Definition:Between (Geometry)

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Definition

Betweenness is one of the undefined terms in Tarski's Geometry.

Intuitively, a point $b$ is between two others $a$ and $c$ if and only if it lies on the line segment $a c$.

However as line segment has not yet been defined, we are not allowed to call upon it at this stage.

We offer an ostensive definition:

In the picture, point $b$ is between the two points $a, c$, and we write:

$\mathsf{B}abc$

However, point $d$ is not between the two points $a$ and $c$, and we write:

$\map \neg {\mathsf B a d c}$

In Euclidean $2$-Space

Define the following coordinates in the $xy$-plane:

\(\ds a\)

\(=\)

\(\ds \tuple {x_1, x_2}\)

\(\ds b\)

\(=\)

\(\ds \tuple {y_1, y_2}\)

\(\ds c\)

\(=\)

\(\ds \tuple {z_1, z_2}\)

where $a, b, c \in \R^2$.

Let:

\(\ds \Delta x_1\)

\(=\)

\(\ds x_3 - x_2\)

\(\ds \Delta x_2\)

\(=\)

\(\ds x_2 - x_1\)

\(\ds \Delta y_1\)

\(=\)

\(\ds y_2 - y_1\)

\(\ds \Delta y_2\)

\(=\)

\(\ds y_3 - y_2\)

Then:

$\mathsf{B}abc \dashv \vdash \paren {\Delta x_1 \Delta y_1 = \Delta x_2 \Delta y_2} \land$

$\paren {0 \le \Delta x_1 \Delta y_1 \land 0 \le \Delta x_2 \Delta y_2}$

As a justification of this definition, consider the case where $\Delta x_1, \Delta x_2 \ne 0$.

\(\ds \Delta x_1 \Delta y_1\)

\(=\)

\(\ds \Delta x_2 \Delta y_2\)

\(\ds \leadsto \ \ \)

\(\ds \frac {\Delta y_1}{\Delta x_2}\)

\(=\)

\(\ds \frac {\Delta y_2}{\Delta x_1}\)

Hence, the right triangles with hypotenuses $ab$ and $bc$ are similar.

Furthermore, the hypotenuses are parallel, because they have the same slope.

They are similarly oriented because $\Delta x$ is by construction parallel to the $x$-axis, $\Delta y$ to the $y$-axis.

They are touching because there are only three points under consideration.

Lastly, the inequalities assure that vertex $b$ lies between the two triangles, because otherwise the inequalities wouldn't hold.

Work In Progress In particular: I didn't mean prove it here, I meant raise a page where the definition is shown to be justified. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{WIP}} from the code.

In Euclidean $n$-Space

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In the Euclidean $n$-dimensional case, consider three points $A$,$B$ and $C$. We say that $B$ is between $A$ and $C$ if and only if:

$\norm {\vec{AB} } < \norm {\vec{AC} }$

and:

$\vec{AB} \cdot \vec{AC} = \norm {\vec{AB} } \norm {\vec{AC} }$

For intuition behind definition see Definition:Between (Geometry)/N-dimensional Euclidean space Intuition

Work In Progress In particular: This is a personal suggestion of a definition by an author. This may not be an official definition, but the intuition behind the definition is justified. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{WIP}} from the code.

Sources

• June 1999: Alfred Tarski and Steven Givant: Tarski's System of Geometry (Bull. Symb. Log. Vol. 5, no. 2: pp. 175 – 214) : p. $201$