Hjelmslev Theorem: What is this about?
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(The black segments in the applet can be dragged by their end points or by any interior point.)
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Copyright © 1996-2012 Alexander BogomolnyThe applet illustrates what's known as Hjelmslev's theorem, named after the notable Danish mathematician Johannes Hjelmslev (1873-1950):
| If two line segments I1 and I2 are related by an isometry f: P1 → P2, then the midpoints between the corresponding points P1, P2 are either all different and collinear or all coincide. |
The applet in fact shows more: the relation between the two segments need not be isometric, it may be just affine. Also, it is not necessary to consider the midpoints of P1P2. All the points that are determined by the same linear combination of the corresponding points are either different and collinear or all coincide.
(The segments in the applet can be moved by either dragging one of their endpoints or any point in-between.)
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Points in a segment AB are parametrized by a linear expression:
| P = P(t) = tA + (1 - t)B, |
where 0 ≤ t ≤ 1. If A and B are different, P(t) is 1-1, for
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Thus P(s) = P(t) implies either t = s or A = B.
A mapping f defined on such a segment AB is affine provided
| f(tA + (1 - t)B) = tf(A) + (1 - t)f(B). |
For two segments A1B1, A2B2, an affine mapping f is defined by
| f(P1) = f(tA1 + (1 - t)B1) = tA2 + (1 - t)B2 = P2. |
For a fixed 0 ≤ r ≤ 1, point P = rP1 + (1 - r)P2 lies on the segment P1P2. Further,
| (1) |
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and therefore belongs to the segment joining
| Ar = rA1 + (1 - r)A2 |
with
| Br = rB1 + (1 - r)B2, |
provided of course the two are different. If they are not, all the linear combinations in (1) coincide. For r = 1/2, we obtain the Hjelmslev Theorem.
References
- H. S. M. Coxeter, Introduction to Geometry, John Wiley & Sons, 1961
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