This chapter links to the course’s interactive demos and uses MathJax for its formulas. Read it through the course website (or download the file and open it in a browser); a Canvas inline preview will reach neither the demos nor the math. All chapters.

Rotation: Axis-Angle, Quaternions, Euler Angles

“How about sending me a fourth gimbal for Christmas?” Michael Collins, Apollo 11, when Houston warned of gimbal lock, 1969

A rotation keeps lengths and angles and fixes the origin, which makes it the best-behaved transform in graphics and, at the same time, the hardest to store well. A matrix applies it fast but drifts and cannot be blended; an axis and an angle say what it is; a quaternion stores it in four numbers that compose by multiplication and blend along a sphere; three Euler angles read nicely and lock; and “turn this direction onto that one” names the smallest rotation that does it. This chapter builds the four representations in that order, starting each in two dimensions where it can: what a rotation keeps, checked on a shape; the matrix as the landed axes; Rodrigues’ formula from the projection split of the vectors chapter, as the 2D rotation inside the plane across the axis; the quaternion as the 3D descendant of the unit complex number, with its sandwich worked on a point and the half angle taken apart one factor at a time; composition, blending and Euler angles. The Appendix recovers an axis and angle from a matrix, steps small rotations, converts a quaternion to a matrix and back, extracts Euler angles at the lock, and measures the drift with real single-precision arithmetic. One axis and angle, \((0, 1, 0)\) at \(30^\circ\), runs through the worked examples and the live demo.

1 · What a rotation preserves

Because a rotation is linear and fixes the origin, it is completely determined by where it sends the three basis vectors. That one fact makes rotation matrices readable, which is the next section.

1.1 The three properties, and why they follow from one

1.2 Reflections, and why two of them are a rotation

2 · The rotation matrix: columns are where the axes land

In two dimensions, a point at radius \(r\) and angle \(\varphi\) is \((r\cos\varphi, r\sin\varphi)\); rotating it by \(\theta\) gives \((r\cos(\varphi + \theta), r\sin(\varphi + \theta))\). Expand with the angle-addition formulas and regroup in terms of \(x = r\cos\varphi\), \(y = r\sin\varphi\):

\[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}. \]

Rotation about the \(y\) axis leaves \(y\) alone and spins the \(x\)–\(z\) plane, so the 2D block drops into the \(x, z\) slots:

\[ R_y(\theta) = \begin{pmatrix} \cos\theta & 0 & \sin\theta \\ 0 & 1 & 0 \\ -\sin\theta & 0 & \cos\theta \end{pmatrix}, \qquad R_y(30^\circ) = \begin{pmatrix} 0.866 & 0 & 0.5 \\ 0 & 1 & 0 \\ -0.5 & 0 & 0.866 \end{pmatrix}. \]

Watch the sign pattern: for \(R_y\) the \(+\sin\) sits top-right, the opposite of the textbook \(R_z\). Look down from above, with \(y\) pointing at you, \(z\) to the right and \(x\) up: the turn about \(y\) is the ordinary counter-clockwise 2D rotation with \(z\) in the role of \(x\) and \(x\) in the role of \(y\), so \(z\) tips toward \(x\) and \(x\) tips toward \(-z\). Column 0 says so: \((\cos\theta, 0, -\sin\theta)\).

The three axis rotations for reference, all counter-clockwise when looking down the axis toward the origin:

\[ R_x(\theta) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix}, \quad R_z(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}. \]

3 · Axis-angle: Rodrigues’ formula

A rotation can be about any unit axis \(\hat{\mathbf{n}}\) by any angle \(\theta\). That pair is the most human representation (“turn 40 degrees about this stick”), and applying it to a vector needs only the projection split and one cross product.

