Figure 1.6. The two roads problem

Exercise 1.11. Consider a linkage made of four unit

necting fixed points located at distance d 4; see figu

a) Find the dimension of the configuration space of thi

b) Let d — 3.9. Prove that the configuration space is t

c)* Let d = 1. Prove that the configuration space is th

four handles, that is, a surface of genus 4.

Figure 1.7. A plane linkage

This exercise has convinced you that, although a p

a very simple mechanism, its configuration space may

cated topology. In fact, this topology can be arbitraril

(we do not discuss the exact meaning of this statemen

To conclude this digression, let us mention a very

a line in space, fixed at the origin. The configuration

the real projective plane; see Digression 5.4 for a discuss

is considered in R

n

, then the configuration space is the

space R P

n _ 1

. This space plays a very prominent rol

and topology. Of course, if the line is oriented, then

configuration space is the sphere 5 n - 1 . A