 (I + U) 
 
 = z T (l + U) = £ T 
 
 Obviously | | £_ T - z_ T | | = | |_z T U T | | = 0(0). 
 
 Q.E.D. 
 
 Thus, for small 0, A has at least m-2 distinct right unit eigenvectors I, which 
 
 T m 
 
 are a 0(0) perturbation of the vectors v. M. The plane P = {l. x = k. ) is 
 
 i=3 
 
 T T 
 
 invariant, so the conic intersection of this and x (21 + U - U ) = k is a 
 
 locus of the rotation. As -> 0, this tends to a circle in a plane perpendicular 
 
 to the subspace of invariant points. 
 
: rm AEC-427 
 
 (6/68) 
 
 AECM 3201 
 
 U.S. ATOMIC ENERGY COMMISSION 
 
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 DISPOSITION OF SCIENTIFIC AND TECHNICAL DOCUMENT 
 
 ( See Instructions on Reverse Side ) 
 
 1. AEC REPORT NO. 
 
 COO-2383-0008 
 
 2. TITLE 
 
 SOME MATRIX RESULTS FOR STABLE m-DIMENSIOWAL ROTATIONS 
 
 3. TYPE OF DOCUMENT (Check one): 
 
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 University of Illinois 
 Urbana, Illinois 6l801 
 
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 April 197^ 
 
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UIUCDCS-R-T^-63U 
 
 3IBLI0GRAPHIC DATA 1- Report No. 
 ' SHEET 
 
 I. Title and Subtitle 
 
 SOME MATRIX RESULTS FOR STABLE m-DIMENSIONAL ROTATIONS 
 
 3. Recipient's Accession No. 
 
 5. Report Date 
 
 April 197*+ 
 
 Author(s) 
 
 C. W. Gear 
 
 8- Performing Organization Rept. 
 No. 
 
 i. Performing Organization Name and Address 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract /Grant No. 
 
 US AEC AT( 11-1)2 383 
 
 2, Sponsoring Organization Name and Address 
 
 US AEC Chicago Operations Office 
 98OO South Cass Avenue 
 Argonne , Illinois 
 
 13. Type of Report & Period 
 Covered 
 
 Preprint -Te chni c al 
 
 14. 
 
 15. Supplementary Notes 
 
 16. Abstracts 
 
 Continuous (real-time) rotation of an object about a 
 line in 3-space can be done using an incremental technique. 
 An equivalent technique in 2-space has a simple modification 
 that makes it stable. The same is shown to be true in m-space, 
 
 17. Key Words and Document Analysis. 17a. Descriptors 
 
 Rotation 
 
 Stability 
 
 Display 
 
 I7b. Idemifiers/Open-Ended Terms 
 
 17c ( 0SAT1 Field/Group 
 
 J18. Availability Statement 
 
 unlimited 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 
 20. Security Class (This 
 
 Page 
 UNCLASSIFIED 
 
 21. No. of Pages 
 
 10 
 
 22. Price 
 
 OHM NTIS-JB (10-70) 
 
 USCOMM-DC 40329-P7 1 
 
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