Definition:Between (Geometry)

From ProofWiki
Jump to navigation Jump to search


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:


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


\(\ds \Delta x_1\) \(=\) \(\ds x_3 - x_2\)
\(\ds \Delta x_2\) \(=\) \(\ds x_2 - x_1\)
\(\ds \Delta y_1\) \(=\) \(\ds y_2 - y_1\)
\(\ds \Delta y_2\) \(=\) \(\ds y_3 - y_2\)


Betweenness(Analytic Def'n).png
$\mathsf{B}abc \dashv \vdash \paren {\Delta x_1 \Delta y_1 = \Delta x_2 \Delta y_2} \land$
$\paren {0 \le \Delta x_1 \Delta y_1 \land 0 \le \Delta x_2 \Delta y_2}$

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.

In Euclidean $n$-Space

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:

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


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

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