mirror of https://github.com/CGAL/cgal
Fixed the documentation and the figures.
This commit is contained in:
Ron Wein
parent
eb95c007b8
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d1a2141f27
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@ -1053,16 +1053,21 @@ Minkowski_sum_2/demo/Minkowski_sum_2/Offset_demo.C -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/approx_offset.fig -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/approx_offset.pstex -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/approx_offset.pstex_t -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/convex_offset.eps -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/convex_offset.fig -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/convex_offset.pdf -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/ex_offset.fig -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/ex_offset.pdf -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/offset_conv.eps -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/offset_conv.fig -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/offset_conv.pdf -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/offset_decomp.eps -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/offset_decomp.fig -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/offset_decomp.pdf -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/onecyc_in.eps -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/onecyc_in.fig -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/onecyc_in.pdf -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/onecyc_out.eps -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/onecyc_out.fig -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/onecyc_out.pdf -text
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Minkowski_sum_2/doc_tex/Minkowski_sum_2/fig/sum_triangles.fig -text
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|
||||
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||||
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||||
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|
||||
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|
||||
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||||
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|
||||
gs clippath
|
||||
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|
||||
eoclip
|
||||
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|
||||
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|
||||
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
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|
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|
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|
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|
||||
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|
||||
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|
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|
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|
||||
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||||
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|
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|
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|
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|
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|
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|
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|
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30.000 slw
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||||
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|
||||
gr
|
||||
|
||||
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|
||||
7.500 slw
|
||||
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|
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30.000 slw
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n 4511.2 5613.8 461.4 178.6028 91.3972 arcn
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gs col21 s gr
|
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gr
|
||||
[] 0 sd
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|
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7.500 slw
|
||||
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gs clippath
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4050 5625 l gs col21 s gr gr
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4835 3426 m 4712 3364 l 4665 3458 l 4788 3519 l 4788 3519 l 4718 3426 l 4835 3426 l cp
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|
||||
[15 15] 15 sd
|
||||
gs clippath
|
||||
6760 5889 m 6858 5987 l 6932 5913 l 6834 5815 l 6834 5815 l 6872 5927 l 6760 5889 l cp
|
||||
eoclip
|
||||
n 5985 5040 m
|
||||
6885 5940 l gs col8 s gr gr
|
||||
[] 0 sd
|
||||
% arrowhead
|
||||
7.500 slw
|
||||
n 6760 5889 m 6872 5927 l 6834 5815 l col8 s
|
||||
% Polyline
|
||||
30.000 slw
|
||||
[15 15] 15 sd
|
||||
gs clippath
|
||||
7702 3723 m 7702 3585 l 7597 3585 l 7597 3723 l 7597 3723 l 7650 3618 l 7702 3723 l cp
|
||||
eoclip
|
||||
n 7650 5625 m
|
||||
7650 3600 l gs col8 s gr gr
|
||||
[] 0 sd
|
||||
% arrowhead
|
||||
7.500 slw
|
||||
n 7702 3723 m 7650 3618 l 7597 3723 l col8 s
|
||||
% Polyline
|
||||
30.000 slw
|
||||
[60] 0 sd
|
||||
gs clippath
|
||||
6739 4460 m 6641 4362 l 6567 4436 l 6665 4534 l 6665 4534 l 6628 4423 l 6739 4460 l cp
|
||||
eoclip
|
||||
n 7515 5310 m
|
||||
6615 4410 l gs col21 s gr gr
|
||||
[] 0 sd
|
||||
% arrowhead
|
||||
7.500 slw
|
||||
n 6739 4460 m 6628 4423 l 6665 4534 l col21 s
|
||||
% Polyline
|
||||
30.000 slw
|
||||
[60] 0 sd
|
||||
gs clippath
|
||||
7077 6127 m 7215 6127 l 7215 6022 l 7077 6022 l 7077 6022 l 7182 6075 l 7077 6127 l cp
|
||||
eoclip
|
||||
n 4500 6075 m
|
||||
7200 6075 l gs col21 s gr gr
|
||||
[] 0 sd
|
||||
% arrowhead
|
||||
7.500 slw
|
||||
n 7077 6127 m 7182 6075 l 7077 6022 l col21 s
|
||||
% here ends figure;
|
||||
%
|
||||
% here starts figure with depth 45
|
||||
% Polyline
|
||||
0 slj
|
||||
1 slc
|
||||
15.000 slw
|
||||
n 4500 5625 m 7200 5625 l 7200 3600 l 6300 4725 l 4500 3825 l
|
||||
4500 5625 l cp gs col0 s gr
|
||||
% Polyline
|
||||
0 slc
|
||||
[60 30 15 30] 0 sd
|
||||
n 6300 4725 m
|
||||
7200 5625 l gs col0 s gr [] 0 sd
|
||||
% here ends figure;
|
||||
$F2psEnd
|
||||
rs
|
||||
showpage
|
||||
%%Trailer
|
||||
%EOF
|
||||
|
|
@ -3,7 +3,7 @@ Landscape
|
|||
Center
|
||||
Metric
|
||||
A4
|
||||
100.00
|
||||
50.00
|
||||
Single
|
||||
-2
|
||||
1200 2
|
||||
|
|
|
|||
Binary file not shown.
