Line equation

As explained in Geometry for Programmers

A line equation is a mathematical relation whose solution set represents a geometric line.

A line equation describes the points belonging to a geometric line. The explicit form y = ax + b is easy to plot and interpret: a controls slope, while b controls the vertical intercept. It cannot represent a strictly vertical line, which requires x = c. section 2.3.1

Choose a representation according to the geometry and dimension:

• Use the general 2D form a₁x + a₂y + a₃ = 0 to represent every line without switching formulas for vertical cases. section 2.3.1
• Use a canonical form with a point and direction coefficients when those geometric ingredients are more useful, especially for later 3D work. section 2.3.1
• Treat ax + by = c as a line in 2D; two such equations form intersecting lines whose common point solves the system. section 3.1.1
• Do not generalize the 2D general form to a 3D line by merely adding z: ax + by + cz = d describes a plane, while a 3D line requires two equations. section 2.3.1

Two-line intersection

The equations Vp − 2Va = 0 and Va + Vp − 450 = 0 represent two lines. Their intersection, (Va, Vp) = (300, 150), is the solution to the system. section 3.1.1

Figure 2.15 An explicit line equation: y = 0.5x + 1

Plot the explicit line y = 0.5x + 1, then compare it with the general and vertical-line cases. section 2.3.1

How Geometry for Programmers covers it

How deep it goes

Explains explicit, general, and canonical forms, then stops short of a single-equation 3D line representation.

Examples

Uses y = 0.5x + 1, vertical lines, a two-trains system, and intersections of nonparallel planes.

What it assumes

Assumes familiarity with variables and coordinates while teaching geometric meaning through slopes, coefficients, and intersections.

What you can do after

Evaluate line representations, interpret linear-system solutions geometrically, and recognize when a 3D equation describes a plane instead.

How it connects

Connects lines with linear equations, linear systems, planes, hyperplanes, and intersections.

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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.