# [cairo-commit] cairo/src cairo-pen.c,1.23,1.24

Bertram Felgenhauer commit at pdx.freedesktop.org
Mon Aug 22 16:29:58 PDT 2005

```Committed by: inte

Update of /cvs/cairo/cairo/src
In directory gabe:/tmp/cvs-serv10557/src

Modified Files:
cairo-pen.c
Log Message:
2005-08-22  Bertram Felgenhauer  <int-e at gmx.de>

* src/cairo-pen.c (_cairo_pen_vertices_needed): use correctly
transposed version of the matrix and fix up the comments above
to use row vector notation.

Index: cairo-pen.c
===================================================================
RCS file: /cvs/cairo/cairo/src/cairo-pen.c,v
retrieving revision 1.23
retrieving revision 1.24
diff -u -d -r1.23 -r1.24
--- cairo-pen.c	7 Apr 2005 17:01:49 -0000	1.23
+++ cairo-pen.c	22 Aug 2005 23:29:56 -0000	1.24
@@ -192,7 +192,7 @@
2.  The question has been posed:  What is the maximum expansion factor
achieved by the linear transformation

-X' = _R_ X
+X' = X _R_

where _R_ is a real-valued 2x2 matrix with entries:

@@ -246,7 +246,9 @@

Thus

-     X'(t) = (a*cos(t) + b*sin(t), c*cos(t) + d*sin(t)) .
+     X'(t) = X(t) * _R_ = (cos(t), sin(t)) * [a b]
+                                             [c d]
+           = (a*cos(t) + c*sin(t), b*cos(t) + d*sin(t)).

Define

@@ -254,22 +256,22 @@

Thus

-     r^2(t) = (a*cos(t) + b*sin(t))^2 + (c*cos(t) + d*sin(t))^2
-            = (a^2 + c^2)*cos^2(t) + (b^2 + d^2)*sin^2(t)
-               + 2*(a*b + c*d)*cos(t)*sin(t)
+     r^2(t) = (a*cos(t) + c*sin(t))^2 + (b*cos(t) + d*sin(t))^2
+            = (a^2 + b^2)*cos^2(t) + (c^2 + d^2)*sin^2(t)
+               + 2*(a*c + b*d)*cos(t)*sin(t)

Now apply the double angle formulae (A) to (C) from above:

r^2(t) = (a^2 + b^2 + c^2 + d^2)/2
-	  + (a^2 - b^2 + c^2 - d^2)*cos(2*t)/2
-	  + (a*b + c*d)*sin(2*t)
+	  + (a^2 + b^2 - c^2 - d^2)*cos(2*t)/2
+	  + (a*c + b*d)*sin(2*t)
= f + g*cos(u) + h*sin(u)

Where

f = (a^2 + b^2 + c^2 + d^2)/2
-     g = (a^2 - b^2 + c^2 - d^2)/2
-     h = (a*b + c*d)
+     g = (a^2 + b^2 - c^2 - d^2)/2
+     h = (a*c + b*d)
u = 2*t

It is clear that MAX[ |X'| ] = sqrt(MAX[ r^2 ]).  Here we determine MAX[ r^2 ]
@@ -377,12 +379,12 @@
double  a = matrix->xx, b = matrix->yx;
double  c = matrix->xy, d = matrix->yy;

-    double  i = a*a + c*c;
-    double  j = b*b + d*d;
+    double  i = a*a + b*b;
+    double  j = c*c + d*d;

double  f = 0.5 * (i + j);
double  g = 0.5 * (i - j);
-    double  h = a*b + c*d;
+    double  h = a*c + b*d;

/*
* compute major and minor axes lengths for

```