algorithmic modeling for Rhino
One of the most simple arches of Persian architecture is the circular arch(6&5). This arch is drawn as follow:
1- Define the span (S) and divide it into 12 equal segments (S/12). This would be six segments for each half-span.
2- The height of the arch will be five units according to the divided segments (5*S/12). This arch is called (6 & 5) because of the span to height ratios, in other words if the half of the span has a 6 unit length the height will have a 5.
2- A line is connected from the impost to the crown of the arch. A perpendicular line is then drawn from the middle of the crown-impost line. This line is extended till it intersects the vertical axis.
3- The intersection is assumed to be the center of the arch and the radius is the distance between this center and the impost or crown. The arch is then drawn and is mirrored to have a full circular arch (6,5).
For drawing a parametric arch we have to define the variables. The goal is to parametrically define the center and radius of the arch and also the angle domain.
the summary for the parametric arch is:
Center of arch : C(0,(d/4*t) 1-t^2)
Radius of arch : (d/4*t) 1+t^2
Angle domain : arcsin ((t^2 - 1) / (t^2 + 1)) to pi/2
You can see the full calculations in my blog : Circular Arch
I have added this arch as the first component of Starfish (A Grasshopper3d plugin I'm preparing). You can download the Grasshopper3d assembly and copy it in the component folder(or simply drag it in Grasshopper canvas). A starfish menu will be added and the CirArch component can be used.