Ray

As explained in Geometry for Programmers

A ray is a geometric object consisting of an endpoint and all points extending indefinitely in one direction from it.

A ray starts at a point P and extends indefinitely in direction d. Its parametric form is R = P + td, with t >= 0; restricting t to nonnegative values distinguishes the ray from the whole line. section 3.8.2

In 3D geometry, rays provide a practical way to test whether directed paths meet triangles. Raytracers emit one ray per screen pixel, while collision systems use similar tests against triangle meshes. section 3.8.1

A ray-triangle intersection test combines two parameterizations:

• Parameterize the ray as R = P + td, allowing t to range from zero toward infinity. section 3.8.2
• Parameterize a triangle ABC as S = A + uAB + vAC, then reject plane points that fall outside the triangle. section 3.8.2
• Solve the resulting linear system to determine whether the ray reaches the triangle and where the intersection occurs. section 3.10

Testing a directed path

For a ray beginning at P = (0.5, 0.5, 0.5, 1) with direction d = (0, 0, 0, -1), the supplied test reports a true intersection condition; reversing the direction to (0, 0, 0, 1) reports false. section 3.10

Figure 3.15 A ray may or may not intersect a triangle in 3D.

Use the figure to compare a ray that intersects a 3D triangle with one that misses it. section 3.8.1

How Geometry for Programmers covers it

How deep it goes

Defines the parameterization, triangle coordinates, filtering constraints, and linear-system solution, but stops short of implementation details.

Examples

Uses 3D ray-triangle figures, parameterized points, and concrete direction tests solved with SymPy.

What it assumes

Assumes point and vector notation plus 3D line equations, while introducing parameter restrictions and triangle-based coordinates.

What you can do after

Apply parametric ray equations and SymPy-based linear systems to test intersections and calculate candidate points.

How it connects

Connects rays to parametric equations, vectors, triangles, linear equations, ray tracing, and collision detection.

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Related concepts

• parametric curveuses · A parametric curve is a curve whose coordinates are expressed as functions of one or more parameters.
• linepart of · A line is a one-dimensional straight geometric set extending indefinitely in both directions, commonly represented by a linear equation.
• position vectoruses · A position vector is a vector from an origin to a point, representing that point’s location through its components.

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.