Re: Fix: NSBezierPath, appendBezierPathWithArcWithCenter:

[Nicola Pero]

> Hi Nicola,
> 
> > Thanks - I read your patch and it made lot of sense - so I applied it.  I
> > did not test it nor did I check on books the formula used to build the
> > bezier path for the last bit of the arc, so I hope you did. :-)
> 
> Yes, I have tested it.
> Anyway, I had to realized another little problem in the same method, so since 
>  
> it isn't committed yet, I will send a new patch covering this remaining issue 
>  
> too.

Well - I already committed your patch, anyway we enjoy further patches. 
:-) 
 
 
> > Ahm - just a quick note - before committing, I sligthly simplified the
> > formula used for F - not that I changed it (unless I made a typo) - just
> > did some trivial algebraic/trigonometric simplifications.  But as I didn't
> > test it, please tell me if I did something wrong.
> 
> Can you send me your changes, so I can test it ?

They should be on CVS already - anyway in case you don't have CVS or
whatever, in attach you find the diff against the original sources (not
against your patch).


Index: NSBezierPath.m
===================================================================
RCS file: /gnustep/gnustep/core/gui/Source/NSBezierPath.m,v
retrieving revision 1.7
retrieving revision 1.8
diff -u -r1.7 -r1.8
--- NSBezierPath.m      2001/01/21 22:32:35     1.7
+++ NSBezierPath.m      2001/03/01 01:47:32     1.8
@@ -828,103 +828,132 @@
     [self lineToPoint: points[i]];
 }
 
-- (void)appendBezierPathWithOvalInRect:(NSRect)aRect
+- (void) appendBezierPathWithOvalInRect: (NSRect)aRect
 {
   [self appendBezierPath: [isa bezierPathWithOvalInRect: aRect]];
 }
 
-- (void)appendBezierPathWithArcWithCenter:(NSPoint)center  
-                                  radius:(float)radius
-                              startAngle:(float)startAngle
-                                endAngle:(float)endAngle
-                               clockwise:(BOOL)clockwise
+/* startAngle and endAngle are in degrees, counterclockwise, from the 
+   x axis */
+- (void) appendBezierPathWithArcWithCenter: (NSPoint)center  
+                                   radius: (float)radius
+                               startAngle: (float)startAngle
+                                 endAngle: (float)endAngle
+                                clockwise: (BOOL)clockwise
 {
-  NSBezierPath *path = [isa bezierPath];
-  float strtangrd, endangrd, diff;
+  float startAngle_rad, endAngle_rad, diff;
   NSPoint p0, p1, p2, p3;
 
-  while(startAngle < 0)
+  while (startAngle < 0)
     startAngle = startAngle + 360;
-  while(startAngle > 360)
+  while (startAngle > 360)
     startAngle = startAngle - 360;
 
-  while(endAngle < 0)
+  while (endAngle < 0)
     endAngle = endAngle + 360;
-  while(endAngle > 360)
+  while (endAngle > 360)
     endAngle = endAngle - 360;
 
-  strtangrd = PI * startAngle / 180;
-  endangrd = PI * endAngle / 180;
+  /* Convert the angles to radians */
+  startAngle_rad = PI * startAngle / 180;
+  endAngle_rad = PI * endAngle / 180;
+
+  /* Start point */
+  p0 = NSMakePoint (center.x + radius * cos (startAngle_rad), 
+                   center.y + radius * sin (startAngle_rad));
+  [self moveToPoint: p0];
 
-  p0 = NSMakePoint(center.x + radius * cos(strtangrd), center.y + radius * 
sin(strtangrd));
-  [path moveToPoint: p0];
-
   if (clockwise)
     {
       diff = -PI / 2;
-      if (strtangrd < endangrd)
-       strtangrd += 2 * PI; 
+      if (startAngle_rad < endAngle_rad)
+       {
+         startAngle_rad += 2 * PI; 
+       }
     }
   else
     {
       diff = PI / 2;
-      if (strtangrd > endangrd)
-       strtangrd -= 2 * PI; 
+      if (startAngle_rad > endAngle_rad)
+       {
+         startAngle_rad -= 2 * PI; 
+       }
     }  
-
-  while (strtangrd > endangrd)
-  {
-      if (strtangrd + diff >= endangrd)
-        {
-         float sin_sta = sin(strtangrd);
-         float cos_sta = cos(strtangrd);
 
-         // FIXME: Add a quarter circle, for clockwise the signs are not 
correct!
-         p1 = NSMakePoint(center.x + radius * (cos_sta - KAPPA * sin_sta), 
-                          center.y + radius * (sin_sta + KAPPA * cos_sta));
-         p2 = NSMakePoint(center.x + radius * (KAPPA * cos_sta -sin_sta),
-                          center.y + radius * (cos_sta + KAPPA * sin_sta));
-         p3 = NSMakePoint(center.x + radius * (-sin_sta), 
-                          center.y + radius * cos_sta);  
-         [path curveToPoint: p3 controlPoint1: p1 controlPoint2: p2];
-         strtangrd += diff;
-       }
-      else
-        {
-         float diff_ang = (endangrd - strtangrd) / 2;
-         float cos_da = cos(diff_ang);
-         float sin_da = sin(diff_ang);
-         double x, y;
-         float sin_sta = sin(strtangrd + diff_ang);
-         float cos_sta = cos(strtangrd + diff_ang);
-
-         x = (4 - cos_da) / 3;
-         y = ((1- cos_da) * (cos_da - 3)) / (3 * sin_da);
-         // FIXME: Add just the missing bit, for clockwise the signs are not 
correct!
-         p1 = NSMakePoint(center.x + radius * (cos_sta * x - sin_sta * y),
-                          center.y + radius * (sin_sta * y + cos_sta * x));  
-         p2 = NSMakePoint(center.x + radius * (cos_sta * x + sin_sta * y), 
-                          center.y + radius * (sin_sta * (-y) + cos_sta * x)); 
 
