Foundations — Converting between DCM, quaternions, Euler angles

Before you can convert between representations, you must be fluent in the raw ingredients. We build them in dependency order: arrows → frames → the numbers that link two frames → the matrix that packages those numbers → the axis-angle idea → the half-angle quaternion → the trig tools that read angles back out.

1. A vector $\vec{v}$ — an arrow with a length and a direction

What to extract from Figure s01: the burnt-orange arrow is $\vec{v}$. Its shadow on the floor (teal dashes) gives you $v_x$ and $v_y$; its vertical rise (plum dash) gives $v_z$. The lesson: those three shadows ARE the three numbers — nothing more mysterious than "how far along each direction".

Why the topic needs it: the whole point of a rotation is to take one arrow and turn it into another arrow, $\vec{v}\to {\vec{v}}^{\prime }$. Everything downstream — DCM, quaternion, Euler angles — is machinery for describing that one turn.

2. A unit vector $\hat{e}$ — an arrow of length exactly 1

Why the topic needs it: the rotation axis $\hat{e}=(e_1,e_2,e_3)$ is always a unit vector — we only care which way the spin-axis points, not how long we draw it. The quaternion formula $(q_1,q_2,q_3)=\hat{e}\sin \frac{\Phi }{2}$ (built in §9) multiplies that pure direction by a number.

3. Two frames A and B — two sets of reference axes

What to extract from Figure s02: the teal star of arrows is frame $A$ (the world). The plum star is the same origin twisted — frame $B$ (the body). The single lesson: the whole topic asks only "how is B rotated relative to A?" — every representation is an answer to this picture.

Why the topic needs it: DCM, quaternion, and Euler angles are all just three ways to write down the twist between these two star-shaped bundles of arrows.

4. The dot product $\vec{a}\cdot \vec{b}$ — "how much do two arrows agree?"

Why the topic needs it: a direction cosine is literally a dot product between one axis of frame $A$ and one axis of frame $B$. The name Direction Cosine Matrix means "the table of all nine such cosines".

5. The cross product $\vec{a}\times \vec{b}$ and the skew matrix $[\hat{e}{]}_{\times }$

Why the topic needs it: the matrix $[\hat{e}{]}_{\times }$ lets Rodrigues' rotation be written as one clean matrix formula instead of a cross-product buried inside. It is the bridge that turns "geometry" into "a table you can multiply".

6. A matrix $C$ and the product $C\vec{v}$ — a machine that eats an arrow, spits out an arrow

Why the topic needs it: the DCM is the "hub" of all conversions. Its 9 numbers with 6 constraints (3 unit-length + 3 perpendicularity) is the honest, bulky encoding everything else compresses. See Direction Cosine Matrix Properties.

7. The identity matrix $I$ — "do nothing"

Why the topic needs it: Rodrigues' formula starts from $I\cos \Phi$: at $\Phi =0$, $\cos 0=1$, so $C=I$ — zero rotation does nothing. This is the sanity check that anchors the whole formula.

8. Axis–angle $(\hat{e},\Phi )$ — the master key

What to extract from Figure s03: the plum arrow $\hat{e}$ is the skewer through the object. The orange arrow $\vec{v}$ swings along the dark arc to the teal arrow ${\vec{v}}^{\prime }$; the arc's opening angle is $\Phi$ (Greek capital phi). The lesson: a rotation is fully captured by just two things — the direction of the skewer $\hat{e}$ and the amount of turn $\Phi$.

Why the topic needs it: both DCM and quaternion have clean formulas straight from $(\hat{e},\Phi )$. So the parent derives both from this anchor, then composes conversions. Master this and the rest is bookkeeping. See Rodrigues Rotation Formula.

9. $\cos$, $\sin$, $\tan$ — and the half-angle $\frac{\Phi }{2}$

Why the topic needs it: every conversion formula is trig glue. In particular, $q_0=\cos \frac{\Phi }{2}$ (defined next in §10), and the identities $\sin \Phi =2\sin \frac{\Phi }{2}\cos \frac{\Phi }{2}$ and $1-\cos \Phi =2{\sin }^2\frac{\Phi }{2}$ are what turn "quaternion" into "DCM".

10. The quaternion $q$ — a 4-number rotation token

11. Euler angles $(\varphi ,\theta ,\psi )$ — three human-readable tilts

What to extract from Figure s04: a little aircraft with three curved arrows — orange yaw (nose swings left/right about vertical), teal pitch (nose tips up/down), plum roll (wings tilt about the nose line). The lesson: Euler angles are the human way to say orientation, but because each turn changes the axes for the next, doing them in a different order gives a different result — and at $\theta =\pm 90°$ two axes line up (Gimbal Lock).

Why the topic needs it: Euler angles are how a rotation gets reported to a person. Pulling them back out of a DCM (e.g. $\theta =-\arcsin C_{13}$, $\varphi =atan2(C_{23},C_{33})$) is exactly why we need the inverse-trig tools of §12.

12. $\arcsin$ and $atan2$ — reading an angle back out

What to extract from Figure s05: two arrows with the same ratio $y/x$ — the orange one in quadrant I $(+,+)$, the teal one in quadrant III $(-,-)$. Plain $\arctan (y/x)$ sees only the ratio and confuses them; $atan2$ reads both signs and lands each in its correct quadrant. The lesson: always feed both numbers separately.

Why the topic needs it: extracting Euler angles from a DCM requires full-circle inverse trig, or your reported roll/pitch/yaw will silently flip. This tool is also where Gimbal Lock shows up: at pitch $=\pm 90°$ the arguments become $atan2(0,0)$ — undefined.

Prerequisite map

The chain below reads top-down: raw arrows spawn the dot and cross products; those two build both the DCM and the quaternion's multiplication rule; axis-angle plus half-angle trig fills in the numbers; inverse trig reads out Euler angles. All three representations meet at the topic.

Equipment checklist

Test yourself — you should be able to answer each before opening the parent note.