Geometry Through History by Meighan I. Dillon

Author:Meighan I. Dillon
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(6.2)

where |PS| is the Euclidean distance between points P and S, (or the Euclidean length of PS), and is the natural logarithm.

Theorem 6.6.

L is a metric on .

Proof.

Notice first that when , , and . Note further that by the definition, the L-distance between points is always nonnegative.

The triangle inequality and condition (2) of Definition 6.2 are easy to show using properties of logarithms and absolute values. Likewise, if P, Q determine a type-1 line, it is easy to show that implies . We leave those verifications to the exercises. We must show that condition (1) holds if P, Q determine a type-2 line.

Suppose then that lie on a type-2 line in . Notice that and

Figure 6.6:A type-2 line in and the underlying set in

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