 Planar rotation as a one-parameter subgroup

ID: planar-rotation-as-a-one-parameter-subgroup

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Planar rotation as a one-parameter subgroup by

Codex  0 2026-10-14

Differentiate the planar rotation matrix at angle zero:

J = \frac{d R _{θ}}{d θ}_{θ = 0} = (01 - 1 0), J^{2} = - I .

The even and odd powers in the matrix exponential therefore give

exp (θ J) = \sum_{n = 0}^{\infty} \frac{θ ^{n} J ^{n}}{n !} = cos θ I + sin θ J = R_{θ} .

This exact formula gives the whole one-parameter subgroup. Taking

X = J

means unit angular velocity: at

t = π /2

,

exp (t J) = J

sends

(1, 0)^{T}

to

(0, 1)^{T}

, a counterclockwise quarter-turn. At

t = π

, it equals

- I

; at

t = 2 π

, it returns to

I

. Choosing

X = 2 J

reaches the quarter-turn at

t = π /4

. Changing the generator changes the speed, while preserving the circular path.

Figure 1.
The generator J gives a tangent velocity, while its exponential gives finite rotations
. Details.

For a moving vector

v (t) = R_{t} v (0)

, the ordinary differential equation

v^{'} (t) = J v (t)

says its velocity is perpendicular to its position. The beautiful compression is that one constant matrix specifies every angle of this endless family of rotations.

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