QuarterCircleSpiral

Excellent posts by John D Cook:
https://www.johndcook.com/blog/2017/05/01/golden-angle/
https://www.johndcook.com/blog/2022/05/09/iterated-golden-rectangles/
https://www.johndcook.com/blog/2022/05/09/logarithmic-spiral/
https://www.johndcook.com/blog/2022/05/10/spiral-arcs/

The Golden Spiral is a logarithmic spiral, with the radius continuously changing with angle theta.

The Golden Spiral can be approximated with circular arcs, in golden rectangles, where each quarter arc has a constant radius, but each successive quarter arc has its radius reduced by the factor of the (golden ratio - 1) = (phi-1) = about 0.618034, which is the "decay factor" The QtrArc spiral is very easy to construct in MoI, and the center of the spiral constructed. Orient Line,Line is very useful.
Adjoining arcs are connected G1, with equal tangents at the connection.
The connection is not G2. The second derivatives are different.

With the Golden rectangles beginning at the Origin, the coordinates of the center of the QtrArc spirals has been calculated by John D Cook.
The center is also located by two "diagonal" lines.
A golden logarithmic spiral can also be created using the LogSpiral5 script, and Orient used to adjust it to the QtrArc ends and the center.
The two spirals are different, but are very close to each other.

Using QtrArc with decay of 85%, 0.85, may approximate violin volutes.
Or multiple decay values may be needed.
http://zhurnal.lib.ru/m/muratow_s_w/violin_design.shtml

The formulas for the Golden Arc Spiral are a bit simpler.
Using John D Cooks slope_intersect formula method for the golden spiral, for a spiral with a different decay factor, e.g. 0.85,
enables the center of QtrArcDecaySpiral to be calculated from the two "diagonals". The initial decay square begins at the origin.
The formula for the coordinates (x0,y0) of the center of the decay spiral are as follows, solving for x0:
x0 = (1+decay)/(decay*decay) For 85% decay x0 = 1.0740203 (+/-).
y0 = (-decay)*(x0) +1 For 85% decay y0 = 0.087083 (+/-).

Construction of the decay arcs in MoI confirms the values.
The logSpiral can also be plotted by calculating associated exponent
value "b", as previously remarked.

- Brian