<span class="mG"></span>2011/6/27 Boguslaw Jackowski <span dir="ltr"><<a href="mailto:B_Jackowski@gust.org.pl">B_Jackowski@gust.org.pl</a>></span><br><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
<br>
Luigi:<div class="im"><br></div>
Then how you would explain the following pattern of the resulting<br>
turning numbers for the first path rotated by an angle between<br>
0 and 90 degree:<br>
<br>
angle range turningnumber<br>
0--1: 2<br>
2--9: 0<br>
10--29: 2<br>
30--36: 0<br>
37--44: 2<br>
45--52: 0<br>
53: 1<br>
54--77: 0<br>
78--85: -1<br>
86--88: 0<br>
89--90: 2<br>
<br></blockquote><div><br>I'm not surprised because at (0,0) (1,1) and (2,0) we don't have C1 continuity (as Dan said).<br><br>At (1,1), we enter with (0,1) direction and then exiting with (0,-1) and this can be done in two different ways:<br>
<br><br>
--->----<br>
| |<br>
| | -180°<br>
^ v <br>
| |<br>
IN OUT<br>
<br>
<br>
___<__<br>
\ /<br>
\ /<br>
\/ +180°<br>
/\<br>
/ \<br>
/ \<br>
/ \<br>
/ \<br>
IN OUT<br>
<br>
<br>For (2,0) and (0,0) we also have two ways :<br>
<br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;">IN</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;">-->------------+</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;"> |</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;"> V</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;"> OUT | -180°</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;">--<------------+</span><br style="font-family: courier new,monospace;">
<br style="font-family: courier new,monospace;">
<br style="font-family: courier new,monospace;">
<br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;">OUT</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;">--<------------+</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;"> |</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;"> ^ +180°</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;"> IN |</span><br style="font-family: courier new,monospace;">
<span style="font-family: courier new,monospace;">-->------------+</span><br style="font-family: courier new,monospace;">
<br style="font-family: courier new,monospace;">
BY combining these ways I believe that we can have turning number in {2,1,0,-1}.<br style="font-family: courier new,monospace;">I've still to discover how these ways are calculated when the rotation change.<br>With (0,0)..<br>
and ..(2,0) we request to MF to calculate the control points (hence the OUT and IN directions) <br>and in this situation this also has some conseguence on the turnig number (I believe).<br><br></div><blockquote class="gmail_quote" style="margin: 0pt 0pt 0pt 0.8ex; border-left: 1px solid rgb(204, 204, 204); padding-left: 1ex;">
Ps. The testing code was:<div class="im"><br>
<br>
p:=(0,0)..{up}(1,1) & (1,1){down} .. (2,0) -- cycle ;<br></div>
for i:=0 upto 90: show (i,turningnumber (p rotated i)); endfor<br>
<br></blockquote></div><br>-- <br>luigi<br><br>