With
Q
1
=
pair
1
+ pair
1

pair
1
· e
∞
(43)
and
Q
2
=
pair
2
− pair
2

pair
2
· e
∞
(44)
we can then directly pick the conformal points Q
1
and
Q
2
out of the point pairs pair
1
and pair
2
.
If (like in subsection 6.2) the second sphere S
2
happens to be only a point (sphere with zero radius),
we can omit the calculation of Q
2
in Equs. (39), (42)
and (44). We simply replace Q
2
= S
2
in Equ. (40).
6.4 Comparison with Welzl’s Bounding
Sphere Algorithm
The iteration of the method suggested in subsection
6.3 (test of inclusion, and if necessary the calcula
tion of the new bounding sphere) results in the ﬁnal
bounding sphere of n points in linear time O(n). The
proposed method is easy to understand and with given
routines for inner and outer products easy to imple
ment. As demonstrated in subsection 6.2 the algo
rithm can be further optimized with Maple. In aver
age Welzl’s algorithm (Welzl, 1991) also runs in as
ymptotically linear time, but the recursion in Welzl’s
algorithm makes it harder to examine and guarantee
the performance time.
7 CONCLUSIONS
We presented a bunch of basics and algorithms ex
pressed in the mathematical framework of conformal
geometric algebra. We are convinced that its easy
handling of geometric objects like spheres, circles or
planes, its easy handling of distances and angles be
tween them as well as its way of ﬁtting and bounding
of geometric objects will provide a promising foun
dation for the analysis of point clouds.
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