Derivation/Calculation From The Normal Form
Note: For simplicity, the following derivation discusses the 3D case. However, it is also applicable in 2D.
In the normal form,
a plane is given by a normal vector as well as an arbitrary position vector of a point . The direction of is chosen to satisfy the following inequality
By dividing the normal vector by its Magnitude, we obtain the unit (or normalized) normal vector
and the above equation can be rewritten as
Substituting
we obtain the Hesse normal form
In this diagram, d is the distance from the origin. Because holds for every point in the plane, it is also true at point Q (the point where the vector from the origin meets the plane E), with, per the definition of the Scalar product
The magnitude of is the shortest distance from the origin to the plane.
Read more about this topic: Hesse Normal Form
Famous quotes containing the words calculation, normal and/or form:
“Common sense is the measure of the possible; it is composed of experience and prevision; it is calculation appled to life.”
—Henri-Frédéric Amiel (1821–1881)
“A normal adolescent is so restless and twitchy and awkward that he can mange to injure his knee—not playing soccer, not playing football—but by falling off his chair in the middle of French class.”
—Judith Viorst (20th century)
“The Republicans hardly need a party and the cumbersome cadre of low-level officials that form one; they have a bankroll as large as the Pentagon’s budget, dozens of fatted PACs, and the well-advertised support of the Christian deity.”
—Barbara Ehrenreich (b. 1941)