Added cairo_arc and cairo_arc_negative.

1 files changed, 44 insertions, 8 deletions

diff --git a/TODO b/TODO
index fc1f9373..0ac2e234 100644
--- a/TODO
+++ b/TODO
@@ -11,12 +11,13 @@ compiled conditionally.
 
 * Verification, profiling, optimization.
 
-
 Some notes on arc support
--------------------------
+=========================
 
-"Approximation of circular arcs by cubic poynomials", Michael Goldapp,
-Computer Aided Geometric Design 8 (1991) 227-238.
+Some general notions
+--------------------
+This is from "Approximation of circular arcs by cubic poynomials",
+Michael Goldapp, Computer Aided Geometric Design 8 (1991) 227-238.
 
 To draw a unit arc from 0 to A with 0 < A < pi/2:
 
@@ -37,9 +38,14 @@ A simpler error function to work with is:
 
 	e(t) = x^2(t) + y^2(t) - 1
 
-And from Dokken[cite]: e(t) ~ 2 abs( rho(t) )
+And from "Good approximation of circles by curvature-continuous Bezier
+curves", Tor Dokken and Morten Daehlen, Computer Aided Geometric
+Design 8 (1990) 22-41, we learn:
+
+	e(t) ~ 2 abs( rho(t) )
 
-A single cubic Bezier spline approximation must have the 4 control points:
+Continuing with Goldapp's analysis, a single cubic Bezier spline
+approximation must have the 4 control points:
 
 	(1, 0)
 	(1, h)
@@ -55,7 +61,38 @@ From which we can determine the maximum error:
 
 	abs( max(e(t)) ) = 4/27 * (sin^6 (A/4)) / (cos^2 (A/4))
 	    t in [0,1]
-	
+
+-----
+
+Now, for Cairo we want to draw an arc of radius R from an angle A to
+an angle B, (where B > A). So the equations above have trivial
+modifications:
+
+The spline control points become
+
+	(R * cos(A), R * sin(A))
+	(R * cos(A) - h * sin(A), R * sin(A) + h * cos (A))
+	(R * cos(B) + h * sin(B), R * sin(B) - h * cos (B))
+	(R * cos(B), R * sin(B))
+
+where	h = 4/3 * R * tan ((B-A)/4)
+
+And the maximum deviation in radius is approximately:
+
+	2/27 * (sin^6 ((B-A)/4) / cos^2 ((B-A)/4))
+
+So now we can get down to writing some C code:
+
+double
+_arc_error_normalized (double angle)
+{
+    return 2/27 * pow (sin (angle / 4), 6) / pow (cos (angle / 4), 2);
+}
+
+And for accurate drawing the following must hold in device space:
+
+	tolerance/radius >= _arc_error_normalized (B-A)
+
 A comparison with PostScript
 ============================
 
@@ -190,4 +227,3 @@ Operators, Rational,Boolean,and Bitwise Operators, Control Operators,
 Type,Attribute,and Conversion Operators, File Operators, Resource
 Operators, Virtual Memory Operators, Miscellaneous Operators,
 Interpreter Parameter Operators, Errors
-