This is the derivation for my main “antigravity spoon” entry.

- `theta = pi * t/(12hr)`
- `vec w = ( x_(w), y_(w) ) = R*( sin theta, -cos theta )`
- `vec s = ( x_(s), y_(s) ) = R*( theta, -1 )`

- `hat u = ( sin theta, -cos theta )`
- `hat e = ( cos theta, sin theta )`
- `vec Delta = vec s – vec w`… `= ( x_(s) – x_(w), y_(s) – y_(w) )`… `= ( R*theta – R*sin theta, -R + R*cos theta )`
- `vec delta = vec Delta / R = ( theta – sin theta, -1 + cos theta )`
- `vec A = vec delta * hat u`… `= (theta – sin theta)*(sin theta) + (-1 + cos theta)*(-cos theta)`… `= (theta * sin theta – sin^2 theta) + (cos theta – cos^2 theta)`
… `= theta * sin theta + cos theta – 1`

✓ checking: there should be no altitude at `theta = 0`: ✓

✓ `vec A(0) = 0 * sin 0 + cos 0 – 1 = 0*0 + 1 – 1 = 0` ✓

- `vec Q = vec delta * hat e`… `= (theta – sin theta)*(cos theta) + (-1 + cos theta)*(sin theta)`… `= (theta * cos theta – sin theta * cos theta) + (-sin theta + cos theta * sin theta)`
… `= theta * cos theta – sin theta`

✓ checking: there should be no eastward travel at `theta = 0`: ✓

✓ `vec Q(0) = 0 * cos 0 – sin 0 = 0*1 + 0 = 0` ✓