1. Split \(\mathbf{v}\) into the part along the axis, which a rotation about that axis cannot move, and the part across it: \(\mathbf{v}_\parallel = (\mathbf{v}\cdot\hat{\mathbf{n}})\hat{\mathbf{n}}\), \(\mathbf{v}_\perp = \mathbf{v} - \mathbf{v}_\parallel\).
2. Build the second axis of the rotation plane: \(\mathbf{w} = \hat{\mathbf{n}}\times\mathbf{v}\) is perpendicular to both \(\hat{\mathbf{n}}\) and \(\mathbf{v}_\perp\) and has the same length as \(\mathbf{v}_\perp\), since \(\|\hat{\mathbf{n}}\times\mathbf{v}\| = \|\mathbf{v}\|\sin\angle(\hat{\mathbf{n}},\mathbf{v}) = \|\mathbf{v}_\perp\|\).
3. Turn inside the plane, a 2D rotation with \(\mathbf{v}_\perp\) and \(\mathbf{w}\) as the two axes: \(\mathbf{v}_\perp' = \cos\theta\,\mathbf{v}_\perp + \sin\theta\,\mathbf{w}\).
4. Add the frozen part back and substitute.

3.1 Rodrigues as a matrix

The cross product with a fixed vector is a linear map, so it has a matrix. For a unit axis \(\hat{\mathbf{n}} = (n_x, n_y, n_z)\), \[ K = \begin{pmatrix} 0 & -n_z & n_y \\ n_z & 0 & -n_x \\ -n_y & n_x & 0 \end{pmatrix}, \qquad K\mathbf{v} = \hat{\mathbf{n}}\times\mathbf{v}, \] and \(K\) is skew-symmetric, \(K^{\mathsf T} = -K\). Two applications give \(K^2\mathbf{v} = \hat{\mathbf{n}}\times(\hat{\mathbf{n}}\times\mathbf{v}) = (\mathbf{v}\cdot\hat{\mathbf{n}})\hat{\mathbf{n}} - \mathbf{v}\), which is minus the perpendicular part of \(\mathbf{v}\). Substituting both into Rodrigues’ formula turns it into a matrix identity:

4 · Quaternions

Why not store the matrix? Nine numbers with six hidden constraints, for three degrees of freedom. Composing matrices in floating point erodes the constraints and shapes begin to shear (Appendix A.5 measures it); averaging two rotation matrices is not a rotation, so orientations cannot be blended. A quaternion has four numbers, one constraint, composes by multiplication, and blends along a sphere.

4.0 What a quaternion is

A complex number \(a + b\,i\) with \(i^2 = -1\) is a point of the plane, and multiplying by the unit complex number \(\cos\theta + i\sin\theta\) turns the plane by \(\theta\): \((x + y\,i)(\cos\theta + i\sin\theta) = (x\cos\theta - y\sin\theta) + (x\sin\theta + y\cos\theta)\,i\), the two rows of the 2D matrix of Section 2. For the point \((2, 1)\) and \(30^\circ\), \((2 + i)(0.866 + 0.5\,i) = 1.232 + 1.866\,i\), the same numbers. Two turns compose by multiplying, the angles add, and the order does not matter; in three dimensions it does (Section 5), so the three-dimensional version needs a product whose order matters.

William Rowan Hamilton spent years looking for the three-dimensional version and found, on 16 October 1843, that it needs three imaginary units. A quaternion is \(w + x\,i + y\,j + z\,k\), multiplied by the rule he carved into Broom Bridge in Dublin,

\[ i^2 = j^2 = k^2 = ijk = -1, \qquad\text{so}\qquad ij = k,\; jk = i,\; ki = j, \qquad ji = -k,\; kj = -i,\; ik = -j . \]

\(w\) is the scalar part and \((x, y, z)\) the vector part; the words entered mathematics here. Writing a quaternion as (vector part, scalar part) and expanding a product term by term with these rules gives

\[ (\mathbf{v}, w)\,(\mathbf{u}, s) = \big(w\,\mathbf{u} + s\,\mathbf{v} + \mathbf{v}\times\mathbf{u},\;\; ws - \mathbf{v}\cdot\mathbf{u}\big), \]

a dot, a cross and two scalings. The cross product is why the product does not commute, and rotations do not commute either.

