A plane in three-space can be defined by one point in the plane and a normal vector orthogonal to the plane.

Vector Equation of a Plane:

Given a point P0(x0, y0, z0) in the plane and a normal vector n, let P(x, y, z) be an arbitrary point in the plane. Let r and r0 be the position vectors of P and P0 respectively. Subtracting r from r0 gives us a vector inside the plane, which is orthogonal to n. Thus:

n . (r - r0) = 0

or

n . r = n . r0

Scalar Equation of a Plane:

Given a point P0(x0, y0, z0) in the plane and a normal vector n = , let P(x, y, z) be an arbitrary point in the plane. The vector equation then becomes:

a(x-x0) + b(y-y0) + c(z-z0) = 0

Linear Equation of a Plane:

ax + by + cz = d

where d = ax0 + by0 + cz0

This node made possible by Calculus Concepts and Contexts by James Stewart.

Plane (?), n. [F., fr. L. platanus, Gr. &?;, fr. &?; broad; -- so called on account of its broad leaves and spreading form. See Place, and cf. Platane, Plantain the tree.] (Bot.)

Any tree of the genus Platanus.

⇒ The Oriental plane (Platanus orientalis) is a native of Asia. It rises with a straight, smooth, branching stem to a great height, with palmated leaves, and long pendulous peduncles, sustaining several heads of small close-sitting flowers. The seeds are downy, and collected into round, rough, hard balls. The Occidental plane (Platanus occidentalis), which grows to a great height, is a native of North America, where it is popularly called sycamore, buttonwood, and buttonball, names also applied to the California species (Platanus racemosa).

Plane (?), a. [L. planus: cf. F. plan. See Plan, a.]

Without elevations or depressions; even; level; flat; lying in, or constituting, a plane; as, a plane surface.

⇒ In science, this word (instead of plain) is almost exclusively used to designate a flat or level surface.

Plane angle, the angle included between two straight lines in a plane. --
Plane chart, Plane curve. See under Chart and Curve. --
Plane figure, a figure all points of which lie in the same plane. If bounded by straight lines it is a rectilinear plane figure, if by curved lines it is a curvilinear plane figure. --
Plane geometry, that part of geometry which treats of the relations and properties of plane figures. --
Plane problem, a problem which can be solved geometrically by the aid of the right line and circle only. --
Plane sailing (Naut.), the method of computing a ship's place and course on the supposition that the earth's surface is a plane. --
Plane scale (Naut.), a scale for the use of navigators, on which are graduated chords, sines, tangents, secants, rhumbs, geographical miles, etc. --
Plane surveying, surveying in which the curvature of the earth is disregarded; ordinary field and topographical surveying of tracts of moderate extent. --
Plane table, an instrument used for plotting the lines of a survey on paper in the field. --
Plane trigonometry, the branch of trigonometry in which its principles are applied to plane triangles.

Plane, n. [F. plane, L. plana. See Plane, v. & a.]

1. (Geom.)

A surface, real or imaginary, in which, if any two points are taken, the straight line which joins them lies wholly in that surface; or a surface, any section of which by a like surface is a straight line; a surface without curvature.

2. (Astron.)

An ideal surface, conceived as coinciding with, or containing, some designated astronomical line, circle, or other curve; as, the plane of an orbit; the plane of the ecliptic, or of the equator.

3. (Mech.)

A block or plate having a perfectly flat surface, used as a standard of flatness; a surface plate.

4. (Joinery)

A tool for smoothing boards or other surfaces of wood, for forming moldings, etc. It consists of a smooth-soled stock, usually of wood, from the under side or face of which projects slightly the steel cutting edge of a chisel, called the iron, which inclines backward, with an apperture in front for the escape of shavings; as, the jack plane; the smoothing plane; the molding plane, etc.

Objective plane (Surv.), the horizontal plane upon which the object which is to be delineated, or whose place is to be determined, is supposed to stand. --
Perspective plane. See Perspective. --
Plane at infinity (Geom.), a plane in which points infinitely distant are conceived as situated. --
Plane iron, the cutting chisel of a joiner's plane. --
Plane of polarization. (Opt.) See Polarization. --
Plane of projection.
(a) The plane on which the projection is made, corresponding to the perspective plane in perspective; -- called also principal plane.
(b) (Descriptive Geom.) One of the planes to which points are referred for the purpose of determining their relative position in space. --
Plane of refraction or reflection (Opt.), the plane in which lie both the incident ray and the refracted or reflected ray.

Plane, v. t. [imp. & p. p. Planed (?); p. pr. & vb. n. Planing.] [Cf. F. planer, L. planare, fr. planus. See Plane, a., Plain, a., and cf. Planish.]

1.

To make smooth; to level; to pare off the inequalities of the surface of, as of a board or other piece of wood, by the use of a plane; as, to plane a plank.

2.

To efface or remove.

He planed away the names . . . written on his tables.
Chaucer.

3.

Figuratively, to make plain or smooth. [R.]

What student came but that you planed her path.
Tennyson.

Plane, v. i.

Of a boat, to lift more or less out of the water while in motion, after the manner of a hydroplane; to hydroplane.

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