Let p and q be variables in the Schläfli symbol {p/q} representing a polygram.
Assume that the 0-th vertex is located at (r⋅cos(0), r⋅sin(0)).
Then, the k-th vertex, where 0 <= k < p, is located at:
(r⋅cos(2π⋅k/p), r⋅sin(2π⋅k/p))
In the trace, a line is drawn by joining the k-th vertex and the (k+q)-th vertex, located at:
(r⋅cos(2π⋅(k+q)/p), r⋅sin(2π⋅(k+q)/p))
In the outline, the above line is intersected by the line joining the k+1-th vertex and the (k-q+1)-th vertex. The k+1-th vertex is located at:
(r⋅cos(2π⋅(k+1)/p), r⋅sin(2π⋅(k+1)/p))
The (k-q+1)-th vertex is located at:
(r⋅cos(2π⋅(k-q+1)/p), r⋅sin(2π⋅(k-q+1)/p))
The equation of the 1st line (k-th to (k+q)-th vertex) is:
y - y₁ y₂ - y₁
------ = -------
x - x₁ x₂ - x₁
y - r⋅sin(2π⋅k/p) r⋅sin(2π⋅(k+q)/p) - r⋅sin(2π⋅k/p)
----------------- = ---------------------------------
x - r⋅cos(2π⋅k/p) r⋅cos(2π⋅(k+q)/p) - r⋅cos(2π⋅k/p)
sin(2π⋅(k+q)/p) - sin(2π⋅k/p)
= -----------------------------
cos(2π⋅(k+q)/p) - cos(2π⋅k/p)
2⋅cos(π⋅(2k+q)/p)⋅sin(π⋅q/p)
= -----------------------------
-2⋅sin(π⋅(2k+q)/p)⋅sin(π⋅q/p)
cos(π⋅(2k+q)/p)
= - ---------------
sin(π⋅(2k+q)/p)
cos(π⋅(2k+q)/p)
y - r⋅sin(2π⋅k/p) = - --------------- ⋅ (x - r⋅cos(2π⋅k/p))
sin(π⋅(2k+q)/p)
cos(π⋅(2k+q)/p) r⋅cos(2π⋅k/p))⋅cos(π⋅(2k+q)/p)
= (- ---------------) ⋅ x + ------------------------------
sin(π⋅(2k+q)/p) sin(π⋅(2k+q)/p)
cos(π⋅(2k+q)/p) r⋅cos(2π⋅k/p))⋅cos(π⋅(2k+q)/p)
y = (- ---------------) ⋅ x + ------------------------------ + r⋅sin(2π⋅k/p)
sin(π⋅(2k+q)/p) sin(π⋅(2k+q)/p)
cos(π⋅(2k+q)/p) r⋅cos(2π⋅k/p))⋅cos(π⋅(2k+q)/p) + r⋅sin(2π⋅k/p)⋅sin(π⋅(2k+q)/p)
= (- ---------------) ⋅ x + --------------------------------------------------------------
sin(π⋅(2k+q)/p) sin(π⋅(2k+q)/p)
cos(π⋅(2k+q)/p) r⋅cos(π⋅(2k+q)/p - 2π⋅k/p))
= (- ---------------) ⋅ x + ---------------------------
sin(π⋅(2k+q)/p) sin(π⋅(2k+q)/p)
cos(π⋅(2k+q)/p) r⋅cos(π⋅q/p))
= (- ---------------) ⋅ x + ---------------
sin(π⋅(2k+q)/p) sin(π⋅(2k+q)/p)
The equation of the 2nd line (k+1-th to (k-q+1)-th vertex) is:
y - y₁ y₂ - y₁
------ = -------
x - x₁ x₂ - x₁
y - r⋅sin(2π⋅(k+1)/p) r⋅sin(2π⋅(k-q+1)/p) - r⋅sin(2π⋅(k+1)/p)
--------------------- = ---------------------------------------
x - r⋅cos(2π⋅(k+1)/p) r⋅cos(2π⋅(k-q+1)/p) - r⋅cos(2π⋅(k+1)/p)
sin(2π⋅(k-q+1)/p) - sin(2π⋅(k+1)/p)
= -----------------------------------
cos(2π⋅(k-q+1)/p) - cos(2π⋅(k+1)/p)
2⋅cos(π⋅(2k-q+2)/p)⋅sin(π⋅q/p)
= -------------------------------
-2⋅sin(π⋅(2k-q+2)/p)⋅sin(π⋅q/p)
cos(π⋅(2k-q+2)/p)
= - -----------------
sin(π⋅(2k-q+2)/p)
cos(π⋅(2k-q+2)/p)
y - r⋅sin(2π⋅(k+1)/p) = - ----------------- ⋅ (x - r⋅cos(2π⋅(k+1)/p))
sin(π⋅(2k-q+2)/p)
cos(π⋅(2k-q+2)/p) r⋅cos(2π⋅(k+1)/p))⋅cos(π⋅(2k-q+2)/p)
