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# 1.5: Isometries, motions, and lines

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In this section, we define lines in a metric space. Once it is done the sentence “We can draw a unique line that passes thru two given points.” becomes rigorous; see (ii) in Section 1.1.

Recall that a map $$f: \mathcal{X} \to \mathcal{Y}$$ is a bijection, if it gives an exact pairing of the elements of two sets. Equivalently, $$f: \mathcal{X} \to \mathcal{Y}$$ is a bijection, if it has an inverse; that is, a map $$g: \mathcal{Y} \to \mathcal{X}$$ such that $$g(f(A)) = A$$ for any $$A \in \mathcal{X}$$ and $$f(g(B)) = B$$ for any $$B \in \mathcal{Y}$$.

Let $$\mathcal{X}$$ and $$\mathcal{Y}$$ be two metric spaces and $$d_{\mathcal{X}}$$, $$d_{\mathcal{Y}}$$ be their metrics. A map

$f: \mathcal{X} \to \mathcal{Y}$

is called distance-preserving if

$d_{\mathcal{Y}} (f(A), f(B)) = d_{\mathcal{X}} (A, B)$

for any $$A, B \in \mathcal{X}$$.

Definition

A bijective distance-preserving map is called an isometry.
Two metric spaces are called isometric if there exists an isometry from one to the other.
The isometry from a metric space to itself is also called a motion of the space.

Exercise $$\PageIndex{1}$$

Show that any distance-preserving map is injective; that is, if $$f: \mathcal{X} \to \mathcal{Y}$$ is a distance-preserving map, then $$f(A) \ne f(B)$$ for any pair of distinct points $$A, B \in \mathcal{X}$$.

Hint

If $$A \ne B$$, then $$d_{\mathcal{X} (A, B) > 0$$. Since $$f$$ is distance-preserving,

$$d_{\mathcal{Y}} (f(A), f(B)) = d_{\mathcal{X}} (A, B)$$.

Therefore, $$d_{\mathcal{Y}} (f(A), f(B)) > 0$$; hence $$f(A) \ne f(B)$$.

Exercise $$\PageIndex{2}$$

Show that if $$f: \mathbb{R} \to \mathbb{R}$$ is a motion of the real line, then either

(a) $$f(x) = f(0) + x$$ for any $$x \in \mathbb{R}$$, or

(b) $$f(x) = f(0) - x$$ for any $$x \in \mathbb{R}$$.

Hint

Set $$f(0) = a$$ and $$f(1) = b$$. Note that $$b = a + 1$$ or $$a - 1$$. Moreover, $$f(x) = a \pm x$$ and at the same time, $$f(x) = b \pm (x - 1)$$ for any $$x$$.

If $$b = a + 1$$, it follows that $$f(x) = a + x$$ for any $$x$$.
The same way, if $$b = a - 1$$, it follows that $$f(x) = a - x$$ for any $$x$$.

Exercise $$\PageIndex{3}$$

Prove that $$(\mathbb{R}^2, d_1)$$ is isometric to $$(\mathbb{R}^2, d_{\infty})$$.

Hint

Show that the map $$(x, y) \mapsto (x + y, x - y)$$ is an isometry $$(\mathbb{R}^2, d_1) \to (\mathbb{R}^2, d_{\infty})$$. That is, you need to check if this map is bijective and distance-preserving.

Advanced Exercise $$\PageIndex{4}$$

Describe all the motions of the Manhattan plane, defined in Section 1.4.

Hint

First prove that two points $$A = (x_A, y_A)$$ and $$B = (x_B, y_B)$$ on the Manhattan plane have a unique midpoint if and only if $$x_A = x_B$$ or $$y_A = y_B$$; compare with the example in Congruent triangles.

Then use above statement to prove that any motion of the Manhattan plane can be written in one of the following two ways:

$$(x, y) \mapsto (\pm x + a, \pm y + b)$$       or       $$(x, y) \mapsto (\pm y + b, \pm x + a)$$,

for some fixed real numbers $$a$$ and $$b$$. (In each case we have 4 choices of signs, so for a fixed pair $$(a, b)$$ we have 8 distinct motions.)

If $$\mathcal{X}$$ is a metric space and $$\mathcal{Y}$$ is a subset of $$\mathcal{X}$$, then a metric on $$\mathcal{Y}$$ can be obtained by restricting the metric from $$\mathcal{X}$$. In other words, the distance between two points of $$\mathcal{Y}$$ is defined to be the distance between these points in $$\mathcal{X}$$. This way any subset of a metric space can be also considered as a metric space.

Definition

A subset $$l$$ of metric space is called a line, if it is isometric to the real line.

A triple of points that lie on one line is called collinear. Note that if $$A$$, $$B$$, and $$C$$ are collinear, $$AC \ge AB$$ and $$AC \ge BC$$, then $$AC = AB + BC$$.

Some metric spaces have no lines, for example discrete metrics. The picture shows examples of lines on the Manhattan plane $$(\mathbb{R}^2, d_1)$$. Exercise $$\PageIndex{5}$$

Consider the graph $$y = |x|$$ in $$\mathbb{R}^2$$. In which of the following spaces (a) $$(\mathbb{R}^2, d_1)$$, (b) $$(\mathbb{R}^2, d_2)$$, (c) $$(\mathbb{R}^2, d_{\infty})$$ does it form a line? Why?

Hint

Assume three points $$A, B$$, and $$C$$ lie on one line, Note that in this case one of the triangle inequalities with the points $$A, B$$, and $$C$$ becomes an equality.

Set $$A = (-1, 1)$$, $$B = (0, 0)$$, and $$C = (1, 1)$$. Show that for $$d_1$$ and $$d_2$$ all the triangle inequalities with the points $$A, B$$, and $$C$$ are strict. It follows that the graph is not a line.

For $$d_{\infty}$$ show that $$(x, |x|) \mapsto x$$ gives the isometry of the graph to $$\mathbb{R}$$. Conclude that the graph is a line in $$(\mathbb{R}^2, d_{\infty})$$.

Exercise $$\PageIndex{6}$$

Show that any motion maps a line to a line.

Hint

Spell the definitions of line and motion.