Euler angles and gimbal lock: the map tears, not the territory

Where you are. You know that rotations about different axes do not commute, so a sequence of turns has to be read in order. This lesson takes the most popular way of writing such a sequence down, three angles, and shows you where it fails.

Point at something and say where

Stand up if you can. Pick something across the room and point at it with a straight arm.

Now describe that direction out loud with two numbers, the way a compass and a spirit level would: how far round the room you are aimed, and how far above level. “Thirty degrees round from the door, twenty degrees up.” Someone else could reproduce it exactly. Point at a few more things. It works every time, and every direction gets its own pair of numbers.

Now point straight up at the ceiling.

Say how far round the room you are aimed.

There is no answer. Every heading describes the same direction once your arm is vertical. The first number has quietly stopped meaning anything, which is worse than it being missing, because it is still there, still a number, still getting logged and averaged and fed into things.

Now the part that actually hurts. Point a degree or so short of vertical, leaning towards the window. Then swing slowly up through the top and the same small distance past it, so you are leaning towards the opposite wall. Your arm moved a couple of degrees, smoothly, without pausing. Your heading flipped by a hundred and eighty.

Nothing happened to your arm. Something happened to the description of your arm.

The idea in one paragraph

Any orientation can be written as three turns about three axes, taken in a fixed order. Three numbers, human-readable, easy to type into a config file: these are Euler angles, and they are the notation everybody reaches for first. They have two problems. The first is that “three angles” does not say which three axes in which order, and there are twenty-four defensible answers to that question, most of which disagree violently. The second is what your arm just did: at certain orientations the three numbers stop being able to name nearby orientations smoothly, one of them stops carrying information, and a tiny real motion demands a huge jump in the numbers. That is called gimbal lock, and the crucial point is that it is a property of the description, not of rotation. The orientation your arm passed through is entirely ordinary. It is the map that tears there, not the territory.

Three numbers means three rings

The name comes from hardware. Mount a ring so it can spin about a vertical axis. Inside it, mount a second ring on a horizontal axis carried by the first. Inside that, a third. Now whatever you bolt into the innermost ring can be aimed anywhere by setting three angles, one per ring.

The order is baked into the metal. The outer ring carries the middle one, so turning the outer ring drags the middle and inner rings along with it; turning the inner ring moves nothing but the payload. That physical asymmetry is the same asymmetry you felt with the phone in lesson 5. Three angles are not three independent facts about an orientation. They are a recipe, and recipes have steps in order.

Twenty-four ways to read three numbers

Count the choices. The first turn can be about x, y or z: three options. The second must be about a different axis than the first: two options. The third must differ from the second, but is allowed to return to the first axis: two options again. That is twelve sequences, six of the xyz kind and six of the zxz kind that come back on themselves.

Then double it, because you still have to say whether each turn is about an axis fixed in the world or an axis carried along by the body. Twenty-four conventions, all in current use, all called “Euler angles”, and several of them called “roll, pitch and yaw” by someone.

This is not a theoretical concern. Take the perfectly innocent triple [30°, 60°, 90°] and read it under all twenty-four conventions:

The median disagreement between two randomly chosen readings of the same three numbers is 86°. Guessing the convention is not a small risk you can take under deadline; it is a coin flip on being nearly opposite.

Where it breaks

Now set the middle angle to 90° and look at what the rings are doing.

The inner ring has been rotated until its axis is parallel to the outer ring’s axis. Turning the first ring and turning the third ring now do exactly the same thing. All three rings still spin freely, so you still have three angles to set, but you can only aim in two independent directions. The third has no ring left to serve it.

The arithmetic agrees, and it is easy to check. In the xyz extrinsic convention, with the middle angle pinned at 90°, the resulting rotation depends only on the difference of the other two:

from scipy.spatial.transform import Rotation as R
import numpy as np

def M(a, b, c):
    return R.from_euler("xyz", [a, b, c], degrees=True).as_matrix()

np.allclose(M(0, 90, 0),  M(35, 90, 35))   # True
np.allclose(M(70, 90, 20), M(50, 90, 0))   # True

Both pairs have the same first-minus-third, and both give byte-identical rotations. Three numbers going in, two degrees of freedom coming out: the map has folded, and a whole line of triples is being crushed onto one orientation.

The libraries know. Ask scipy to convert that orientation back into angles and it hands you a canonical answer and a warning:

R.from_euler("xyz", [35, 90, 35], degrees=True).as_euler("xyz", degrees=True)
# array([ 0., 90.,  0.])
# UserWarning: Gimbal lock detected. Setting third angle to zero
# since it is not possible to uniquely determine all angles.

You put in 35, 90, 35 and got back 0, 90, 0. Nothing is wrong: those are the same orientation. But any code that round-trips angles through a rotation and compares the numbers before and after has just been handed a false alarm, and any code that stores angles as the source of truth has just lost information it can never recover.

The map tears, not the territory

Here is the distinction the whole lesson turns on.