|
|
@ -0,0 +1,484 @@
|
|||
%!PS-Adobe-2.0 EPSF-2.0
|
||||
%%Title: onecyc_in.fig
|
||||
%%Creator: fig2dev Version 3.2 Patchlevel 5-alpha5
|
||||
%%CreationDate: Mon Mar 13 16:13:43 2006
|
||||
%%For: wein@lap-halperin2 (Ron Wein,M-18,+972-3-6406114,+972-9-9580491)
|
||||
%%BoundingBox: 0 0 195 131
|
||||
%Magnification: 0.3000
|
||||
%%EndComments
|
||||
/MyAppDict 100 dict dup begin def
|
||||
/$F2psDict 200 dict def
|
||||
$F2psDict begin
|
||||
$F2psDict /mtrx matrix put
|
||||
/col-1 {0 setgray} bind def
|
||||
/col0 {0.000 0.000 0.000 srgb} bind def
|
||||
/col1 {0.000 0.000 1.000 srgb} bind def
|
||||
/col2 {0.000 1.000 0.000 srgb} bind def
|
||||
/col3 {0.000 1.000 1.000 srgb} bind def
|
||||
/col4 {1.000 0.000 0.000 srgb} bind def
|
||||
/col5 {1.000 0.000 1.000 srgb} bind def
|
||||
/col6 {1.000 1.000 0.000 srgb} bind def
|
||||
/col7 {1.000 1.000 1.000 srgb} bind def
|
||||
/col8 {0.000 0.000 0.560 srgb} bind def
|
||||
/col9 {0.000 0.000 0.690 srgb} bind def
|
||||
/col10 {0.000 0.000 0.820 srgb} bind def
|
||||
/col11 {0.530 0.810 1.000 srgb} bind def
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||||
/col12 {0.000 0.560 0.000 srgb} bind def
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||||
/col13 {0.000 0.690 0.000 srgb} bind def
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||||
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||||
/col15 {0.000 0.560 0.560 srgb} bind def
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||||
/col16 {0.000 0.690 0.690 srgb} bind def
|
||||
/col17 {0.000 0.820 0.820 srgb} bind def
|
||||
/col18 {0.560 0.000 0.000 srgb} bind def
|
||||
/col19 {0.690 0.000 0.000 srgb} bind def
|
||||
/col20 {0.820 0.000 0.000 srgb} bind def
|
||||
/col21 {0.560 0.000 0.560 srgb} bind def
|
||||
/col22 {0.690 0.000 0.690 srgb} bind def
|
||||
/col23 {0.820 0.000 0.820 srgb} bind def
|
||||
/col24 {0.500 0.190 0.000 srgb} bind def
|
||||
/col25 {0.630 0.250 0.000 srgb} bind def
|
||||
/col26 {0.750 0.380 0.000 srgb} bind def
|
||||
/col27 {1.000 0.500 0.500 srgb} bind def
|
||||
/col28 {1.000 0.630 0.630 srgb} bind def
|
||||
/col29 {1.000 0.750 0.750 srgb} bind def
|
||||
/col30 {1.000 0.880 0.880 srgb} bind def
|
||||
/col31 {1.000 0.840 0.000 srgb} bind def
|
||||
|
||||
end
|
||||
save
|
||||
newpath 0 131 moveto 0 0 lineto 195 0 lineto 195 131 lineto closepath clip newpath
|
||||
-21.4 146.0 translate
|
||||
1 -1 scale
|
||||
|
||||
% This junk string is used by the show operators
|
||||
/PATsstr 1 string def
|
||||
/PATawidthshow { % cx cy cchar rx ry string
|
||||
% Loop over each character in the string
|
||||
{ % cx cy cchar rx ry char
|
||||
% Show the character
|
||||
dup % cx cy cchar rx ry char char
|
||||
PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char)
|
||||
false charpath % cx cy cchar rx ry char
|
||||
/clip load PATdraw
|
||||
% Move past the character (charpath modified the
|
||||
% current point)
|
||||
currentpoint % cx cy cchar rx ry char x y
|
||||
newpath
|
||||
moveto % cx cy cchar rx ry char
|
