Sign of Half-Plane is Well-Defined

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\LL$ be a straight line embedded in a cartesian plane $\CC$, given by the equation:

$l x + m y + n = 0$

Let $\HH_1$ and $\HH_2$ be the half-planes into which $\LL$ divides $\CC$.

Let the sign of a point $P = \tuple {x_1, y_1}$ in $\CC$ be defined as the sign of the expression $l x_1 + m y_1 + n$.


Then the sign of $\HH_1$ and $\HH_2$ is well-defined in the sense that:

all points in one half-plane $\HH \in \set {\HH_1, \HH_2}$ have the same sign
all points in $\HH_1$ are of the opposite sign from the points in $\HH_2$
all points on $\LL$ itself have sign $0$.


Proof

By definition of $\LL$, if $P$ is on $\LL$ then $l x_1 + m y_1 + n = 0$.

Similarly, if $P$ is not on $\LL$ then $l x_1 + m y_1 + n \ne 0$.


Let $P = \tuple {x_1, y_1}$ and $Q = \tuple {x_2, y_2}$ be two points not on $\LL$ such that the line $PQ$ intersects $\LL$ at $R = \tuple {x, y}$.

Let $PR : RQ = k$.

Then from Joachimsthal's Section-Formulae:

\(\ds x\) \(=\) \(\ds \dfrac {k x_2 + x_1} {k + 1}\)
\(\ds y\) \(=\) \(\ds \dfrac {k y_2 + y_1} {k + 1}\)
\(\ds \leadsto \ \ \) \(\ds l \paren {k x_2 + x_1} + m \paren {k y_2 + y_1} + n \paren {k + 1}\) \(=\) \(\ds 0\) as these values satisfy the equation of $\LL$
\(\ds \leadsto \ \ \) \(\ds k\) \(=\) \(\ds -\dfrac {l x_1 + m y_1 + n} {l x_2 + m y_2 + n}\)
\(\ds \leadsto \ \ \) \(\ds k\) \(=\) \(\ds -\dfrac {u_1} {u_2}\) where $u_1 = l x_1 + m y_1 + n$ and $u_2 = l x_2 + m y_2 + n$


If $u_1$ and $u_2$ have the same sign, then $k$ is negative.

By definition of the position-ratio of $R$, it then follows that $R$ is not on the ine segment $PQ$.

Hence $P$ and $Q$ are in the same one of the half-planes defined by $\LL$.


Similarly, if $u_1$ and $u_2$ have the opposite signs, then $k$ is positive.

Again by definition of the position-ratio of $R$, it then follows that $R$ is on the ine segment $PQ$.

That is, $\LL$ intersects the ine segment $PQ$.

That is, $P$ and $Q$ are on opposite sides of $\LL$.

Hence $P$ and $Q$ are in opposite half-planes.

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Sign_of_Half-Plane_is_Well-Defined&oldid=512639"