Points, Lines, and Planes

Three starting objects

Imagine marking a location on a tabletop, tracing a straight path through it, and considering the whole flat surface of the table. These suggest a point, a line, and a plane.

We take point, line, and plane as undefined terms: starting ideas whose behavior we describe and accept, rather than define using simpler geometric objects. The descriptions below help you picture them. They are not definitions built from other terms.

A pencil dot has width, but the point it represents does not. A drawn line stops at the edge of the page, but the geometric line continues. A slanted parallelogram can represent a plane; its outline is a drawing boundary, not an edge of the plane.

Points are collinear when one line contains them. They are coplanar when one plane contains them. Three points that do not all lie on one line are noncollinear.

A line, a ray, and a segment

On a straight road, you could describe the whole path, the part beginning at one location and continuing in one direction, or just the part between two locations. Geometry gives each a different name.

We take the order of points along a line, including betweenness, as another starting idea. For distinct points occurring in order $A,B,C$, the point $B$ is between $A$ and $C$. This order is available before we define a segment.

A segment $\overline{AB}$ consists of endpoints $A$ and $B$ and all points between them. A ray $\overrightarrow{AB}$ begins at endpoint $A$, passes through the distinct point $B$, and continues without end in that direction.

The first letter of a ray matters: it names the endpoint. The second chooses a direction, not a stopping place. If $A$, $B$, and $C$ occur in that order on a line, rays $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are the same ray. Rays $\overrightarrow{BA}$ and $\overrightarrow{BC}$ are opposite rays: they share endpoint $B$ and together form the whole line.

Facts we accept at the start

Two distinct pinholes fix where a straight thread must run. Three pinholes all on that thread still leave a flat sheet free to turn around it. A third pinhole off the thread fixes the sheet’s position. These models help explain why the conditions below matter, although a physical model is not a proof.

An incidence relationship says which points belong to which lines or planes. We accept the following as starting facts, also called postulates. A postulate is a fact we agree to use without proving it within this course.

We also accept that every line contains at least two distinct points, every plane contains at least three noncollinear points, and three-dimensional space contains points that do not all lie in one plane. These prevent the objects from collapsing into smaller ones.

One point alone does not determine a unique line. Three collinear points do not determine a unique plane: many planes can turn around their common line.

Two distinct lines share at most one point. Two distinct points already fix one line, so they leave no choice of a second different line through both. This is a proved result from accepted line uniqueness.

Lines in space

Two straight lines drawn on a sheet may meet or stay apart. Space gives us another possibility: they can pass in different directions without sharing a plane.

• Intersecting lines share exactly one point. If their meeting makes a square corner, they are perpendicular, written $\ell \perp m$. The term right angle also names that square corner; the next angle lesson develops its measurement.
• Parallel lines are distinct, coplanar lines with no common point, written $\ell \parallel m$.
• Skew lines are lines that do not lie together in any plane. They are not parallel, and we prove below that they cannot intersect.
• Coincident lines are two names for the same line. They share every point, rather than exactly one point.

The rectangular prism below lets you compare actual relationships. For this model, take the familiar box facts as givens: every face is a rectangle with parallel opposite edge lines and square corners, and opposite faces lie in parallel planes. Its edges are segments, but statements about their lines refer to the complete lines extending beyond the edges.

More generally, we can prove that two distinct intersecting lines are coplanar. Take their meeting point $V$, a point $X$ on the first line other than $V$, and a point $Y$ on the second line other than $V$.

Distinct lines already share $V$ and can have no other common point, so $Y$ is off the first line. The unique line through $V$ and $X$ is that first line, which misses $Y$. Thus $V,X,Y$ are noncollinear.

Their unique plane contains both lines by the line-in-plane foundation. A meeting therefore requires a common plane: lines that are not coplanar cannot meet.

A crossing of strokes in a perspective picture does not establish an intersection. Use the stated shape, labels, givens, and markings. Do not infer a square corner or parallelism from appearance alone.

Two planes

The floor and a wall suggest two different planes meeting along a line. Two shelves at different heights suggest parallel planes. Two names for the plane of one shelf describe coincident planes.

• Intersecting planes are distinct planes whose intersection, the set of all their common points, is a line. They cannot share just one isolated point or have another common point off that line.
• Parallel planes are distinct planes with no common point.
• Coincident planes are the same plane under different names, so every point of either belongs to both.

In ordinary three-dimensional Euclidean space, those are the possibilities for two planes. The word distinct is essential: the intersection-line foundation does not say that a plane meets itself in just a line.

For the prism, the planes of front face $ABFE$ and right face $BCGF$ meet along line $BF$. The planes of front face $ABFE$ and back face $DCGH$ are parallel. Plane $ABF$ and plane $AEF$ are coincident: each triple is noncollinear and lies in the same front-face plane.

What supports each statement?

This reference will grow through the course. A definition fixes a word’s meaning; an accepted foundation supplies a starting fact; a proved result follows from facts already available. An experimental observation reports what a drawing or measurement suggests.

We work in ordinary Euclidean space. The accepted foundations describe that setting; the pictures help you see it, but do not replace the conditions.

Common mistakes