Line Segments

A segment is the simplest geometric object after a point. Pick two points, draw the straight piece between them, and you have a segment. The non-obvious thing, the thing this section exists to teach, is that this segment has a clean algebraic description: every point on it is a weighted average of the two endpoints, controlled by a single parameter $t$ between $0$ and $1$. The parametric idea you learn here scales up to rays, lines, curves, and even multi-dimensional surfaces.

The parametric form

Given endpoints $A$ and $B$, the segment from $A$ to $B$ consists of all points

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

Check the endpoints: $P(0) = A$ (start) and $P(1) = B$ (end). For $t$ strictly between $0$ and $1$ you sweep continuously across the interior. An equivalent rearrangement uses the direction vector $B - A$:

$P(t) = A + t(B - A).$

This makes the geometry transparent: start at $A$, move $t$ of the way toward $B$.

• Computer Graphics: Every line of code that draws a line segment between two pixels, CAD, game UI, data plots, uses the parametric form $P(t) = P_0 + t(P_1 - P_0)$ for t \in [0, 1], rasterised one pixel per step.
• Game Design: Pathfinding for a unit moving from A to B clamps its position to the segment $\overline{AB}$ until it reaches B; the parameter $t$ here is the fraction of the route completed.
• Animation: Linear interpolation (lerp) between two key positions is segment parametrisation: at $t = 0.5$, the character is exactly halfway between start and end. Every smooth motion in modern animation is built from chained lerps.

Switch to "segment" mode in the widget. Drag the anchor $A$; adjust slope and intercept; the line between the green and orange dots is the segment from $A$ to the next-unit point $B$.

The midpoint and other proportional points

The parameter $t$ literally represents the fraction of the way from $A$ to $B$. Some special values:

• $t = 0$: the point $A$.
• $t = 1/4$: a quarter of the way from $A$ to $B$.
• $t = 1/2$: the midpoint, $\tfrac{1}{2}(A + B)$, same as the midpoint formula.
• $t = 2/3$: two-thirds of the way from $A$ to $B$.
• $t = 1$: the point $B$.

For any rational $p/q$, the point $P(p/q)$ divides the segment in ratio $p : (q - p)$ from $A$. This is how segment bisection, trisection, and general division calculations all reduce to one formula.

From point-set to parametric formula

The leap of insight here is that a one-dimensional geometric set (a segment, a curve in the plane, etc.) can be described by a one-dimensional algebraic parameter. The segment is the image of [0, 1] under the function $t \mapsto (1-t)A + tB$. This idea, encoding geometry as a parametric function, is what allows you to describe curves you cannot easily draw, and it returns in calculus and in physics (motion of a particle along a trajectory).

Try it

• For $A = (1, 2)$ and $B = (7, 5)$ at $t = 1/3$: $P = (2/3)(1, 2) + (1/3)(7, 5) = (3, 3)$.
• For $A = (2, -4)$ and $B = (6, 8)$ at $t = 1/2$: midpoint is $(4, 2)$, matching the midpoint formula.
• For $A = (0, -2)$ and $B = (8, 6)$ at $t = 3/4$: $P = (\tfrac{1}{4})(0, -2) + (\tfrac{3}{4})(8, 6) = (6, 4)$.

A trap to watch for

The parameter range $0 \leq t \leq 1$ is what makes the formula represent a segment rather than an entire line. If you let $t$ go negative or above $1$, you get points outside the segment, on the ray extending past $A$ or past $B$. Common confusion: the formula $P(t) = (1 - t)A + tB$ only describes the segment when you restrict $t$; the same formula with unrestricted $t \in \mathbb{R}$ describes the full line. Always check which object you mean. The next two sections formalise rays and lines using exactly this idea.

What you now know

You can describe any segment by a parametric formula, compute the point at any fraction along it (midpoint, quarter, ratio division), and connect the parametric description to the midpoint formula from §8.2. The next section relaxes the parameter range to $t \geq 0$ to describe rays.

Quick check

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References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 10, §1, segments via the parametric form.
• Euclid (c. 300 BCE). Elements, Book I, Definitions 3-4 (Heath translation, Dover, 1956). The axiomatic origin of "segment" between two points.
• Coxeter, H. S. M. (1969). Introduction to Geometry, 2nd ed. Wiley. §13 covers convex combinations, segments as the simplest example.
• Stewart, J. (2015). Calculus, 8th ed. Cengage. Chapter 10 develops parametric curves as a generalisation of this idea.