= (- -----------------) ⋅ x + ------------------------------------
sin(π⋅(2k-q+2)/p) sin(π⋅(2k-q+2)/p)
cos(π⋅(2k-q+2)/p) r⋅cos(2π⋅(k+1)/p))⋅cos(π⋅(2k-q+2)/p)
y = (- -----------------) ⋅ x + ------------------------------------ + r⋅sin(2π⋅(k+1)/p)
sin(π⋅(2k-q+2)/p) sin(π⋅(2k-q+2)/p)
cos(π⋅(2k-q+2)/p) r⋅cos(2π⋅(k+1)/p))⋅cos(π⋅(2k-q+2)/p) + r⋅sin(2π⋅(k+1)/p)⋅sin(π⋅(2k-q+2)/p)
= (- -----------------) ⋅ x + --------------------------------------------------------------------------
sin(π⋅(2k-q+2)/p) sin(π⋅(2k-q+2)/p)
cos(π⋅(2k-q+2)/p) r⋅cos(π⋅(2k-q+2)/p - 2π⋅(k+1)/p))
= (- -----------------) ⋅ x + ---------------------------------
sin(π⋅(2k-q+2)/p) sin(π⋅(2k-q+2)/p)
cos(π⋅(2k-q+2)/p) r⋅cos(π⋅q/p))
= (- -----------------) ⋅ x + -----------------
sin(π⋅(2k-q+2)/p) sin(π⋅(2k-q+2)/p)
Line 1: y = m₁⋅x + c₁
Line 2: y = m₂⋅x + c₂
Let x be the x-coordinate of the intersect. Then:
m₁⋅x + c₁ = m₂⋅x + c₂
c₂ - c₁
x = - -------
m₂ - m₁
r⋅cos(π⋅q/p)) r⋅cos(π⋅q/p)) / cos(π⋅(2k-q+2)/p) cos(π⋅(2k+q)/p)
= - ( ----------------- - --------------- ) / ( (- -----------------) - (- ---------------) )
sin(π⋅(2k-q+2)/p) sin(π⋅(2k+q)/p) / sin(π⋅(2k-q+2)/p) sin(π⋅(2k+q)/p)
r⋅cos(π⋅q/p)) r⋅cos(π⋅q/p)) / cos(π⋅(2k-q+2)/p) cos(π⋅(2k+q)/p)
= ( ----------------- - --------------- ) / ( ----------------- - --------------- )
sin(π⋅(2k-q+2)/p) sin(π⋅(2k+q)/p) / sin(π⋅(2k-q+2)/p) sin(π⋅(2k+q)/p)
r⋅cos(π⋅q/p))⋅sin(π⋅(2k+q)/p) - r⋅cos(π⋅q/p))⋅sin(π⋅(2k-q+2)/p)
= ---------------------------------------------------------------------
cos(π⋅(2k-q+2)/p)⋅sin(π⋅(2k+q)/p) - cos(π⋅(2k+q)/p)⋅sin(π⋅(2k-q+2)/p)
r⋅cos(π⋅q/p))⋅[sin(π⋅(2k+q)/p) - sin(π⋅(2k-q+2)/p)]
= ---------------------------------------------------------------------
sin(π⋅(2k+q)/p)⋅cos(π⋅(2k-q+2)/p) - cos(π⋅(2k+q)/p)⋅sin(π⋅(2k-q+2)/p)
r⋅cos(π⋅q/p))⋅2⋅cos((1/2)⋅(π⋅(2k+q)/p + π⋅(2k-q+2)/p))⋅sin((1/2)⋅(π⋅(2k+q)/p - π⋅(2k-q+2)/p))
= ---------------------------------------------------------------------------------------------
sin(π⋅(2k+q)/p - π⋅(2k-q+2)/p)
2⋅r⋅cos(π⋅q/p))⋅cos(π⋅(2k+1)/p)⋅sin(π⋅(q-1)/p)
= ----------------------------------------------
sin(π⋅(2q-2)/p)
2⋅r⋅cos(π⋅q/p))⋅cos(π⋅(2k+1)/p)⋅sin(π⋅(q-1)/p)
= ----------------------------------------------
2⋅sin(π⋅(q-1)/p)⋅cos(π⋅(q-1)/p)
r⋅cos(π⋅q/p))⋅cos(π⋅(2k+1)/p)
= -----------------------------
cos(π⋅(q-1)/p)
Then, substitute the value of x into Line 1:
y = m₁⋅x + c₁
cos(π⋅(2k+q)/p) r⋅cos(π⋅q/p))⋅cos(π⋅(2k+1)/p) r⋅cos(π⋅q/p))
= (- ---------------) ⋅ ----------------------------- + ---------------
sin(π⋅(2k+q)/p) cos(π⋅(q-1)/p) sin(π⋅(2k+q)/p)
r⋅cos(π⋅q/p))⋅cos(π⋅(2k+q)/p)⋅cos(π⋅(2k+1)/p) r⋅cos(π⋅q/p))
= - --------------------------------------------- + ---------------
sin(π⋅(2k+q)/p)⋅cos(π⋅(q-1)/p) sin(π⋅(2k+q)/p)