[coldcity_again]

I've been thinking about this since [1] the other day, but I still love how rotation by small angles lets you drop trig entirely.

Let α represent a roll rotation, and β a pitch rotation.

Let R(α) be:

    ( cos α   sin α   0)
    (-sin α   cos α   0)
    (   0       0     1)

Let R(β) be:

    (1     0       0   )
    (0   cos β   -sin β)
    (0   sin β    cos β)

Combine them:

    R(β).R(α) = (    cos α           sin α          0   )
                ((-sin α*cos β)   (cos α*cos β)   -sin β)
                ((-sin α*sin β)   (cos α*sin β)    cos β)

But! For small α and β, just approximate:

    ( 1   α   0)
    (-α   1  -β)
    ( 0   β   1)

So now:

    x' = x + αy
    y' = y - αx - βz
    z' = z + βy

[1]https://news.ycombinator.com/item?id=47348192

[coldcity_again]

If you just see the conclusion I think it's hard to immediately grok how rotation can arise from this.

This is a great technique for cheaply doing 3D starfields etc on 8-bit machines.

Look ma, no sine table!

[srean]

A related interesting fact is that small angular motions compose almost like vectors, order does not matter (i.e. they are commutative). This makes differential kinematics easier to deal with when dealing with polar or cylindrical coordinate systems.

Large angular deflections while being linear transforms, do not in general commute.

It will spoil the linear relation in your elegant expression, but a slightly better approximation for cos for small θ is

    1 - 0.5θ²

[coldcity_again]

Great point, thanks.