I found this an interesting problem so I gave it a little time.
I derived an expression that calculates the spacing distance for rectangles for use with arraypath when used on a concave arc. The user inputs the arc radius, the rectangle length, and its width. I created an Excel file with the expression as well as a VLISP program. In Excel it looks like this:
=(2*RCave^2*(1-COS(2*ATAN((L/2)/((RCave^2-(L/2)^2)^0.5-W))) ) )^0.5
Where RCave = arc radius, L = the rectangle length, and W = the rectangle width.
The workflow for laying out the stones is as follows given the green polyline in the following sequence.
- The rectangles in this example have a length of 4 and a width of 3. The concave radius is about 14.5. Using arraypath with a distance of 4 yields this:
Note how the bottom left corner touch the green polyline. It is better to use the midpoint of the rectangle as the base point so I shortened the path by L/2. I did this by creating a circle with a radius of L/2 = 2 at the start of the path and then trimming to it. This is the result of using arraypath again on the trimmed path. The number of copies is enough to bring the shape to the approximate beginning of the concave portion of the path.
- The path is exploded to isolate the concave portion. Using the VLISP program or Excel, the concave distance spacing is determined to be about 5.027
Command: STONECALC
Enter Concave Radius. 14.51
Enter stone length. 4
Enter stone width. 3
angle = 19.9501°
s = 5.02683
The angle value is the angle that could be used in a polar array if desired.
- A circle of half this distance is created at the bottom right hand corner of the last rectangle to find the midpoint of the first box on the concave arc. Its orientation is determined by a line from that point to the arc’s center (cyan line). The align command is used to orient the magenta box that will be used with arraypath.
- arraypath is then used to create the remainder of the rectangles using the spacing length of 5.027.
This process could be streamlined a bit. For example, the radius could be determined by selecting the arc and then fed into the arraypath command along with the spacing and number of items.
(defun c:StoneCalc (/)
(setq R (getreal "\nEnter Concave Radius. ")
L (getreal "\nEnter stone length. ")
W (getreal "\nEnter stone width. ")
)
(setq gamma
(* 2
(atan
(/ (/ L 2) (- (expt (- (expt R 2) (expt (/ L 2) 2)) 0.5) W))
)
)
)
(setq gammad (/ (* gamma 180.0) pi))
(setq s (expt (* 2 (expt R 2) (- 1 (cos gamma))) 0.5))
(princ "\nangle = ")
(princ gammad)
(princ "°\ns = ")
(princ s)
(princ)
)
lee.minardi
