Lesson · Geometry

Intro to Geometry

Geometry is the study of objects in space. This lesson starts from the very beginning — what a “space” even is — and builds up through points, coordinates and line segments to describing movement itself as a function.

Space, points, and coordinates

Geometry can be thought of as the study of objects in space. But what space? Like most things in math, it depends on context. We characterize a space by how many dimensions it has: one‑dimensional space (1D) is a number line; two‑dimensional space (2D) is an infinite flat plane; and three‑dimensional space (3D) is the space we live in. We can go further — 4D, 5D, even 999D — we just can't easily picture them.

What determines a space's dimensionality? The number of orthogonal components it has — its independent directions. We usually visualize 2D space as the Cartesian plane, shown here with a single point on it:

Now take that point and shift it. If we move it left or right, its y‑coordinate never changes; if we move it up or down, its x‑coordinate never changes. That's what orthogonal means: left/right movement is independent of up/down movement, and vice‑versa. Here is the same starting point (in coral) shifted three units in each of the four directions (in blue):

We labelled each location with its coordinates. In 2D we write (x, y) — the left/right position first, the up/down position second — so (4, −7) means 4 units right of center and 7 units below it. In 3D we add a height, (x, y, z). In general, N‑dimensional space needs N numbers per location — which is how we can still talk about 6D or 99D space even when we can't draw it.

Line segments

A single point is the most basic object in space, but not very interesting on its own. Plot two points and connect them with a straight line, and you get a line segment. Because a segment is fixed entirely by its two endpoints, we can describe any segment just by naming them. Here is the segment with endpoints (1, −1) and (4, 3):

Now move the segment left by 2 — that is, subtract 2 from the x‑coordinate of each endpoint, giving the new endpoints (−1, −1) and (2, 3):

Movement as a function

Could we describe that movement with a function? Recall that a function needs three things: an input, a rule, and an output. Here both the input and output are line segments, and the rule, in plain English, is “add 3 to the y‑coordinate of each endpoint.”

To write it as an equation we need names. Call the input segment L. It has two endpoints, so name them e₀ and e₁. Each endpoint is a point, so it has coordinates: let e₀ = (x₀, y₀) and e₁ = (x₁, y₁), and write the segment as e₀e₁. Naming our function “up”:

That's a little clunky. Since a segment is just its two endpoints, let's write it in coordinate form — L = {(x₀, y₀), (x₁, y₁)} — and rewrite the function:

Now plug in the example segment {(3, 1), (6, 0)}:

And graphing the result confirms the function does exactly what we designed — it moves the whole segment up by 3, leaving its left/right position untouched:

Ready to practice?

The printable version has practice problems to work through — graphing points and segments, and writing your own movement functions (including a couple of challenges, like a function that flips a point across the x‑axis).