The half angle is the one surprise. It comes from how a quaternion rotates a point: write the point as the quaternion \((\mathbf{p}, 0)\) and multiply on both sides, a sandwich, \(p' = q\,p\,q^{-1}\). Split \(\mathbf{p}\) as in Section 3. On the part across the axis, \(\hat{\mathbf{n}}\cdot\mathbf{p} = 0\), so the product rule above gives \(q\,(\mathbf{p}, 0) = (\cos\tfrac\theta2\,\mathbf{p} + \sin\tfrac\theta2\,\hat{\mathbf{n}}\times\mathbf{p},\; 0)\): the left factor turns it by \(\theta/2\), and the right factor \(q^{-1}\) turns it by another \(\theta/2\). On the part along the axis, the left factor tilts it into the fourth (\(w\)) component and the right factor tilts it back, leaving it unchanged. So the point turns by \(\theta\) while \(q\) itself has turned only \(\theta/2\). A one-sided product \(q\,p\) would turn the across part by half the angle and smear the along part into \(w\); the sandwich is what makes the halves add.

Expanded into vector operations (with \(\mathbf{q}_v = (x, y, z)\)) the sandwich needs no trigonometry at all:

\[ \mathbf{p}' \;=\; \mathbf{p} \;+\; 2w\,(\mathbf{q}_v\times\mathbf{p}) \;+\; 2\,\mathbf{q}_v\times(\mathbf{q}_v\times\mathbf{p}) . \]

4.1 The double cover

\(q\) and \(-q\) give the same rotation: the sandwich \((-q)\,p\,(-q)^{-1}\) uses the quaternion twice, and the two signs cancel. So \((0, -0.259, 0, -0.966)\) is also \(R_y(30^\circ)\); it is the quaternion of \(R_y(390^\circ)\), a full extra turn, and \(\sin 195^\circ, \cos 195^\circ\) are exactly those numbers. Every rotation has two quaternions, antipodal on the unit sphere. This matters in exactly one place, interpolation: the arc from \(q_1\) to \(q_2\) and the arc from \(q_1\) to \(-q_2\) reach the same orientation, but one is short and the other goes the long way round. Section 5 tests the sign of \(q_1\cdot q_2\) for that reason. The name for the two-to-one relation is the double cover: the unit quaternions cover the rotations twice.

5 · Composition and interpolation

Do rotation \(q_1\), then \(q_2\): multiply, \(q = q_2\,q_1\), with the one nearest the point applied first, the same right-to-left reading as matrices. The product is

\[ \begin{aligned} (q_1 q_2)_x &= \phantom{-}x_1 w_2 + y_1 z_2 - z_1 y_2 + w_1 x_2, \\ (q_1 q_2)_y &= -x_1 z_2 + y_1 w_2 + z_1 x_2 + w_1 y_2, \\ (q_1 q_2)_z &= \phantom{-}x_1 y_2 - y_1 x_2 + z_1 w_2 + w_1 z_2, \\ (q_1 q_2)_w &= -x_1 x_2 - y_1 y_2 - z_1 z_2 + w_1 w_2, \end{aligned} \]

sixteen multiplies against a 3×3 product’s twenty-seven, and the result stays a rotation after a single renormalization. In vector form the same product is \(\mathbf{q}_v = w_1\mathbf{q}_{2v} + w_2\mathbf{q}_{1v} + \mathbf{q}_{1v}\times\mathbf{q}_{2v}\), \(w = w_1 w_2 - \mathbf{q}_{1v}\cdot\mathbf{q}_{2v}\): a cross product sits inside it, which is why it is not commutative, exactly as rotations are not. Pitch then yaw is not yaw then pitch.

5.1 From one direction to another

A fifth way to name a rotation: “turn this direction onto that one”, as when a head turns toward a target. For unit vectors \(\mathbf{a}\) and \(\mathbf{b}\), the smallest such rotation has axis \(\mathbf{a}\times\mathbf{b}\) and angle \(\arccos(\mathbf{a}\cdot\mathbf{b})\), and its quaternion needs no trigonometry: since \(1 + \mathbf{a}\cdot\mathbf{b} = 1 + \cos\theta = 2\cos^2\tfrac\theta2\) and \(\|\mathbf{a}\times\mathbf{b}\| = \sin\theta = 2\sin\tfrac\theta2\cos\tfrac\theta2\), both carry the factor \(2\cos\tfrac\theta2\), and

\[ q = \mathrm{normalize}\big(\mathbf{a}\times\mathbf{b},\; 1 + \mathbf{a}\cdot\mathbf{b}\big) \]

is exactly \((\sin\tfrac\theta2\,\hat{\mathbf{n}}, \cos\tfrac\theta2)\). Unity’s Quaternion.FromToRotation and three.js’s setFromUnitVectors compute it. It names one rotation among many: any extra twist about \(\mathbf{b}\) also sends \(\mathbf{a}\) onto \(\mathbf{b}\), and this one has none.

