Pál Joints

set height

1

translate by

k

1

k

1

0

rotate by

r

1

if

k

1

= 1

r

1

0

translate by

k

2

if

r

1

= 1

k

2

0

rotate if

k

2

= 1

0

area swept out

area = 0.463648

This Demonstration shows a method called Pál joints that lets you move a segment to a parallel segment through an area approaching 0. If the height is

h

, then the area swept out by the segment is

arctan(1/(h+1))≈1/(h+1)

. Therefore, given two parallel lines, a unit line segment can be moved continuously from one to the other sweeping out a set of arbitrarily small measure.