Definition:Point at Infinity/Homogeneous Cartesian Coordinates
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Definition
The point at infinity is expressed in homogeneous Cartesian coordinates by an ordered triple in the form:
- $\tuple {X, Y, Z}$
where:
- $Z = 0$
- $X$ and $Y$ are arbitrary.
Point on Line
Let $\LL$ be a straight line embedded in a cartesian plane $\CC$.
Let $\LL$ be given in homogeneous Cartesian coordinates by the equations:
- $l X + m Y + n Z = 0$
The point at infinity is expressed in homogeneous Cartesian coordinates by an ordered triple in the form:
- $\tuple {-m, l, n}$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $9$. Parallel lines. Points at infinity