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2
twodimensional sphere S .
iii) The configuration space of a planar double pendulum is the direct
2
1
1
product of two circles, i.e. the torus T
= S
x S .
iv) The configuration space of a spherical double pendulum is the direct
2
2
product of two spheres S
x S .
v) A rigid segment in the plane has for its configuration space the direct
2
1
product
R
R
x S , which is homeomorphic to the open solid torus.
As we see in the above examples, the configuration space of a mechanical
system is not necessarily homeomorphic to a linear space, but in each case
the points of the configuration space have a neighborhood homeomorphic to an
open ball.
In the following chain of definitions we
fix a positive integer
n.
Definition. Let X be an arbitrary set. A local parameterization of X is an


n
injective mapping
v
:
W
L
X from an open subset
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 Spring '09
 SEDITA

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