Different Coordinate Frames

Source

Lecture: 1 — Geometry
Simulation: Simulation 2 · Different Coordinate Frames (lecture visual aid)
Concept: The same geometric point has different coordinate labels in different coordinate systems — coordinates describe how we label space, not an intrinsic property of the point.
Serves: Use Cartesian (and polar) coordinates to describe points and trajectories; understand coordinates as labels rather than physical observables; geometry as a mathematical description of space.
Inquiry: Given an (x, y) point, what are its coordinates in the (x′, y′) system? (Set up so it can be read off the grid.)
Reflection: The coordinates changed every time you moved the axes, but the point didn't move — so what exactly are coordinates describing?

Pedagogy principles applied

ID	Principle	How it shows up here
A1	Single concept	One idea only: coordinates are frame-dependent labels, not the point itself.
A2	Minimalism	Only (x, y), (x′, y′), r, r′ and θ are shown — no velocity, time or polar readouts.
A3	Formal layer off by default	Rotation-matrix equations live behind the ⚛︎ The Physics sheet, closed on load.
B1	Standard terminology	"Frame S", "Frame S′", "rotation angle θ" — no invented jargon.
B3	Symbols defined	x, y / x′, y′ / θ defined in the Explore panel and the formal panel.
B4	Descriptive headers	Left header "Frame S"; right header "Frame S′ (rotated by θ)".
C1	Numbers are referenced	Every readout is read or compared in the staged build-up steps.
C2	Contextual labels	(x, y) in blue/S, (x′, y′) in red/S′, r labelled "distance from origin", θ in degrees.
D1	Visual dominance	The two grid panels fill the hero — the largest element on screen.
D2	Legibility	Grid/tick labels ≥12px; coordinate readouts at 14px.
D3	Labelled entities	Each marker is a distinct shape + colour with its coordinate pair printed adjacent.
D5	No clipping	Coordinate labels carry a backdrop and clamp inside the panel for every θ and edge case.
D8	Labels, not narration	Grids carry only axis labels and coordinates; explanation lives in side panels.
G2	Invariant labelled	r is named "invariant under rotation"; r = r′ is shown beneath the panels.
G3	Real re-render	Changing θ rotates the whole S′ grid (axes + gridlines), not just a label swap.
G4c	Identity preserved	Each marker keeps its symbol, colour and physical position as θ changes.
J1	Consistent notation	Primes throughout; θ always in degrees — never radians.
J3	Model correctness	(x′, y′) from the passive-rotation formulae; right-panel clicks back-transform correctly.

Teach me

Speed

Frame S — axes x, y (unrotated)

Frame S′ — axes x′, y′ (rotated by θ)

Guided Inquiry

1 · Two frames, one marker

Let’s meet our two frames. Frame S labels the plane with axes x and y. Frame S′ uses axes x′ and y′, rotated counter-clockwise by an angle θ about the same origin O. A single marker sits at one physical location, and each frame gives it its own pair of numbers.

2 · Which pair is the true one?

Here’s the first puzzle. Frame S and Frame S′ report different pairs of numbers for the same marker. So which pair do you think is the marker’s true position?

3 · What survives a rotation?

Now we’ll set the frame turning. Suppose Frame S′ is rotated continuously while the marker stays where it is. Write r = √(x² + y²) and r′ = √(x′² + y′²) for the marker’s distance from the origin in each frame. What do you think happens to its S′ numbers?

4 · The frames at 90°

Time to put some numbers on it. Right now θ is 0°, so the two frames agree and both report (3.0, 2.0). Predict: when θ reaches 90°, what pair will Frame S′ report for the same marker? Take a moment to picture the axes turning.

5 · A frame that empties one axis

Here’s a neat one. Frame S′ now sits at θ = 90°. Turn it back toward 0°, until the marker’s y′ reads 0 — the marker then lies on the x′ axis itself. What do you think x′ reads there?

6 · Invariants

Let’s name what you’ve found. Across every rotation you set, x′ and y′ changed while r didn’t. A quantity that comes out the same in every frame is called an invariant — and in special relativity, the spacetime interval plays exactly the role r plays here.

Symbols: x, y — coordinates in Frame S · x′, y′ — coordinates in Frame S′ · θ — counter-clockwise rotation from S to S′ · r — distance from the origin.

Listen

Controls

▾

Rotation angle θ θ = 30°

0°360°

Passive rotation of axes by θ (S → S′)

Matrix form

Invariant — distance from the origin is the same in every frame

Inverse — read off (x, y) from a click on the S′ grid

Symbols

x, y — coordinates in Frame S · x′, y′ — coordinates in Frame S′ · θ — counter-clockwise rotation angle from S to S′ (degrees). Coordinates are dimensionless grid units (−5 … +5).

The Physics

Coordinate frames · what this sim shows, the physics behind it, and every control

At a glance

One marker sits on a plane and is labelled twice, using two sets of axes that share an origin, one turned by an angle θ. It is the simplest setting for a theme the whole course returns to: two descriptions of the same thing can differ in some numbers and agree in others.

What you'll learn

How one marker gets two different coordinate pairs from two frames that share an origin.
How to compute the pair in the turned frame from the first pair and the angle θ.
How to tell which numbers depend on the choice of axes and which do not.

One marker (★), two sets of axes through the same origin O. Frame S′ is Frame S turned counter-clockwise by θ.

What's on screen

Frame S (left panel, blue axes x and y): the reference axes.
Frame S′ (right panel, red axes x′ and y′): the same plane, with its axes and grid turned by θ about O.
★ marker: drawn at the same spot in both panels. The box beside it prints its coordinate pair in that panel's frame: (x, y) on the left, (x′, y′) on the right.
r and r′ readouts (under the panels): the marker's distance from the origin, as each frame computes it.