5.2 Blending orientations

To ease a camera from orientation \(q_1\) to \(q_2\), walk the short arc between them on the unit sphere at constant angular speed (slerp, spherical linear interpolation). The unit quaternions are the points \((x, y, z, w)\) of length 1, a sphere in four dimensions; the shortest path between two of them is a great arc, which lies in the plane through \(q_1\), \(q_2\) and the origin, so every slerp happens on an ordinary circle. A straight-line blend leaves the sphere: halfway between two orientations \(120^\circ\) apart (\(60^\circ\) apart on the sphere), the average quaternion has length \(0.866\), and renormalizing it gives a rotation that moves fast in the middle and slow at the ends.

\[ \mathrm{slerp}(q_1, q_2, t) = \frac{\sin\big((1-t)\Omega\big)}{\sin\Omega}\,q_1 + \frac{\sin(t\,\Omega)}{\sin\Omega}\,q_2, \qquad \cos\Omega = q_1\cdot q_2 . \]

\(\Omega\) is the angle between the two quaternions in four dimensions, half the turn between the orientations (the half angle again).

(If \(q_1\cdot q_2 < 0\), negate one of them first: \(q\) and \(-q\) are the same rotation, and the short arc is the one between the nearer pair. For the pair below, using \(-q_2\) would give \(\cos\Omega = -0.5\), \(\Omega = 120^\circ\), and a swing of \(240^\circ\) the other way round.) Stepping an animation by composing the same small quaternion each frame is a discretized slerp; it is how a turntable, a door, or an orbiting camera is animated without ever touching a matrix, and the animation chapter measures slerp against the normalized-lerp shortcut, nlerp (normalize the straight blend): it walks the same arc at an uneven speed, \(27.8^\circ\) at \(t = 0.25\) where slerp has turned \(30^\circ\) for the pair below, with no sine or arccosine to evaluate.

6 · Euler angles and gimbal lock

The vocabulary comes from vehicles. Yaw turns about the up axis (look left or right), pitch about the side or wing axis (nose up or down), roll about the forward axis (bank). In Unity, with \(y\) up and \(z\) forward, yaw is about \(y\), pitch about \(x\) and roll about \(z\); aerospace puts \(x\) forward and \(z\) down, so there roll is about \(x\), pitch about \(y\) and yaw about \(z\). Same words, different letters.

The human way to say an orientation: three rotations about three axes, in a fixed order. Unity’s Quaternion.Euler(x, y, z) applies \(Z\) first, then \(X\), then \(Y\), so the world-space product is \(R_y\,R_x\,R_z\). Three readable numbers, “pitch 20, yaw 45, roll 0”, but the order is a convention, and different tools pick different ones, which is the usual cause of an imported model arriving rotated wrong.

6.1 Conventions

Twelve orders are possible (six with three distinct axes, six that repeat the first), each read either intrinsically (each rotation about the body’s own, already-rotated axis) or extrinsically (each about a fixed world axis). The two readings are the same product written in opposite order: intrinsic \(X\), then \(Y\), then \(Z\) is the matrix \(R_x R_y R_z\), which is also extrinsic \(Z\), then \(Y\), then \(X\). A library’s name for its order does not say which reading it means; the matrix does.

system	order name	matrix on a column vector	reads as
Unity	ZXY	\(R_y\,R_x\,R_z\)	\(Z\) first about the world axes, then \(X\), then \(Y\) (about the body: \(Y\), \(X\), \(Z\))
three.js default, and the library’s makeTRS	XYZ	\(R_x\,R_y\,R_z\)	\(Z\) first about the world axes, then \(Y\), then \(X\) (about the body: \(X\), \(Y\), \(Z\))
aerospace (yaw, pitch, roll)	ZYX	\(R_z\,R_y\,R_x\)	roll about world \(x\) first, then pitch, then yaw

6.2 Gimbals and the lock

A gimbal is a ring on two pins: it turns about one axis, the line through its pins. Nest three, each pinned inside the last, and the outer ring turns everything inside it while the inner one turns only the object. Three nested gimbals are a physical Euler-angle triple read intrinsically: the outer angle is the leftmost factor of the product, the inner angle the rightmost. The outer ring is pinned to the ground, so its axis is a world axis; the object is fixed to the inner ring, so that ring’s axis is one of the object’s own. Gyroscopes, camera heads and spacecraft inertial platforms are built this way, which is where the next failure got its name.

