Euclidean Metric & Distance

Source

Lecture: 2 — Vectors
Simulation: Simulation 2 — Euclidean space & metric (guided inquiry); a zoom toggle between the finite (Δ) and infinitesimal (d) displacement pictures.
Concept: The Euclidean metric (Kronecker delta δᵢⱼ) is the rule that turns coordinate differences into distance: ds² = δᵢⱼ dxⁱ dxʲ = dx² + dy².
Serves: Describe Euclidean space as a geometric structure equipped with a metric; use the Kronecker delta as the metric in Cartesian coordinates; compute inner products and norms.
Inquiry: Will the straight-line Δs equal, exceed, or fall short of the grid-path |Δx|+|Δy|? Does translating both points change Δs? When does equality hold?
Probe: Prediction (a) "Δs = |Δx|+|Δy|" targets the taxicab/Manhattan-distance confusion, rebutted only when the student selects it.

Some notes are hidden while a guided-inquiry prediction is still open.

Pedagogy principles applied

ID	Principle	How it shows up here
A1	One concept	Everything serves "use the metric to get a distance." No second example.
A2	Every element serves it	Grid, triangle, inputs, readout and zoom toggle all build the distance.
A3/A4	Formal tools optional	The δᵢⱼ matrix & metric equation live in the zoomed-in view and the ⚛︎ The Physics sheet (opened on demand).
B1	Standard terms	"metric", "Kronecker delta", "displacement", "coordinate difference".
B3	Symbols defined on use	δᵢⱼ glossed on first appearance; ds (infinitesimal) vs Δs (finite) distinguished.
B4	Honest headers	"Displacement on a 2D grid" / "Infinitesimal displacement & the metric".
C1	Numbers read/compared	Student reads P₁,P₂ coordinates and compares computed Δs to the visual hypotenuse.
C2	Numbers carry units	"Δx = 3 (grid units)", "Δs = 5.00 (grid units)".
D1	Visual is largest	The plane dominates; inputs & readout are compact in the aside.
D2	Readable text	Canvas labels ≥12px; panel prose ≥14px.
D3	Shape + adjacent label	P₁, P₂ are filled circles with coordinate labels beside them.
D5	No clipping	Points snap & clamp to the grid; labels are bounds-checked; both views fit.
D6	Incremental complexity	Concrete numbers first; formalism added only on zoom-in.
D7	No lockstep gating	Zoom, dragging, inputs and prediction all work from load.
D8	Canvas = labels only	Narration lives in the panels, not painted on the grid.
F2	Conceptual prediction	Predict a qualitative comparison, not a calculation.
F9–F14	Phase-appropriate wording	Open prompt; formative feedback ("try typing your values").
J1	One notation source	δᵢⱼ, dxⁱ (superscript indices), Δ for finite differences throughout.

Euclidean Metric & Distance

Teach me

Speed

The metric at a point

δᵢⱼ (Kronecker delta) = 1 when i = j, 0 otherwise — this 2×2 identity matrix is the Euclidean metric.

Guided Inquiry

1 · Two points on a plane

Let’s start with two points. P₁ and P₂ sit on a Cartesian grid, and if you drag either one it snaps to whole grid units. Their separation is carried entirely by the two coordinate differences Δx = x₂ − x₁ and Δy = y₂ − y₁. Turning that pair of numbers into a single distance takes a rule — and a rule is exactly what this plane comes equipped with.

2 · From P₁ to P₂

Here’s a warm-up. Travelling from P₁ to P₂ along the grid lines covers |Δx| + |Δy|. How do you think the length of the straight segment joining the two points compares with that?

Take a look at the sidebar: it now carries the calculator that applies this rule, and the readout that reports Δs.

3 · Moving the pair

Now we’ll move things around. Note the present Δx, Δy and Δs, then rebuild the same Δx and Δy somewhere else on the grid. What do you think Δs does?

4 · When is the distance zero?

Here’s a subtler one. The rule Δs² = Δx² + Δy² returns a number for every pair of points. For which separations do you think it returns exactly zero?

A rule with this property — a positive length for every distinct pair, zero only for coincident points — is called positive-definite. Every distance you’ve ever measured rests on it.

5 · The rule has a name

Let’s give the rule its proper name. Written for infinitesimal steps, it reads ds² = dx² + dy², and the array of coefficients in front of those steps — the Kronecker delta δᵢⱼ, ones on the diagonal, zeros off it — is the metric. A space is Euclidean precisely because it carries this metric.

6 · One sign away

One last leap. Spacetime keeps the shape of this rule but flips a single sign: ds² = −(c dt)² + dx². Judging from how the Euclidean rule behaved, what do you think becomes possible there that can’t happen on this plane?

That single sign is the whole difference. Euclidean geometry keeps invariance and positive-definiteness together; spacetime keeps the invariance and drops the positive-definiteness — and everything strange about relativistic geometry follows from that.

