Transformations - Transformations and Projections in Computer Graphics - page 47

n=3;
A=[.5774,-.5774,-.5774; .5774,.7886,-.2115; .5774,-.2115,.7886]
% Rotation from 1,1,1 to x-axis
Q=eye(n);
for j=1:n-1,
for i=n:-1:j+1,
T=eye(n);
D=sqrt(A(j,j)^2+A(i,j)^2);
cos=A(j,j)/D; sin=A(i,j)/D;
T(j,j)=cos; T(j,i)=sin; T(i,j)=-sin; T(i,i)=cos; T
A=T*A;
Q=Q*T';
end;
end;
Q
A

Figure 1.28: Computing Three Givens Matrices.

T1=[0.7071,0,0.7071; 0,1,0; -0.7071,0,0.7071];
T2=[0.8165,0.5774,0; -0.5774,0.8165,0; 0,0,1];
T3=[1,0,0; 0,0.9660,0.2587; 0,-0.2587,0.9660];
p=[1;1;1];
a=T1*p
b=T2*a
c=T3*b
Figure 1.29: Rotating Point (1 , 1 , 1) to the x Axis.

J. Wallace Givens, Jr. (1910-1993) pioneered the use of plane rotations in the early
days of automatic matrix computations. Givens graduated from Lynchburg College
in 1928, and he completed his Ph.D. at Princeton University in 1936. After spending
three years at the Institute for Advanced Study in Princeton as an assistant of Oswald
Veblen, Givens accepted an appointment at Cornell University, but later moved to
Northwestern University. In addition to his academic career, Givens was the director
of the Applied Mathematics Division at Argonne National Laboratory and, like his
counterpart Alston Scott Householder at Oak Ridge National Laboratory, Givens
served as an early president of SIAM. He published his work on the rotations in 1958.
—Carl D. Meyer

dimensions is similar to multiplying two complex numbers because the product
( a, b ) cd
−

=( ac

− bd, ad + bc )

dc