ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002
rotations by c-€ = 0. The remaining velocity vector fields
are said to belong to infinitesimal helical motions. At all
instants, the velocity vector field of a smooth motion be-
longs to one of the three cases, if it is nonzero.
It turns out that it is useful to study path normals of mo-
tions, i.e. lines that are orthogonal to the velocity vector
of one of their points. Here it is convenient to describe
lines by their Plücker coordinates. If a line G contains
a point p and is parallel to the vector v, then the pair
(gg) — (v, p x v) is called its Plücker coordinate vector.
It is easy to see that g does not depend on the particular
choice of p. The Plücker coordinate vector is unique only
up to scalar multiples. Its two components g, g are not in-
dependent, but fulfill the relation g - g — 0. A point x is
contained in G if and only if x x g equals g.
Connections between Plücker coordinates of lines and ve-
locity vector fields are shown by the following two lem-
mas. For a more detailed treatment, see (Pottmann Wall-
ner, 2001).
Lemma 1 4 Jine with Plücker coordinates (g, g) is a path
normal of a smooth motion (c, c), if and only if e-g- €: g =
0.
Lemma 2 /f (c, c) represents the velocity vector field of a
uniform rotation or helical motion, then the Plücker coor-
dinates (g, g) of the axis, the angular velocity w and the
pitch p are reconstructed by
(g,g) = (c, € — pe). (2)
p-e-e/c, w=|lel,
2.) Approximation of a set of lines by a linear line
complex
We consider a set of lines Ny, Ns, .. ., represented by their
Plücker coordinates (n;,m;). Here, the direction vectors
shall be normalized, n? — 1.
We would like to approximate these lines with a linear
complex C' which consists of all lines (x, X) which sat-
isfy the linear equation c- X -- €- x — 0. C shall be
represented by the coefficients (c, c) of this equation. Us-
ing as a deviation measure of a line N to a linear com-
plex C the so-called moment M(N, C) (Pottmann Wall-
ner, 2001), the minimization of the squared sum of mo-
ments 5 / M (Ni, C) amounts to the minimization problem
F(e,€) - V (e-n;-€.nj? 2 min, | (lJel| ^ 1). (3)
F' is a quadratic function of six real arguments, and the
side condition ||e|| = 1 is also quadratic. We can therefore
rewrite Equ. (3) in the form
(6, €) - K - (e, €) — min, (¢,97-D(c,®) =1, (4
with two (6 x 6)-matrices K and D. The matrix D has
nonzero entries only in its upper left 3 x 3 corner. The
solution of this problem is straightforward. The minimum
is assumed for (c, €) which fulfills
(K — AD) - (e, €) = (0,0),
det( —AD) 2 0,
llell = 1,
A minimal. (5)
This means that we have to choose the smallest solution
A of the cubic equation det(X — AD) = 0 and solve the
equation (K — AD)(c, €) — (0, 0). Details can be found
in (Pottmann Randrup, 1998, Pottmann Wallner, 2001).
3 APPROXIMATION IN THE SET OF PLANES
Problems in geometric computing which involve sets of
planes can sometimes be transformed to problems for
points by application of a projective duality. This is true
as long as only projective and algebraic properties are in-
volved. As soon as we perform approximation, we need
distance measures, also in the space of planes.
We have shown in earlier papers how to solve this problem
by introducing a Euclidean metric in the space of planes
(see e.g. (Pottmann Wallner, 2001)): For that, it is neces-
sary to remove a bundle of planes, which in our approach
are the planes parallel to the z-axis in an appropriate Carte-
sian coordinate system (z, y, z). Planes not parallel to the
z-axis can be written in the form
Z — uo 4 u1z 4 usy. (6)
We see that (uo, u1, u») are affine coordinates in the result-
ing affine space A* of planes not parallel to the z-axis.
We will now introduce a Euclidean metric in A*. Thereby
we make sure that the deviation between two planes shall
be measured within some region of interest. This region
shall be captured by its projection I" onto the xy-plane.
For a positive measure 1 in R? we define the distance d "
between planes À = (ao, a1, a») and B = (bo, by, bs) as
d, (A, B) = ||(ao — bo) + (a1 — b1)z + (az — b2)yll 2),
(7)
Le. the L?(4)-distance of the linear functions whose
graphs are A and B. This, of course, makes sense only if
the linear function which represents the difference between
the two planes is in L? (11). We will always assume that the
measure 4 1s such that all linear and quadratic functions
possess finite integral.
Figure 1: To the definition of the deviation of two planes.
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