Listen

View & readout

▾

Distance Δs = ? grid units

Compute Δs from the metric

▾

Read coordinates from the grid, then type Δx = x₂ − x₁ and Δy = y₂ − y₁.

Δx =

grid units

Δy =

grid units

Enter Δx and Δy to see the metric in action.

The formal tools stay closed while a guided-inquiry prediction is open. Commit to the question, or switch the guided inquiry off.

Metric (infinitesimal)

Einstein summation over i, j ∈ {1, 2}. The metric squares-and-sums coordinate differences.

Kronecker delta = the Euclidean metric

δᵢⱼ = 1 when i = j, 0 otherwise. In Cartesian coordinates the metric is just the identity.

Finite displacement (what you compute)

Signed differences; squaring means the sign never affects Δs.

Norm / inner product

The same metric measures the length of any vector: ‖v‖² = δᵢⱼ vⁱ vʲ.

What stays invariant

Rotate (3,4) by 90° → (−4,3): components change, the distance 5 does not. Translating both points leaves Δs unchanged too.

The Physics

Euclidean space and the metric · what this sim shows, the physics behind it, and every control

At a glance

This sim turns two points on a grid into one distance, Δs, using a rule applied to the coordinate differences Δx and Δy. Change one sign in that rule and it becomes the interval of spacetime, so this is the first step toward relativistic geometry.

What you'll learn

How the two differences Δx and Δy are turned into a single distance.
What the metric δᵢⱼ is, and what its entries mean.
How the distance responds when you move the pair or turn the axes, and which numbers do change.
What changes in the rule when space becomes spacetime.

Two points, the two coordinate differences, and the straight segment between them.

What's on screen

P₁ (cyan) and P₂ (amber): the two points, each labelled with its coordinates. They open at (1, 2) and (4, 6).
Dashed legs Δx and Δy, meeting at a right angle.
Blue segment: the straight line from P₁ to P₂. Its length is Δs, shown in a violet label once you compute it.
View & readout and Compute Δs from the metric panels: the zoom toggle, the Distance readout, and the calculator.

See itPut the pair at its opening positions. Watch for: the two dashed legs meeting at a right angle, with the blue segment joining P₁ and P₂.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Drag P₁, P₂	Moves a point on the grid. It snaps to whole units, and cannot be dropped on the other point. Works in the Zoomed out view.	x, y: 0–8 opens at (1, 2) and (4, 6)
Zoomed out / Zoomed in	Zoomed out shows the grid and both points. Zoomed in shows a schematic triangle of tiny steps dx, dy, ds, with the metric written out. It is not a magnified copy of the grid triangle.	2 views opens Zoomed out
Distance (readout)	Shows Δs in grid units once the Δx and Δy you typed match the points on the grid.	readout opens "Δs = ?"
Δx, Δy (Compute Δs from the metric)	Type the differences x₂ − x₁ and y₂ − y₁. The sim works out Δs² and Δs step by step, and warns if your numbers do not match the grid. Moving a point clears them.	any whole number opens empty
Speed	The scene is static, so nothing visible changes.	0.25×–4× opens at 1×
↻ Reset	Puts P₁ and P₂ back at their opening positions and clears the calculator.	top bar

Fixed: lengths are in grid units, and the grid runs from 0 to 8 on both axes.

The straight distance from squared legs

The blue segment is the hypotenuse of a right triangle whose legs are Δx and Δy:

Signed differences; squaring means the sign never affects Δs.

Δx Δy

These sliders fill in the sim's Compute Δs boxes.

Square each difference, add, then take the root. For P₁ (1, 2) and P₂ (4, 6), Δx = 3 and Δy = 4, so Δs = √(9 + 16) = 5 grid units.
Walking along the grid lines costs |Δx| + |Δy| = 7. Squaring before adding is what separates 5 from 7.
The sign of a difference only records direction. Δx = −3 and Δx = +3 give the same Δs.

See itType Δx = 3 and Δy = 4 for the opening pair. Watch for: Distance reading Δs = 5.00 grid units, set against a grid path of 7.

The metric is the rule for tiny steps

Written for infinitesimal steps, the rule is ds² = δᵢⱼ dxⁱ dxʲ. The array δᵢⱼ is the metric:

Einstein summation over i, j ∈ {1, 2}.

The Kronecker delta: in Cartesian coordinates the metric is the identity.

Einstein notation: an index repeated once up and once down is summed. In 2D, δᵢⱼ dxⁱ dxʲ has four terms, and the two with i ≠ j are multiplied by 0.
The 1s on the diagonal give dx² + dy². The 0s off the diagonal mean there is no dx dy cross term, which is what orthonormal axes ê₁, ê₂ provide.
It is called the metric because it measures lengths and angles, through norms and inner products.