||||
% Reposition by cx,cy if the character in the string is cchar
|
||||
3 index eq { % cx cy cchar rx ry
|
||||
4 index 4 index rmoveto
|
||||
} if
|
||||
% Reposition all characters by rx ry
|
||||
2 copy rmoveto % cx cy cchar rx ry
|
||||
} forall
|
||||
pop pop pop pop pop % -
|
||||
currentpoint
|
||||
newpath
|
||||
moveto
|
||||
} bind def
|
||||
/PATcg {
|
||||
7 dict dup begin
|
||||
/lw currentlinewidth def
|
||||
/lc currentlinecap def
|
||||
/lj currentlinejoin def
|
||||
/ml currentmiterlimit def
|
||||
/ds [ currentdash ] def
|
||||
/cc [ currentrgbcolor ] def
|
||||
/cm matrix currentmatrix def
|
||||
end
|
||||
} bind def
|
||||
% PATdraw - calculates the boundaries of the object and
|
||||
% fills it with the current pattern
|
||||
/PATdraw { % proc
|
||||
save exch
|
||||
PATpcalc % proc nw nh px py
|
||||
5 -1 roll exec % nw nh px py
|
||||
newpath
|
||||
PATfill % -
|
||||
restore
|
||||
} bind def
|
||||
% PATfill - performs the tiling for the shape
|
||||
/PATfill { % nw nh px py PATfill -
|
||||
PATDict /CurrentPattern get dup begin
|
||||
setfont
|
||||
% Set the coordinate system to Pattern Space
|
||||
PatternGState PATsg
|
||||
% Set the color for uncolored pattezns
|
||||
PaintType 2 eq { PATDict /PColor get PATsc } if
|
||||
% Create the string for showing
|
||||
3 index string % nw nh px py str
|
||||
% Loop for each of the pattern sources
|
||||
0 1 Multi 1 sub { % nw nh px py str source
|
||||
% Move to the starting location
|
||||
3 index 3 index % nw nh px py str source px py
|
||||
moveto % nw nh px py str source
|
||||
% For multiple sources, set the appropriate color
|
||||
Multi 1 ne { dup PC exch get PATsc } if
|
||||
% Set the appropriate string for the source
|
||||
0 1 7 index 1 sub { 2 index exch 2 index put } for pop
|
||||
% Loop over the number of vertical cells
|
||||
3 index % nw nh px py str nh
|
||||
{ % nw nh px py str
|
||||
currentpoint % nw nh px py str cx cy
|
||||
2 index oldshow % nw nh px py str cx cy
|
||||
YStep add moveto % nw nh px py str
|
||||
} repeat % nw nh px py str
|
||||
} for
|
||||
5 { pop } repeat
|
||||
end
|
||||
} bind def
|
||||
|
||||
% PATkshow - kshow with the current pattezn
|
||||
/PATkshow { % proc string
|
||||
exch bind % string proc
|
||||
1 index 0 get % string proc char
|
||||
% Loop over all but the last character in the string
|
||||
0 1 4 index length 2 sub {
|
||||
% string proc char idx
|
||||
% Find the n+1th character in the string
|
||||
3 index exch 1 add get % string proc char char+1
|
||||
exch 2 copy % strinq proc char+1 char char+1 char
|
||||
% Now show the nth character
|
||||
PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr)
|
||||
false charpath % string proc char+1 char char+1
|
||||
/clip load PATdraw
|
||||
% Move past the character (charpath modified the current point)
|
||||
currentpoint newpath moveto
|
||||
% Execute the user proc (should consume char and char+1)
|
||||
mark 3 1 roll % string proc char+1 mark char char+1
|
||||
4 index exec % string proc char+1 mark...
