Hjelmslev's Theorem:
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A Mathematical Droodle

(The black segments in the applet can be dragged by their end points or by any interior point.)

Explanation

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Copyright © 1996-2018 Alexander Bogomolny

The applet illustrates what's known as Hjelmslev's theorem, named after the notable Danish mathematician Johannes Hjelmslev (1873-1950):

If two line segments $I_{1}$ and $I_{2}$ are related by an isometry $f: P_{1} \rightarrow P_{2},$ then the midpoints between the corresponding points $P_{1},$ $P_{2}$ are either all different and collinear or all coincide.

Hjelmslev's theorem

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 $P_{1}P_{2}.$ All the points that are determined by the same linear combination of the corresponding points are either different and collinear or all coincide.

Points in a segment $AB$ are parametrized by a linear expression:

$P = P(t) = tA + (1 - t)B,$

where $0 \le t \le 1.$ If $A$ and $B$ are different, $P(t)$ is 1-1, for

$\begin{align} P(t) - P(s) &= (t - s)A + (s - t)B\\ &= (t - s)(A - B). \end{align}$

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 $A_{1}B_{1},$ $A_{2}B_{2},$ an affine mapping $f$ is defined by

$f(P_{1}) = f(tA_{1} + (1 - t)B_{1}) = tA_{2} + (1 - t)B_{2} = P_{2}.$

For a fixed $r,$ $0 \le r \le 1,$ point $P = rP_{1} + (1 - r)P_{2}$ lies on the segment $P_{1}P_{2}.$ Further,

(1)

$\begin{align} P &= rP_{1} + (1 - r)P_{2}\\ &= r[tA_{1} + (1 - t)B_{1}] + (1 - r)[tA_{2} + (1 - t)B_{2}]\\ &= t[rA_{1} + (1 - r)A_{2}] + (1 - t)[rB_{1} + (1 - r)B_{2}]\\ &= tA_{r} + (1 - t)B_{r} \end{align}$

and therefore belongs to the segment joining

$A_{r} = rA_{1} + (1 - r)A_{2}$

with

$B_{r} = rB_{1} + (1 - r)B_{2},$

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's Theorem.

(A JavaScript illustration is available elsewhere.)

References

  1. H. S. M. Coxeter, Introduction to Geometry, John Wiley & Sons, 1961

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Copyright © 1996-2018 Alexander Bogomolny
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