Round caps coverage extimate explanation

Comment on how the round caps coverage has been computed, explaining
the complete procedure. The comments doesn't contain intermediate
(verbose and ugly) results, but when executed in a symbolic math
program (sage, for example) computes the expected results.

Reviewed-by: M. Joonas Pihlaja <jpihlaja@cc.helsinki.fi>

src/cairo-stroke-style.c

@@ -145,6 +145,22 @@ _cairo_stroke_style_dash_period (const cairo_stroke_style_t *style)
 /*
  * Coefficient of the linear approximation (minimizing square difference)
  * of the surface covered by round caps
+ *
+ * This can be computed in the following way:
+ * the area inside the circle with radius w/2 and the region -d/2 <= x <= d/2 is:
+ *   f(w,d) = 2 * integrate (sqrt (w*w/4 - x*x), x, -d/2, d/2)
+ * The square difference to a generic linear approximation (c*d) in the range (0,w) would be:
+ *   integrate ((f(w,d) - c*d)^2, d, 0, w)
+ * To minimize this difference it is sufficient to find a solution of the differential with
+ * respect to c:
+ *   solve ( diff (integrate ((f(w,d) - c*d)^2, d, 0, w), c), c)
+ * Which leads to c = 9/32*pi*w
+ * Since we're not interested in the true area, but just in a coverage extimate,
+ * we always divide the real area by the line width (w).
+ * The same computation for square caps would be
+ *   f(w,d) = 2 * integrate(w/2, x, -d/2, d/2)
+ *   c = 1*w
+ * but in this case it would not be an approximation, since f is already linear in d.
  */
 #define ROUND_MINSQ_APPROXIMATION (9*M_PI/32)