Line Intersection Calculator

What does this line intersection calculator compute?

Two lines A₁x + B₁y + C₁ = 0 and A₂x + B₂y + C₂ = 0 meet at one point when A₁B₂ − A₂B₁ ≠ 0, are parallel and empty when that determinant is 0 and the lines differ, and coincide when they are the same line.

The published formula or method is A₁x + B₁y + C₁ = 0 ∩ A₂x + B₂y + C₂ = 0. The following references support the stated mathematics or method:

• Margaret Stevens — The Maths Workbook (University of Oxford): Linear-function slopes and point-slope equations, and simultaneous linear equations in two variables. Line intersection is the corresponding system solution; vertical lines use an implicit rather than finite-slope representation. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.

Which inputs and limits should you check?

Keep both points and line coefficients in the same coordinate system. Parallel, coincident, or degenerate lines require their own result classifications.

The worked example uses (0, 0)–(2, 2) and (0, 2)–(2, 0) with Δ ≠ 0. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.geometry.coordinate v1.8.0 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Numerical domain and interpretation

Coordinates are Cartesian values in one consistent length unit. Inputs accept finite decimal numbers or decimal strings; nonzero coordinates and published numeric outputs must be normal binary64 (absolute value at least 2⁻¹⁰²²). Nonzero text that converts to zero is rejected as NUMERIC_UNDERFLOW. NUMERIC_RANGE means translate or rescale your inputs. Explicit empty fields are errors, not defaults.

All seven operations use the same engine. Distance and midpoint allow identical points; slope and line operations require distinct defining points. A vertical line has null slope and intercept. Intersections concern infinite lines, not segments: parallel distinct lines return empty, coincident lines return coincident, and both have null intersection coordinates.

Exact dyadic predicates classify the normalized binary64 directions without an absolute geometric epsilon. The principal undirected angle is atan2(|d₁×d₂|, |d₁·d₂|), in [0°,90°], with radians also available. A rounded angle of90° does not itself imply an exact perpendicular classification. Nearly parallel lines are sensitive to input rounding; exact predicates do not recover decimal information lost before computation. Typical well-conditioned scalar comparisons use relative2e−12; no input perturbation or intersection-position bound is claimed.

Line coefficients use integer gcd only for exact safe integers; otherwise they share a power-of-two normalization and canonical sign. Display equations are floating presentations. The uniformly scaled figure is schematic; close labels may merge, and an unrepresentable plotting range is explicitly omitted. Complete numerical results and raw supplied inputs are available in exports.

Browser calculations run locally. A shared URL sends its parameters to the server and may appear in service logs; REST/MCP also send inputs. This page makes no claim about a fixed log retention period or processing region.