A physical three-ring gimbal at lock really has lost something. Its payload cannot instantaneously rotate about the missing direction, because there is no ring that produces that motion. That is a mechanical fact about metal, and it is why real hardware treats gimbal lock as an emergency.

Three numbers on paper have lost nothing of the kind. The orientation at pitch 90° is a perfectly ordinary orientation with perfectly ordinary neighbours in every direction. Rotation itself is smooth there; you swung your arm through it without noticing. What broke is the chart you were using to name orientations, in exactly the same way that a flat map of the Earth must distort or tear at the poles while the Earth stays round.

And this is not a defect of one badly chosen convention that a cleverer one would fix. Every three-number scheme has this problem somewhere; the conventions differ only in where they put the tear. The reason is topological, and it fits in a sentence: the space of orientations is closed and bounded, with no edges and nothing running off to infinity, and no patch of ordinary three-number space has that shape. Something must tear or fold when you lay one over the other, exactly as when you try to peel a globe flat. Three numbers will always work beautifully in a neighbourhood and always fail somewhere. Which is the argument for quaternions: four numbers with one constraint, which is a curved space that matches the shape of the problem instead of fighting it.

What the tear costs you in motion

Take a straight, uniform physical motion and watch what happens to the numbers. Pitch upward at a steady one degree per step, straight through vertical, and read the xyz angles at each step:

The step from 89° to 91° is two degrees of real motion; the numbers move by about 255° when you measure the jump across all three. Feed that into anything that differentiates, interpolates, or computes a loss against a target, and you get a violent spike from a motion that was perfectly smooth.

It is not only the crossing. Approaching the tear, the numbers get progressively more hysterical, and there is an exact statement of how fast. The three angle rates map to an actual turn rate through a 3×3 matrix whose determinant is exactly $\cos(\text{middle angle})$. In plain language: as the middle angle approaches 90° that matrix goes flat, so inverting it, which is what you do every time you ask “what angle rates give me the motion I want”, blows up. Worst case, one unit of physical turn rate costs you about 2.7 units of angle rate at a middle angle of 60°, about 81 at 89°, and about 810 at 89.9°. The three angles are describing a mild motion using enormous, nearly cancelling changes, and floating point notices long before you reach the singularity itself.

When three angles are still the right answer

None of this makes Euler angles a mistake. It makes them a presentation format.

They are the best notation in existence for a human reading an orientation off a screen, for typing a camera pose into a scene file by hand, and for arguing about a pose out loud. Nobody has ever debugged a wrist orientation faster by reading four quaternion components. And when the hardware really is a three-ring mount, its three angles are not a representation at all: they are measurements of three physical joints, and their lock is a real one you must plan around.

The rule that follows is worth memorising once and never revisiting:

Convert at the boundary, in one place, with the convention named in the function signature. Never let three angles be the thing your system actually stores.

Review

Check yourself

Do this

No new code file; twenty minutes at a REPL and on paper.

1. Measure the convention chaos yourself. In a Python session, take one triple and read it every way:

import itertools, numpy as np
from scipy.spatial.transform import Rotation as R

seqs = [a+b+c for a, b, c in itertools.product("xyz", repeat=3) if a != b and b != c]
rots = {s: R.from_euler(s, [30, 60, 90], degrees=True) for s in seqs + [s.upper() for s in seqs]}

Then, for every pair, compute the angle of the relative rotation with np.degrees(np.linalg.norm((r1 * r2.inv()).as_rotvec())). Confirm the numbers in the table above: 24 conventions, 21 distinct orientations, and 237 of the 276 pairs more than 45° apart. That relative-rotation angle is the same “how far apart are two orientations” tool you built in lesson 5, and it is the right way to compare orientations for the rest of your career.

2. Find the fold. Fix the middle angle at 90° and vary the other two, checking as_matrix() for equality. Work out from the results which combination of the first and third angles the orientation actually depends on, then confirm it with two triples you choose yourself. You will have derived the collapse rather than read it.

3. Watch the numbers jump while the motion does not. Sweep the middle angle from 80° to 100° in one-degree steps, converting each rotation back with as_euler("xyz", degrees=True), and print the change at each step. Then compute the true angle between consecutive orientations with the relative-rotation formula. One column stays at 1° the whole way; the other detonates once. Sit with those two columns side by side, because that pair of behaviours is the lesson.

4. On paper, audit one interface you own. Find somewhere in a system you work on where an orientation, a heading, an angle or a rotation crosses a boundary between two components. Write down what convention each side assumes, and how you know. If you cannot answer for either side, you have found the same bug this lesson is about, before it found you.

What you can now do

You can write down an orientation as three angles and state the convention that makes them mean anything, including whether the axes are fixed or carried along. You can recognise gimbal lock from its symptoms: a round-trip test that suddenly fails, a stored angle that stops carrying information, joint rates that spike near one pose, a 180° jump in a log during a smooth motion. You can say exactly what is lost, which is the coordinates and not the rotation, and explain why no three-number scheme escapes it. And you know the rule that follows: compute in quaternions, display in angles, and name the convention every single time.