Definition:Position-Ratio of Point
Definition
Let $P$ and $Q$ be points in space.
Let $R$ be a point on $PQ$ dividing $PQ$ in the ratio:
- $PR : RQ = l : m$
where:
- $l$ is the length of the vector $PR$ in the direction of $PQ$
- $m$ is the length of the vector $RQ$ in the direction of $PQ$.
Then $PR : RQ$ is referred to as the position-ratio of $R$ with respect to the base-points $P$ and $Q$.
When $R$ is between $P$ and $QA$, the position-ratio is positive.
When $R$ is outside the segment $PQ$, the position-ratio is negative:
Value of Position-Ratio
The value of the position-ratio can be expressed as follows:
Let $k$ denote the position-ratio of $R$.
Then:
- $k = \dfrac {PQ} {RQ} - 1$
Examples
Let $P$ and $Q$ be points.
Let $R$ be a point on the straight line passing through $P$ and $Q$.
Let $k$ denote the position-ratio of $R$.
$R$ Approaching $P$
As $R$ approaches $P$, $k \to 0$.
$R$ Approaching $Q$
As $R$ approaches $Q$ where $R$ is on the line segment $PQ$, $k \to +\infty$.
As $R$ approaches $Q$ where $R$ is not on the line segment $PQ$, $k \to -\infty$.
$R$ Approaching Infinity
As $R$ approaches the point at infinity on the left, $k \to -1$ from above.
As $R$ approaches the point at infinity on the right, $k \to -1$ from below.
$R$ Midway between $P$ and $Q$
When $R$ is on the midpoint of the line segment $PQ$, $k = +1$.
This is the centroid of $P$ and $Q$.
Also see
- Results about position-ratios can be found here.
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text I$. Coordinates: $11$. Position-ratio of a point