-         p3 = NSMakePoint(center.x + radius * cos(endangrd), 
-                          center.y + radius * sin(endangrd));
-         [path curveToPoint: p3 controlPoint1: p1 controlPoint2: p2];
-         break;
-       }
+  while ((clockwise) ? (startAngle_rad > endAngle_rad) 
+        : (startAngle_rad < endAngle_rad))
+    {
+    /* Add a quarter circle */
+    if ((clockwise) ? (startAngle_rad + diff >= endAngle_rad) 
+       : (startAngle_rad + diff <= endAngle_rad))
+      {
+       float sin_start = sin (startAngle_rad);
+       float cos_start = cos (startAngle_rad);
+       float sign = (clockwise) ? -1.0 : 1.0;
+       
+       p1 = NSMakePoint (center.x 
+                           + radius * (cos_start - KAPPA * sin_start * sign), 
+                         center.y 
+                           + radius * (sin_start + KAPPA * cos_start * sign));
+       p2 = NSMakePoint (center.x 
+                           + radius * (-sin_start * sign + KAPPA * cos_start),
+                         center.y 
+                           + radius * (cos_start * sign + KAPPA * sin_start));
+       p3 = NSMakePoint (center.x + radius * (-sin_start * sign),
+                         center.y + radius *   cos_start * sign);
+       
+       [self curveToPoint: p3  controlPoint1: p1  controlPoint2: p2];
+       startAngle_rad += diff;
+      }
+    else
+      {
+       /* Add the missing bit
+        * We require that the arc be less than a semicircle.
+        * The arc may go either clockwise or counterclockwise.
+        * The approximation is a very simple one: a single curve
+        * whose middle two control points are a fraction F of the way
+        * to the intersection of the tangents, where
+        *      F = (4/3) / (1 + sqrt (1 + (d / r)^2))
+        * where r is the radius and d is the distance from either tangent
+        * point to the intersection of the tangents. This produces
+        * a curve whose center point, as well as its ends, lies on
+        * the desired arc.
+        */
+       NSPoint ps = [self currentPoint];
+       /* tangent is the tangent of half the angle */
+       float tangent = tan ((endAngle_rad - startAngle_rad) / 2);
+       /* trad is the distance from either tangent point to the
+          intersection of the tangents */
+       float trad = radius * tangent;
+       /* pt is the intersection of the tangents */
+       NSPoint pt = NSMakePoint (ps.x - trad * sin (startAngle_rad),
+                                 ps.y + trad * cos (startAngle_rad));
+       /* This is F - in this expression we need to compute 
+          (trad/radius)^2, which is simply tangent^2 */
+       float f = (4.0 / 3.0) / (1.0 + sqrt (1.0 +  (tangent * tangent)));
+       
+       p1 = NSMakePoint (ps.x + (pt.x - ps.x) * f, ps.y + (pt.y - ps.y) * f);
+       p3 = NSMakePoint(center.x + radius * cos (endAngle_rad),
+                        center.y + radius * sin (endAngle_rad));
+       p2 = NSMakePoint (p3.x + (pt.x - p3.x) * f, p3.y + (pt.y - p3.y) * f);
+       [self curveToPoint: p3  controlPoint1: p1  controlPoint2: p2];
+       break;
+      }
   }
 }
 
-- (void)appendBezierPathWithArcWithCenter:(NSPoint)center  
-                                  radius:(float)radius
-                              startAngle:(float)startAngle
-                                endAngle:(float)endAngle
-{
-  [self appendBezierPathWithArcWithCenter: center radius: radius
-       startAngle: startAngle endAngle: endAngle clockwise: NO];
+- (void) appendBezierPathWithArcWithCenter: (NSPoint)center  
+                                   radius: (float)radius
+                               startAngle: (float)startAngle
+                                 endAngle: (float)endAngle
+{
+  [self appendBezierPathWithArcWithCenter: center  radius: radius
+       startAngle: startAngle  endAngle: endAngle  clockwise: NO];
 }
 
-- (void)appendBezierPathWithArcFromPoint:(NSPoint)point1
-                                toPoint:(NSPoint)point2
-                                 radius:(float)radius
+- (void) appendBezierPathWithArcFromPoint: (NSPoint)point1
+                                 toPoint: (NSPoint)point2
+                                  radius: (float)radius
 {
   // TODO
 }
@@ -1326,9 +1355,9 @@
   INVALIDATE_CACHE();
 }
 
-- (void)curveToPoint:(NSPoint)aPoint 
-                controlPoint1:(NSPoint)controlPoint1
-                controlPoint2:(NSPoint)controlPoint2
+- (void) curveToPoint: (NSPoint)aPoint 
+       controlPoint1: (NSPoint)controlPoint1
+       controlPoint2: (NSPoint)controlPoint2
 {
   PathElement elem;
   
@@ -1666,3 +1695,4 @@
       [path lineToPoint: coeff[3]];
     }
 }
+