|
||||
cleartomark % string proc char+1
|
||||
} for
|
||||
% Now display the last character
|
||||
PATsstr dup 0 4 -1 roll put % string proc (char+1)
|
||||
false charpath % string proc
|
||||
/clip load PATdraw
|
||||
neewath
|
||||
pop pop % -
|
||||
} bind def
|
||||
% PATmp - the makepattern equivalent
|
||||
/PATmp { % patdict patmtx PATmp patinstance
|
||||
exch dup length 7 add % We will add 6 new entries plus 1 FID
|
||||
dict copy % Create a new dictionary
|
||||
begin
|
||||
% Matrix to install when painting the pattern
|
||||
TilingType PATtcalc
|
||||
/PatternGState PATcg def
|
||||
PatternGState /cm 3 -1 roll put
|
||||
% Check for multi pattern sources (Level 1 fast color patterns)
|
||||
currentdict /Multi known not { /Multi 1 def } if
|
||||
% Font dictionary definitions
|
||||
/FontType 3 def
|
||||
% Create a dummy encoding vector
|
||||
/Encoding 256 array def
|
||||
3 string 0 1 255 {
|
||||
Encoding exch dup 3 index cvs cvn put } for pop
|
||||
/FontMatrix matrix def
|
||||
/FontBBox BBox def
|
||||
/BuildChar {
|
||||
mark 3 1 roll % mark dict char
|
||||
exch begin
|
||||
Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata]
|
||||
PaintType 2 eq Multi 1 ne or
|
||||
{ XStep 0 FontBBox aload pop setcachedevice }
|
||||
{ XStep 0 setcharwidth } ifelse
|
||||
currentdict % mark [paintdata] dict
|
||||
/PaintProc load % mark [paintdata] dict paintproc
|
||||
end
|
||||
gsave
|
||||
false PATredef exec true PATredef
|
||||
grestore
|
||||
cleartomark % -
|
||||
} bind def
|
||||
currentdict
|
||||
end % newdict
|
||||
/foo exch % /foo newlict
|
||||
definefont % newfont
|
||||
} bind def
|
||||
% PATpcalc - calculates the starting point and width/height
|
||||
% of the tile fill for the shape
|
||||
/PATpcalc { % - PATpcalc nw nh px py
|
||||
PATDict /CurrentPattern get begin
|
||||
gsave
|
||||
% Set up the coordinate system to Pattern Space
|
||||
% and lock down pattern
|
||||
PatternGState /cm get setmatrix
|
||||
BBox aload pop pop pop translate
|
||||
% Determine the bounding box of the shape
|
||||
pathbbox % llx lly urx ury
|
||||
grestore
|
||||
% Determine (nw, nh) the # of cells to paint width and height
|
||||
PatHeight div ceiling % llx lly urx qh
|
||||
4 1 roll % qh llx lly urx
|
||||
PatWidth div ceiling % qh llx lly qw
|
||||
4 1 roll % qw qh llx lly
|
||||
PatHeight div floor % qw qh llx ph
|
||||
4 1 roll % ph qw qh llx
|
||||
PatWidth div floor % ph qw qh pw
|
||||
4 1 roll % pw ph qw qh
|
||||
2 index sub cvi abs % pw ph qs qh-ph
|
||||
exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph
|
||||
% Determine the starting point of the pattern fill
|
||||
%(px, py)
|
||||
4 2 roll % nw nh pw ph
|
||||
PatHeight mul % nw nh pw py
|
||||
exch % nw nh py pw
|
||||
PatWidth mul exch % nw nh px py
|
||||
end
|
||||
} bind def
|
||||
|
||||
% Save the original routines so that we can use them later on
|
||||
/oldfill /fill load def
|
||||
/oldeofill /eofill load def
|
||||
/oldstroke /stroke load def
|
||||
/oldshow /show load def
|
||||
/oldashow /ashow load def
|
||||
/oldwidthshow /widthshow load def
|
||||
/oldawidthshow /awidthshow load def
|
||||
/oldkshow /kshow load def
|
||||
|
||||
% These defs are necessary so that subsequent procs don't bind in
|
||||
% the originals
|
||||
/fill { oldfill } bind def
|
||||
/eofill { oldeofill } bind def
|
||||
/stroke { oldstroke } bind def
|
||||
/show { oldshow } bind def
