Given just a few points on a log spiral, how to find the center? Considering that the circle is a degenerate case of the log spiral, is there a way to generalize the method for finding circle centers (i.e. via Euclid III.1)?
I see here that the center of the golden log spiral is found at the point where the long diagonals of the inscribing golden rectangles meet. Alternatively, inscribed golden triangles can be used. Perhaps there is some way to generalize this method to all log spirals?
A logarithmic spiral has the property that rays from the center cut the spiral at the same angle $\alpha$. So if you had tangents at two points $p_1, p_2$ on the curve, you could hypothesize a center $(x,y)$, compute the angles $\alpha_1, \alpha_2$, and require $\alpha_1=\alpha_2$.
Red rays make different angles with tangents; green rays make same angle, $107^\circ$.
That gives you one equation in the two unknowns $(x,y)$. So, barring degeneracies, tangents at three distinct points would permit solving for $(x,y)$.
This approach requires enough points of the spiral to determine (or approximate) three distinct tangents.