New metrics for rigid body motion interpolation - helix

metric (i.e. the kinetic energy) at the Lie algebra so(n) (or se(n , 1) and extending it through left
(right) translations will be equivalent to inheriting the appropriate metric from the ambient space.
3.1 A metric in the ambient space
Let W be a symmetric positive definite n n matrix. For any A 2 GL(n; IR) and any X; Y 2
T A GL(n; IR), define
W;A = Tr(X T YW)=Tr(WX T Y )=Tr(YWX T ) (10)
where the first subscript stands for the chosen matrix weight and the second denotes the point at
which the form is defined. By definition, form (10) is the same at all points in GL(n; IR). Whythe
second subscript then? This will become useful when studying the invariance of the above form. It
is clear that form (10) is quadratic in the entries of X and Y .Letx; y 2 IR n2 be the column vectors
obtained by abutting the transposed lines from X and Y . Then,
W;A = x T Wy;
where
W = blockdiagonal(W T ;W T ;:::;W T ); W 2 IR n2 n 2 :
It is easy to see that W is symmetric and positive definite if and only if W is symmetric and positive
definite. Therefore, (10) is a metric when W is symmetric and positive definite. Some interesting
results are in order.
Proposition 3.1
Metric (10) defined on GL(n; IR) is left invariant when restricted to SO(n).
Metric (10) restricted to SO(n) is bi-invariant if W = I; > 0, I is the n n identity
matrix.
Proof: Let any A; B 2 GL(n; IR) and any X; Y 2 T A GL(n; RR). Then, we have
W;A = Tr(X T YW); W;BA = Tr(X T B T BY W)
from which we conclude that the metric is invariant under left translations with elements B 2
SO(n). When restricted to SO(n), (i.e A 2 SO(n)), the metric becomes left invariant. Biinvariance
is right invariance when the metric is left invariant. Note that
W;A = Tr(YWX T ); W;AB = Tr(YBWB T X T )
9