The sim's Zoomed in view draws this schematically: it is not a magnification of the 3-4-5 triangle.

See itSwitch to the Zoomed in view. Watch for: the box on the canvas, with ds² = δᵢⱼ dxⁱ dxʲ = dx² + dy² beside a 2 × 2 matrix of ones and zeros.

Sliding the pair changes coordinates, not Δs

Coordinates are labels that depend on where the origin sits. The rule uses only differences, so moving both points by the same amount changes x₁, y₁, x₂ and y₂ but leaves Δx and Δy alone.

Slide the opening pair by (+3, −1). P₁ goes from (1, 2) to (4, 1), and P₂ from (4, 6) to (7, 5). Still Δx = 3 and Δy = 4, so Δs = 5.
Moving the origin is the same thing seen from the other side: every coordinate shifts by one fixed amount, and the differences do not.
Distance belongs to the separation, not to where the pair happens to sit.

See itCompute Δs for the opening pair, then slide both points. Watch for: both points change coordinates, while Distance stays at Δs = 5.00.

Turning the axes changes components, not length

The same metric gives the length of any vector, and the angle between two. Observers whose Cartesian axes are rotated relative to each other write different components for the same vector, and get the same length.

The inner product and the squared length of a vector.

Turn the figure, or turn the axes: the components change, Δs does not.

Turn the figure. Rotating the separation (3, 4) by 90° gives (−4, 3). Both components change, and Δs = 5 either way.
Turn the axes. Bob writes a particle's velocity as (v/√2)(1, 1). Alice's axes are rotated, so she writes (v, 0). Bob finds ½v² + ½v² = v², and Alice finds v²: the same speed.
This works because δᵢⱼ is the same array in every Cartesian frame, so the number it returns is agreed on by all of them.

See itTurn the opening separation through 90°. Watch for: the blue segment swinging to point up and to the left, while Distance still reads Δs = 5.00.

Zero separation means one point, until a sign flips

Both terms in Δs² = Δx² + Δy² are squares, so neither can be negative. Δs = 0 only when Δx = 0 and Δy = 0: P₁ and P₂ are the same point. A rule with this property is positive-definite. Sharing a grid line is not enough: with Δx = 0 and Δy = 4, Δs = 4.

Spacetime keeps the shape of the rule and flips one sign. In spacetime the time term is negative and the space terms are positive (c is the speed of light):

The same form with one sign flipped.

With a minus sign the two terms can cancel. Two distinct events with Δx = cΔt have Δs² = 0: the separation a light ray covers.
Only differences still appear, so the interval is unchanged by sliding both events. What is lost is positive-definiteness.

This sim draws only the plane. The spacetime rule is stated here, not shown.

See itPut P₂ straight above P₁ and compute Δs. Watch for: Δx = 0, yet Distance reads Δs = 4.00, not zero.

Where it matters

Distances from coordinates. Graphics, surveying and robotics all find the straight-line distance between two points this way. Rotating the camera or the map leaves those distances unchanged.
Speed in any frame. Velocity components depend on the axes, but the speed does not, as in the Bob and Alice example.
Spacetime. The interval Δs² = −c²Δt² + Δx² + Δy² + Δz² is the same rule with one sign flipped, and it is the quantity every inertial observer agrees on.

Go deeper: the metric in other coordinates

In polar coordinates the same plane has ds² = dr² + r² dθ². The metric's components are no longer the identity.
So δᵢⱼ is unchanged under the moves between Cartesian frames, such as translations and rotations. That is a narrower claim than "invariant in every coordinate system".

Summary

In one line. Squaring and adding the coordinate differences gives a distance, Δs² = δᵢⱼ Δxⁱ Δxʲ, that does not change when the pair is moved or the axes are turned. Spacetime keeps this form with one sign flipped.

Key points

Δs = √(Δx² + Δy²). The grid path |Δx| + |Δy| is longer unless one leg is zero.
δᵢⱼ, with ones on the diagonal and zeros elsewhere, is the metric of Cartesian coordinates.
Only differences enter, and δᵢⱼ is the same in every Cartesian frame, so Δs survives sliding and turning.
Δs = 0 only for coincident points. In spacetime, one sign flip lets distinct events have zero interval.

Common mix-ups

"The distance is |Δx| + |Δy|." That is the grid path, 7 for the opening pair. The metric squares before it adds, so Δs = 5.
"Different components mean a different length." Components depend on the axes. The length does not.
"δᵢⱼ never changes." It is unchanged between Cartesian frames. In polar coordinates the components of the metric change.

Things to try on your own

Find a pair of points whose Δs equals |Δx| + |Δy|. Describe what all such pairs have in common.
Start from Δx = 3 and Δy = 4. Double Δx and halve Δy, drag P₂ to match, and compute Δs. Compare it with 5.