Bugfix from Graydon Hoare.

1 files changed, 0 insertions, 84 deletions

diff --git a/TODO b/TODO
index 0ac2e234..f99f5cc5 100644
--- a/TODO
+++ b/TODO
@@ -2,8 +2,6 @@
 
 * Implement text support for the image backend.
 
-* Add arc support.
-
 * Re-implement pattern support with a more PostScript-like API.
 
 * Virtualize the backend interface so that the various backends can be
@@ -11,88 +9,6 @@ compiled conditionally.
 
 * Verification, profiling, optimization.
 
-Some notes on arc support
-=========================
-
-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:
-
-	Y
-	
-	|   .
-	|  /  .
-	| /    .
-	|/A    .
-	+------.-- X
-	0      1
-
-The deviation in radius is given by:
-
-	rho(t) = sqrt ( x^2(t) + y^2(t) ) - 1
-
-A simpler error function to work with is:
-
-	e(t) = x^2(t) + y^2(t) - 1
-
-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) )
-
-Continuing with Goldapp's analysis, a single cubic Bezier spline
-approximation must have the 4 control points:
-
-	(1, 0)
-	(1, h)
-	(cos(A) + h * sin(A), sin(A) - h * cos(A))
-	(cos(A), sin(A))
-
-Various approximations can be determined by selecting the value of
-h. A convenient value, (though not optimal in terms of error), is:
-
-	h = 4/3 * tan(A/4)
-
-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
 ============================