At a glance
Two points sit fixed on a plane, and you choose the coordinate axes that describe them. This sim lets you shift and turn those axes and compare the numbers you read in each choice, the idea that relativity later applies to space and time.
What you'll learn
- How the sim turns A and B into coordinates in shifted and turned axes.
- How the coordinates, the differences Δx′ and Δy′, and the distance d each respond to a shift of the origin and to a turn of the axes.
- How the Euclidean metric combines two differences into one length.
- What it means for a quantity to be a scalar, and why physics is written with scalars.
What's on screen
- A (cyan dot): the first point. Drag it anywhere on the plane.
- B (amber dot): the second point, also draggable.
- White line: the straight line from A to B. Its length is d.
- Grey axes with unit ticks and an origin: the coordinate system. They read x, y, O until you move them, then x′, y′, O′.
- Dashed legs: Δx′ (green) along x′, then Δy′ (pink) along y′, ending at B.
- Readouts (top right): the coordinates of A and B, Δx′, Δy′ and d, in the current axes.
The controls
Press ⌖ on a row to see where that control sits in the sim.
| Control | What it changes | Range | |
|---|---|---|---|
| Origin shift Tx | Puts the new origin at x = Tx, measured in the original axes. | −5 to 5 opens at 0 | |
| Origin shift Ty | Puts the new origin at y = Ty, measured in the original axes. | −5 to 5 opens at 0 | |
| Axis rotation θ | Turns both axes anticlockwise through θ, about the origin. While the sim plays, θ also advances on its own. | 0°–360° opens at 0° | |
| View | How the dashed legs are drawn. 1 · Basics: faint and unlabelled. 2 · Components and 3 · Metric: bold, with their values. | 1, 2 or 3 opens at 3 | |
| Points A and B | Drag either dot on the plane to move it. | −8 to 8 on each axis open at (−1, −1) and (2, 3) | |
| Readouts | The coordinates of A and B, Δx′, Δy′ and d, in the current axes. | panel | |
| ▶ Play / ⏸ Pause | Starts or stops the automatic turning of the axes. | top bar | |
| Speed | Scales the automatic turning: 20° per second at 1×. | 0.25×–4× opens at 1× | |
| ↻ Reset | Puts A, B, Tx, Ty and θ back to their opening values. | top bar |
Fixed: lengths are in grid units, with no physical scale.
The sim moves the axes, not the points
A and B have fixed positions on the plane. You control the coordinate system that describes them: Tx and Ty put its origin at (Tx, Ty), and θ turns both axes anticlockwise. Moving the axes while the points stay put is called a passive transformation.
Every number on screen is read in the current axes, which is why the sim adds primes once you move them. The two differences are Δx′ = x′B − x′A and Δy′ = y′B − y′A. They are the legs of a right triangle whose hypotenuse is the line AB, and d is the length of that line.
A shift of the origin cancels in every difference
Moving the origin changes both points' coordinates by the same amount, Tx in x and Ty in y. Subtracting one point from the other removes that common amount:
Δx′ = (xB − Tx) − (xA − Tx) = xB − xA = Δx, and likewise Δy′ = Δy.
- The coordinates change. With Tx = 3 and Ty = −2 the default points A = (−1, −1) and B = (2, 3) read A′ = (−4, 1) and B′ = (−1, 5).
- The differences do not. They stay Δx′ = 3 and Δy′ = 4. A coordinate says where a point is relative to the origin you chose. A difference compares the two points, so it never refers to the origin.
- The distance d is built from these differences (section 06), so a shift leaves it unchanged too.
A turn changes the legs but not the sum of their squares
Once the shift has cancelled, only θ is left. A turn mixes the original differences Δx and Δy:
At θ = 30° the default legs 3 and 4 become 4.60 and 1.96. Add the squares and the cross terms cancel, leaving cos²θ + sin²θ = 1:
These sliders move the sim too. After dragging A or B, reopen the sheet to update.
The line above recomputes d from the legs in your current axes. The sim's own d readout is computed from the points' original coordinates.
The metric turns two differences into one length
The rule d² = Δx′² + Δy′² is the Euclidean metric δij: Pythagoras' theorem, applied to the components in whatever Cartesian axes you use.
- d is never negative. It is the square root of a sum of squares, and it is 0 only when Δx′ = Δy′ = 0, that is, when A and B are the same point.
- Δx′ and Δy′ are components of one vector, the step from A to B, so they depend on the axes. d is a scalar: a single number that every choice of Cartesian axes agrees on. For the default points, d = √(3² + 4²) = 5.
Where it matters
- Speed is a scalar too. One particle has V = (v/√2)(1, 1) in Bob's axes and Ṽ = (v, 0) in Alice's. The components differ, but V₁² + V₂² = v² in both.
- Mapping and surveying. Two survey teams that set up different axes record different coordinates for the same two landmarks. In the same units, over a patch small enough to treat as flat, they agree on the distance between them.
- Spacetime. Lecture 5 keeps this recipe and changes the metric. The interval is Δs² = −c²Δt² + Δx² + Δy² + Δz², with c the speed of light, and every inertial observer agrees on it. Δs² < 0 is timelike, = 0 lightlike, > 0 spacelike.
Go deeper: the recipe belongs to Cartesian axes
δij survives shifts, turns and reflections of Cartesian axes. It does not survive every change of coordinates. Take the points (1, 0) and (0, 1), whose true distance has d² = 2. Written in polar coordinates (r, φ) they are (1, 0°) and (1, 90°), and adding the squared polar differences gives Δr² + Δφ² = 0 + (π/2)² ≈ 2.47. Polar components need a different metric.
Summary
Key points
- A shift changes the coordinates but not the differences.
- A turn changes Δx′ and Δy′, but cos²θ + sin²θ = 1 keeps their sum of squares.
- d is a scalar. It is never negative, and it is 0 only when A and B coincide.
Common mix-ups
- "The coordinates changed, so the distance must have changed." The length of AB does not depend on which axes you read it in.
- "Δx and Δy are invariant." Only under a shift of the origin. A turn mixes them.
- "Turning the axes turns the points." Nothing on the plane moves. The axes turn and A and B stay where they were.
Things to try on your own
- With the default points, set Tx = 3 and Ty = −2, then check A′ and B′ against x′ = x − Tx and y′ = y − Ty.
- Set θ to 53° and read Δx′ and Δy′. The x′ axis now lies almost along AB. Δx′ is close to d and Δy′ is close to 0.
- Drag A toward B and watch Δx′, Δy′ and d shrink together.