< User talk:Paul WormerRevision as of 06:00, 30 March 2010 by imported>Paul Wormer
PD Image Equation for plane. X is arbitary point in plane; and are collinear.
In analytic geometry several closely related algebraic equations are known for a plane in three-dimensional Euclidean space. One such equation is illustrated in the figure. Point X is an arbitrary point in the plane and O (the origin) is outside the plane. The point A in the plane is chosen such that vector
is orthogonal to the plane. The collinear vector
is a unit (length 1) vector normal (perpendicular) to the plane. Evidently a is the distance of O to the plane. The following relation holds for an arbitrary point X in the plane
This equation for the plane can be rewritten in terms of coordinates with respect to a Cartesian frame with origin in O. Dropping arrows and hat for component vectors (real triples), we find
with
Conversely, given the following equation for a plane
writing
it follows that
Hence
where f , a, and n are collinear.