6.3 What locks: the angles, not the airplane

The two axes that line up at the lock are not two of the airplane’s axes. Those stay perpendicular at every pose, and turning the airplane about any of them always works: a program that keeps the orientation as a quaternion or a matrix and multiplies in a small turn about the body’s own axis, \(R \leftarrow R\,R_{\text{axis}}(\delta)\), can turn any way from any pose and never locks. That is how engines and physics apply rotation (Unity’s transform.rotation * Quaternion.AngleAxis(d, Vector3.up), three.js’s object.rotateY(d)). The axes that line up are the outer ring’s, fixed in the world, and the inner ring’s, fixed in the airplane.

What fails is the description by three angles. A program whose state is the triple \((x, y, z)\) (an inspector, an animation curve on Euler channels, a camera kept as yaw and pitch) has to express every turn as changes of the three numbers, and near the lock some turns need enormous changes.

A physical gimbal is that program in hardware: its three ring angles are the state. To hold an inertial platform still while the spacecraft turns about the dead direction, the gimbal motors would have to turn \(1/\cos x\) times as fast as the craft, so the Apollo crews flew around the lock orientation. Nothing is wrong with the orientation at pitch \(90^\circ\); the object is fine. It is the chart \((\text{yaw}, \text{pitch}, \text{roll})\) that goes singular there (a chart is a set of coordinates laid over the rotations, as latitude and longitude are laid over the globe), the way longitude is undefined at the poles: one step at the North Pole can change your longitude by \(90^\circ\). A quaternion names an axis and an angle with no nested frames, so there is no configuration where two of its parts align. That is the engineering reason engines store the quaternion and show Euler angles only for editing: transform.rotation is a quaternion under the hood, always. Cameras that do keep yaw and pitch as their state clamp the pitch short of \(\pm 90^\circ\).

7 · Choosing a representation

	matrix \(3\times3\)	axis-angle	quaternion	Euler angles
numbers stored	9	4 (or 3, as \(\theta\hat{\mathbf{n}}\))	4	3
constraints	6	1 (unit axis)	1 (unit length)	0
apply to a point	9 multiplies	via Rodrigues, about 20	via the sandwich, about 20; or convert once	convert first
compose	27 multiplies	convert first	16 multiplies	convert first
invert	transpose	negate \(\theta\)	conjugate	negate and reverse the order
interpolate	no	awkward	slerp	badly (Section 6, and the animation chapter)
singularities	none	axis undefined at \(\theta = 0\)	none (a double cover)	gimbal lock
drift under composition	shears	none stored	length only, one divide fixes it	none stored
readable by a person	columns are axes	yes	no	yes

The practice that follows from the table: store and interpolate a quaternion, compose quaternions, convert to a matrix once per frame for the GPU, show Euler angles in the inspector, and use axis-angle when the physics hands you an angular velocity. Every conversion in that pipeline is in this chapter. The fifth name, “turn this direction onto that one” (Section 5.1), is a way to build a rotation rather than to store one; it produces a quaternion.

8 · In code

9 · Exercises

Appendix

A.1 The log map: axis and angle from a matrix

The trace of a matrix is the sum of its diagonal entries. For a rotation it has a picture. Diagonal entry \(i\) is \(\mathbf{e}_i\cdot(R\mathbf{e}_i)\): how much of axis \(i\) is still pointing along itself after the turn. Measure with a frame lined up with the rotation, one axis along \(\hat{\mathbf{n}}\) and two in the plane across it: the axis keeps all of itself (1), and each in-plane direction keeps \(\cos\theta\) of itself. The trace is the same in every orthonormal frame (a change of frame is \(R\mapsto Q^{\mathsf T}RQ\), and the trace of a product does not change when its factors are cycled), so \(\mathrm{tr}\,R = 1 + 2\cos\theta\) whatever the axis. For \(R_y(30^\circ)\): \(0.866 + 1 + 0.866 = 2.732 = 1 + 2(0.866)\).