See itTurn Frame S′ to 30°. Watch for: the ★ at the same spot in both panels, with a different label beside it in each.

The controls

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

Control	What it changes	Range
Rotation angle θ	How far Frame S′ is turned counter-clockwise from Frame S. Dragging it pauses the automatic turning.	0°–360° opens at 30°
The ★ marker	Click or drag in either panel to move it. The other panel follows. Positions within 0.22 of a whole number snap to it.	±6 in the panel you drag opens at (3, 2)
r and r′ readouts	The marker's distance from the origin, √(x² + y²) in Frame S and √(x′² + y′²) in Frame S′, to one decimal place.	panel
▶ Play / ⏸ Pause	Turns Frame S′ automatically at 10° per second, counter-clockwise, wrapping from 360° to 0°. Pause holds the angle.	top bar
Speed	Scales the automatic turning. It changes how fast θ advances, not what the angles mean.	0.25×–4× opens at 1×
↻ Reset	Puts θ back to 30° and the marker back to (3, 2), and stops the automatic turning.	top bar

Fixed: both panels share one origin and one scale. Coordinates are plain grid units, and each grid square is 1 × 1 in its own frame.

Coordinates depend on the axes

To set up Cartesian coordinates you choose an origin O, an x-axis through it, and a y-axis at 90° to the x-axis. A point's coordinates are then where the perpendiculars from the point meet the two axes, measured from O.

Frame S′ keeps the origin and turns the axes by θ. The marker has not moved, but the axes it is measured against have, so the lengths along them are different. The two panels print different pairs for the same point.

Key idea. A coordinate pair is not a property of the point alone. It is the point's position measured along a chosen set of axes, a label we assign. Nothing in the plane makes S's axes the real ones: calling them "unrotated" is a convention.

See itFrame S′ starts at θ = 0°, then turns to your angle. Watch for: the right-hand label changing as S′ turns, while the ★ stays put and the left label never changes.

How the numbers change when the axes turn

The axes turn by +θ and the marker does not (a passive rotation, as in Lecture 3). The pair in Frame S′ follows from the pair in Frame S:

S′ coordinates from S coordinates; θ is counter-clockwise from S to S′.

Each coordinate is the marker's projection onto one axis: x′ onto the x′ axis, y′ onto the y′ axis.
Turning the axes by +θ changes the pair exactly as turning the marker by −θ would, with the axes held fixed.
At θ = 90° the x′ axis lies along y and the y′ axis along −x, so x′ = y and y′ = −x.
The rule runs backwards too: x = x′ cos θ − y′ sin θ, y = x′ sin θ + y′ cos θ. Each pair fixes the other.

θ = 60° and the marker at (3, 2): x′ = 3.2, y′ = −1.6.

See itSet θ = 90° and read both labels. Watch for: the S′ label showing the S label's second number first, then the first number with its sign flipped.

The marker's distance from O stays the same

Write r = √(x² + y²) and r′ = √(x′² + y′²) for the distance from O to the marker in each frame. Put the rotation formulas into r′²: the cross terms cancel, and cos²θ + sin²θ = 1.

True for every θ.

This slider turns Frame S′ in the sim too.

x′ and y′ trade value as θ changes, but their squares always add up to the same r².
So r belongs to the marker, not to the axes. A quantity that every rotated set of axes agrees on is called an invariant (the course also says scalar).
When the marker lies on the x′ axis, y′ = 0 and x′ = ±r: one component carries the whole distance.

See itLet Frame S′ turn on its own. Watch for: the S′ label's numbers keep changing while r and r′ stay equal.

See itTurn S′ until the marker lies on its x′ axis. Watch for: the second number of the S′ label reading about 0, and the first equal to r.

Where it matters

Choosing convenient axes. On an inclined plane, turn the axes by the slope angle so that x′ runs along the slope. A displacement along the slope then has y′ = 0, the same alignment as in section 05.
Separation of two points. Take P = (1, 0) and Q = (1, 1) in Frame S. With the axes turned by 45°, P = (1/√2, −1/√2) and Q = (√2, 0). The numbers differ; the separation is 1 in both.
Special relativity. Observers moving past each other also assign different coordinates to one event. The quantity they agree on, playing the role r plays here, is the spacetime interval, Δs² = −c²Δt² + Δx² + Δy² + Δz² (c is the speed of light).

Go deeper: what "invariant" is relative to

Lecture 3 also allows the origin to move: x̃ = R(θ)x + x̄. A shift of O changes every point's coordinates, so r changes with it.
The difference between two points' coordinates is unaffected by the shift and is only turned by R(θ). That is why the separation survives both changes, while r survives only rotation about O.

Summary

In one line. A coordinate pair is a label tied to a set of axes. Turning the axes changes the pair by a rotation, and leaves the marker's distance from O unchanged.

Key points

One marker, two pairs: x′ = x cos θ + y sin θ and y′ = −x sin θ + y cos θ, and the reverse rule recovers (x, y).
r′ = r at every θ, so distance from O is an invariant under rotation about O.
"Invariant" means unchanged under a stated change: r survives turning the axes, not moving O.

Common mix-ups

"One of the two pairs is the marker's real position." Each is its position relative to its own axes.
"Turning the axes moves the marker." That would be an active rotation. Here the marker stays and only its labels change.
"At θ = 90° you just swap x and y." You swap and flip a sign: x′ = y and y′ = −x, so (3, 2) becomes (2, −3).

Things to try on your own

Drag the ★ to (4, 3) in the left panel. Find the whole-degree θ where y′ is closest to 0, and check that x′ is close to r = 5.
With the ★ at (3, 2), predict the S′ label at θ = 180°, then set θ to check.