Improve dash pattern approximation

Dash pattern approximation was taking the caps style into account
only for coverage computation, but not when computing the new
pattern, thus the computed approximation had a higher coverage if
the caps style was SQUARE or ROUND.

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

1 files changed, 54 insertions, 2 deletions

diff --git a/src/cairo-stroke-style.c b/src/cairo-stroke-style.c
index 28dc5ffb7..9a18e740f 100644
--- a/src/cairo-stroke-style.c
+++ b/src/cairo-stroke-style.c
@@ -251,8 +251,60 @@ _cairo_stroke_style_dash_approximate (const cairo_stroke_style_t *style,
 
     *num_dashes = 2;
 
-    dashes[0] = scale * coverage;
-    dashes[1] = scale * (1.0 - coverage);
+    /*
+     * We want to create a new dash pattern with the same relative coverage,
+     * but composed of just 2 elements with total length equal to scale.
+     * Based on the formula in _cairo_stroke_style_dash_stroked:
+     * scale * coverage = dashes[0] + cap_scale * MIN (dashes[1], line_width)
+     *                  = MIN (dashes[0] + cap_scale * (scale - dashes[0]),
+     *                         dashes[0] + cap_scale * line_width) = 
+     *                  = MIN (dashes[0] * (1 - cap_scale) + cap_scale * scale,
+     *	                       dashes[0] + cap_scale * line_width)
+     *
+     * Solving both cases we get:
+     *   dashes[0] = scale * (coverage - cap_scale) / (1 - cap_scale)
+     *	  when scale - dashes[0] <= line_width
+     *	dashes[0] = scale * coverage - cap_scale * line_width
+     *	  when scale - dashes[0] > line_width.
+     *
+     * Comparing the two cases we get:
+     *   second > first
+     *   second > scale * (coverage - cap_scale) / (1 - cap_scale)
+     *   second - cap_scale * second - scale * coverage + scale * cap_scale > 0
+     * 	 (scale * coverage - cap_scale * line_width) - cap_scale * second - scale * coverage + scale * cap_scale > 0
+     *   - line_width - second + scale > 0
+     *   scale - second > line_width
+     * which is the condition for the second solution to be the valid one.
+     * So when second > first, the second solution is the correct one (i.e.
+     * the solution is always MAX (first, second).
+     */
+    switch (style->line_cap) {
+    default:
+        ASSERT_NOT_REACHED;
+	dashes[0] = 0.0;
+	break;
+
+    case CAIRO_LINE_CAP_BUTT:
+        /* Simplified formula (substituting 0 for cap_scale): */
+        dashes[0] = scale * coverage;
+	break;
+
+    case CAIRO_LINE_CAP_ROUND:
+        dashes[0] = MAX(scale * (coverage - ROUND_MINSQ_APPROXIMATION) / (1.0 - ROUND_MINSQ_APPROXIMATION),
+			scale * coverage - ROUND_MINSQ_APPROXIMATION * style->line_width);
+	break;
+
+    case CAIRO_LINE_CAP_SQUARE:
+        /*
+	 * Special attention is needed to handle the case cap_scale == 1 (since the first solution
+	 * is either indeterminate or -inf in this case). Since dash lengths are always >=0, using
+	 * 0 as first solution always leads to the correct solution.
+	 */
+        dashes[0] = MAX(0.0, scale * coverage - style->line_width);
+	break;
+    }
+
+    dashes[1] = scale - dashes[0];
 
     *dash_offset = on ? 0.0 : dashes[0];
 }