Plane

As explained in Geometry for Programmers

A plane is a flat two-dimensional geometric surface extending indefinitely in two independent directions.

A plane is a flat two-dimensional geometric surface extending indefinitely in two independent directions. In three dimensions, an equation such as 0.3x + 0.2y - z + 3 = 0 describes one by assigning z to every combination of x and y. section 2.3.1

Planes provide a practical geometric language for describing intersections. Two nonparallel planes intersect in a straight line; parallel or coinciding planes do not produce a single intersection line. section 2.3.1

Common representations connect planes with broader geometric constructions:

• An implicit equation describes which points belong to the plane without constructing them one at a time. section 2.3.1
• A three-dimensional line can be represented as the intersection of two planes. section 2.3.1
• A surface can be created by transforming the x-y plane, much like bending an infinite sheet of metal. section 8.2.1
• A parametric surface uses two parameters, commonly written as f(u, v) → (x, y, z). section 8.2.1

A plane equation

The equation z = 0.3x + 0.2y + 3 describes a plane. Rewriting it as 0.3x + 0.2y - z + 3 = 0 makes its implicit form explicit and supports geometric reasoning about intersections. section 2.3.1

Figure 2.17 This plane is z = 0.3x + 0.2x + 3 or 0.3x + 0.2y - z + 3 = 0.

View the illustrated plane equation and its equivalent explicit and implicit forms. section 2.3.1

How Geometry for Programmers covers it

How deep it goes

Defines implicit and explicit equations, explains plane intersections, and connects planes to transformations without developing construction algorithms.

Examples

Uses the plane z = 0.3x + 0.2y + 3, plane intersections, and transformation of the x-y plane into a surface.

What it assumes

Assumes intuition about lines and planes in two and three dimensions while formalizing that intuition with equations.

What you can do after

Evaluate plane equations, recognize their geometric form, and understand lines as intersections of nonparallel planes.

How it connects

Connects planes with lines, linear algebra, systems of linear equations, transformations, and surfaces.

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Related concepts

• hyperplaneincludes · A hyperplane is a flat affine subspace with dimension one less than its ambient space, capable of dividing that space into regions.
• surfaceused by · A surface is a two-dimensional geometric set, often embedded in three-dimensional space or representing a boundary or relation.
• lineproduces · A line is a contiguous sequence of text characters ending at a line boundary or the end of a text stream.
• arrayused by · An array is an indexed, ordered collection of elements arranged in one or more dimensions, with elements accessed by position.

This explainer was generated by Manning's knowledge graph from the text of Geometry for Programmers. The section links point to the parts of the book it draws on.