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 $(\mathbb{R}^2, d_1)$.

$截屏2021-01-28 下午12.49.09.png$

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.