Linear Equations and Inequalities

Core Idea

Every cut above is one linear inequality. The line is the equation — n·p = d, satisfied by exactly the points on the boundary. The claimed side is the inequality — n·p ≤ d, satisfied by half of all space. Neither is geometry by itself; the gold region is what happens when several claims are stacked: the points that survive every test at once. That is the whole trick, and graphics runs on it — a view frustum is six such claims in 3D, and every scattered object that fails even one of them never reaches the GPU.

Where Exactly, or Which Side

The two forms answer different questions:

• Equation — ax + by + cz = d — answers “where exactly?” Its solutions form an object one dimension down: a plane in 3D, a line in 2D, a point in 1D. Nothing thick, nothing filled — a boundary.
• Inequality — ax + by + cz ≤ d — answers “which side?” Its solutions are a half-space: an infinite region, everything on one side of that boundary (plus the boundary itself, if the ≤ is soft).

An engine asks the second question constantly and the first one rarely. Exact intersections (a ray hitting a plane at one point) are equation problems; visibility, containment, culling, and collision are all side-of-the-boundary problems.

The Coefficients Are the Normal

The unglamorous form ax + by + cz = d hides the geometry. Group the coefficients into a vector and it snaps into focus:

n · p = d        where n = (a, b, c)

The coefficient vector is the plane’s normal — perpendicular to the boundary, pointing toward the positive side. And the leftover expression n·p − d is not just a test, it’s a measurement: zero on the plane, positive on the side n points to, negative behind — and when n is unit length, its magnitude is the exact distance to the boundary.

That single dot product is the atom of spatial reasoning: one multiply-add per axis, and you know which side a point is on and how far away it sits. Evaluate it over all of space instead of at one point and you have a signed distance field — the plane SDF p.y from that article is exactly n·p − d with n = (0,1,0), d = 0.

Geometry by Stacking Claims

One inequality claims half of space. Intersect a few claims and the surviving region becomes a shape — this is how graphics builds its workhorse volumes without storing any of them as meshes:

• A triangle is three half-planes. The inside-a-triangle test — the heart of rasterization’s coverage question — is three edge inequalities that must all agree; the barycentric inside-test 0 ≤ α, β, γ in meshes is the same three claims in different clothes.
• An AABB is six axis-aligned inequalities. x_min ≤ x ≤ x_max and friends — which is why box collision checks are so cheap: six sign comparisons, no geometry touched.
• The view frustum is six half-spaces. Left, right, top, bottom, near, far — each a plane equation with its inequality facing inward. Frustum culling tests a bounding volume against the six: entirely outside any single plane means the object is skipped before a vertex is processed. The actual per-vertex clipping mechanics live in clip space.
• A circle’s interior is one (non-linear) claim. length(p) < r — the same which-side idea with a curved boundary, which is the doorway to SDFs: keep the measurement, not just the verdict.

One property makes all of this survive motion: affine transformations preserve sidedness. Transform a shape and its points don’t leak across its planes — an inside point stays inside — so claims tested in whatever space is cheapest are still honest.

Connections

This is the load-bearing math one layer under everything else here: the plane test n·p − d is a dot product wearing work clothes; promote the verdict to a measurement and you get signed distance; stack three claims and you get the rasterizer’s edge test; stack six and you get the frustum whose clipping story belongs to clip space. The matrices that move geometry between spaces leave all of these verdicts intact — which is the quiet reason the whole pipeline can keep asking “which side?” in whichever space is cheapest.