Rays and Half-Lines

A ray is a segment that does not stop. Stand at a point $A$, pick a direction (say, toward another point $B$), and walk forever. The trail you trace out is a ray, it has a starting endpoint but no terminal one. We have already done the algebraic work for segments; this section adapts it to rays simply by letting the parameter $t$ grow without bound.

The parametric form

Given a starting point $A$ and a "direction" point $B$, the ray from $A$ through $B$ is

$P(t) = (1 - t)A + tB \;=\; A + t(B - A) \quad \text{for } t \geq 0.$

At $t = 0$ you are at $A$; at $t = 1$ you reach $B$; for $t > 1$ you continue past $B$ in the same direction. The constraint $t \geq 0$ is what distinguishes a ray from a segment (which had $0 \leq t \leq 1$) or a full line (which has $t \in \mathbb{R}$).

• Computer Graphics: A ray traced from a camera through a pixel into a scene is mathematically $P(t) = \text{origin} + t \cdot \text{direction}$ with $t \geq 0$; ray-tracing renderers test each ray for intersection with scene geometry to colour the pixel.
• Physics: A laser beam is a physical ray: it starts at the emitter and continues in a straight direction (assuming no obstacles or atmosphere); the $t \geq 0$ constraint excludes "behind the laser" from any meaningful question.
• Optics: Snell's law of refraction traces light rays from the source through interfaces: each segment is a ray with $t \geq 0$ until it hits the next surface, where a new ray is spawned.

Switch to "ray" mode. The widget shows a ray starting at the green anchor and extending in the slope direction. The dashed extension is the "opposite" direction the ray does not include.

Direction vector

The vector $B - A$ is the direction vector of the ray. Any positive scalar multiple of it represents the same direction, so the same ray. Example: with $A = (1, 2)$ and $B = (4, 6)$, the direction vector is $(3, 4)$, and the ray is

$P(t) = (1 + 3t,\; 2 + 4t) \quad \text{for } t \geq 0.$

At $t = 2$ you are at $(7, 10)$, on the same ray but beyond $B$.

Rays from the origin and same-direction tests

A ray from the origin in the direction of a nonzero point $P$ is the set $\{tP : t \geq 0\}$. Two nonzero points $P$ and $Q$ lie on the same ray from the origin precisely when $Q = cP$ for some positive scalar $c$. If $c$ is negative, they lie on opposite rays through the origin, the same straight line, but pointing the wrong way.

Why rays matter

Rays are the building block for two geometric notions you have already met:

• An angle (§5.2) is the wedge between two rays sharing a common endpoint (the vertex).
• A convex set (the kind of region a parallelogram or disc encloses) is characterised by which rays it contains.

Beyond geometry, rays show up in linear programming, convex optimization, and the geometry of light. Half-lines are everywhere.

Try it

• Ray from $A = (2, 1)$ toward $B = (5, 3)$. Direction $(3, 2)$. At $t = 3$: $P = (2, 1) + 3(3, 2) = (11, 7)$.
• Ray from origin through $(3, -1)$: every point on it has the form $(3t, -t)$ for $t \geq 0$. Try $t = 0.5$: $(1.5, -0.5)$.
• Do $(4, 6)$ and $(6, 9)$ lie on the same ray from the origin? Yes, $(6, 9) = 1.5 \cdot (4, 6)$ and $1.5 > 0$.

A trap to watch for

A common confusion is between a ray and its "reverse" ray. The ray from $A$ toward $B$ uses $t \geq 0$. The ray from $A$ in the opposite direction is a different ray, requiring $t \leq 0$ in the same parametrisation. Two rays sharing the same endpoint but pointing opposite ways together form a line. So "the line $AB$" includes both rays, while "the ray from $A$ through $B$" is only one of them. Watch the direction language carefully on a problem, "from $A$ to $B$" and "from $B$ to $A$" describe different rays.

What you now know

You can describe any ray by a parametric formula, identify its direction vector, and decide whether two points lie on the same ray from a given origin. The next section relaxes the parameter range completely, $t \in \mathbb{R}$, to describe lines.

Quick check

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References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 10, §2, rays via the parametric form.
• Coxeter, H. S. M. (1969). Introduction to Geometry, 2nd ed. Wiley. §13 develops half-lines and convex cones.
• Rockafellar, R. T. (1970). Convex Analysis. Princeton University Press. §1 defines rays and uses them for convex set theory.
• Hartshorne, R. (2000). Geometry: Euclid and Beyond. Springer. Chapter 1 reconciles the parametric and synthetic definitions of a ray.