The same fact, from the matrix form: the trace of \(R\) is \(3 + 0 + (1 - \cos\theta)\,\mathrm{tr}(K^2)\), and \(\mathrm{tr}(K^2) = -2\) for a unit axis, so \(\mathrm{tr}\,R = 1 + 2\cos\theta\). The symmetric part of \(R\) is \(I + (1 - \cos\theta)K^2\) and the antisymmetric part (half of \(R - R^{\mathsf T}\), the matrix minus its transpose, which keeps only the off-diagonal differences) is \(\sin\theta\,K\), from which the axis can be read entry by entry:

A.2 Small angles, and the angular velocity

For small \(\theta\) (in radians), \(\sin\theta\approx\theta\) and \(1 - \cos\theta\approx\theta^2/2\), so \(R \approx I + \theta K + \tfrac12\theta^2K^2\), and to first order (keeping the terms proportional to \(\theta\) and dropping \(\theta^2, \theta^3, \ldots\)) \(R\mathbf{v}\approx\mathbf{v} + \theta\,\hat{\mathbf{n}}\times\mathbf{v}\): a small turn moves every point along the tangent \(\hat{\mathbf{n}}\times\mathbf{v}\) of its circle. That is the physics of spinning: a body turning about \(\hat{\mathbf{n}}\) at rate \(\omega\) has angular velocity \(\boldsymbol\omega = \omega\hat{\mathbf{n}}\), and a point at \(\mathbf{p}\) on it moves with velocity \(\boldsymbol\omega\times\mathbf{p}\). The exact rotation is the limit of many small ones, \(R = \exp(\theta K)\), the matrix exponential, defined by the same power series as \(e^x\): \(\exp(A) = I + A + A^2/2! + A^3/3! + \cdots\). For \(A = \theta K\), \(K^3 = -K\) folds every power back onto \(K\) or \(K^2\), the odd terms sum to \(\sin\theta\,K\) and the even ones to \((1 - \cos\theta)K^2\), which is Rodrigues again. That is why the axis-angle vector \(\theta\hat{\mathbf{n}}\) is called the exponential coordinate of a rotation and Appendix A.1 its logarithm.

A.3 Quaternion to matrix, and matrix to quaternion

Apply the sandwich to the three basis vectors and the columns of the matrix come out as polynomials in the four components. That is how an engine turns the stored quaternion into the matrix the GPU needs, every frame, with no trigonometry:

A.4 Extracting Euler angles from a matrix

Multiplying out \(R_y(y)\,R_x(x)\,R_z(z)\) puts single trigonometric factors in three places: the entry \((1, 2)\) is \(-\sin x\), the entries \((0, 2)\) and \((2, 2)\) are \(\sin y\cos x\) and \(\cos y\cos x\), and the entries \((1, 0)\) and \((1, 1)\) are \(\cos x\sin z\) and \(\cos x\cos z\). So, with \(\mathrm{atan2}(s, c)\) the angle whose sine and cosine are in the ratio \(s : c\), in all four quadrants,

\[ x = \arcsin(-R_{12}), \qquad y = \mathrm{atan2}(R_{02}, R_{22}), \qquad z = \mathrm{atan2}(R_{10}, R_{11}), \]

and the extraction is exact as long as \(\cos x \ne 0\).

Even away from the lock, a matrix has two Euler triples: \((20^\circ, 45^\circ, 10^\circ)\) and \((160^\circ, 225^\circ, 190^\circ)\) produce the same matrix, since \(x\mapsto180^\circ - x\) with \(180^\circ\) added to the other two is the same orientation. The extraction above always returns the one with \(x\in[-90^\circ, 90^\circ]\).

A.5 Drift, measured

The claim “matrices drift” can be measured. Compose the same \(1^\circ\) rotation 200,000 times in single-precision arithmetic three ways: as a \(3\times3\) matrix product, as a quaternion product, and as a quaternion product renormalized after every step.

The matrix needs six constraints re-imposed (a full re-orthonormalization, which is the Gram–Schmidt of the vectors chapter applied to its columns); the quaternion needs one (divide by its length), and that is cheap enough to do every frame. A game running at 60 frames per second accumulates 200,000 compositions in under an hour.

A.6 Exercises