|
||||
/ashow { oldashow } bind def
|
||||
/widthshow { oldwidthshow } bind def
|
||||
/awidthshow { oldawidthshow } bind def
|
||||
/kshow { oldkshow } bind def
|
||||
/PATredef {
|
||||
MyAppDict begin
|
||||
{
|
||||
/fill { /clip load PATdraw newpath } bind def
|
||||
/eofill { /eoclip load PATdraw newpath } bind def
|
||||
/stroke { PATstroke } bind def
|
||||
/show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def
|
||||
/ashow { 0 0 null 6 3 roll PATawidthshow }
|
||||
bind def
|
||||
/widthshow { 0 0 3 -1 roll PATawidthshow }
|
||||
bind def
|
||||
/awidthshow { PATawidthshow } bind def
|
||||
/kshow { PATkshow } bind def
|
||||
} {
|
||||
/fill { oldfill } bind def
|
||||
/eofill { oldeofill } bind def
|
||||
/stroke { oldstroke } bind def
|
||||
/show { oldshow } bind def
|
||||
/ashow { oldashow } bind def
|
||||
/widthshow { oldwidthshow } bind def
|
||||
/awidthshow { oldawidthshow } bind def
|
||||
/kshow { oldkshow } bind def
|
||||
} ifelse
|
||||
end
|
||||
} bind def
|
||||
false PATredef
|
||||
% Conditionally define setcmykcolor if not available
|
||||
/setcmykcolor where { pop } {
|
||||
/setcmykcolor {
|
||||
1 sub 4 1 roll
|
||||
3 {
|
||||
3 index add neg dup 0 lt { pop 0 } if 3 1 roll
|
||||
} repeat
|
||||
setrgbcolor - pop
|
||||
} bind def
|
||||
} ifelse
|
||||
/PATsc { % colorarray
|
||||
aload length % c1 ... cn length
|
||||
dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor
|
||||
} ifelse } ifelse
|
||||
} bind def
|
||||
/PATsg { % dict
|
||||
begin
|
||||
lw setlinewidth
|
||||
lc setlinecap
|
||||
lj setlinejoin
|
||||
ml setmiterlimit
|
||||
ds aload pop setdash
|
||||
cc aload pop setrgbcolor
|
||||
cm setmatrix
|
||||
end
|
||||
} bind def
|
||||
|
||||
/PATDict 3 dict def
|
||||
/PATsp {
|
||||
true PATredef
|
||||
PATDict begin
|
||||
/CurrentPattern exch def
|
||||
% If it's an uncolored pattern, save the color
|
||||
CurrentPattern /PaintType get 2 eq {
|
||||
/PColor exch def
|
||||
} if
|
||||
/CColor [ currentrgbcolor ] def
|
||||
end
|
||||
} bind def
|
||||
% PATstroke - stroke with the current pattern
|
||||
/PATstroke {
|
||||
countdictstack
|
||||
save
|
||||
mark
|
||||
{
|
||||
currentpoint strokepath moveto
|
||||
PATpcalc % proc nw nh px py
|
||||
clip newpath PATfill
|
||||
} stopped {
|
||||
(*** PATstroke Warning: Path is too complex, stroking
|
||||
with gray) =
|
||||
cleartomark
|
||||
restore
|
||||
countdictstack exch sub dup 0 gt
|
||||
{ { end } repeat } { pop } ifelse
|
||||
gsave 0.5 setgray oldstroke grestore
|
||||
} { pop restore pop } ifelse
|
||||
newpath
|
||||
} bind def
|
||||
/PATtcalc { % modmtx tilingtype PATtcalc tilematrix
|
||||
% Note: tiling types 2 and 3 are not supported
|
||||
gsave
|
||||
exch concat % tilingtype
|
||||
matrix currentmatrix exch % cmtx tilingtype
|
||||
% Tiling type 1 and 3: constant spacing
|
||||
2 ne {
|
||||
% Distort the pattern so that it occupies
|
||||
% an integral number of device pixels
|
||||
dup 4 get exch dup 5 get exch % tx ty cmtx
|
||||
XStep 0 dtransform
|
||||
round exch round exch % tx ty cmtx dx.x dx.y
|
||||
XStep div exch XStep div exch % tx ty cmtx a b
|
||||
0 YStep dtransform
|
||||
round exch round exch % tx ty cmtx a b dy.x dy.y
|
||||
YStep div exch YStep div exch % tx ty cmtx a b c d
|
||||
7 -3 roll astore % { a b c d tx ty }
|
||||
} if
|
||||
grestore
|
||||
} bind def
|
||||
/PATusp {
|
||||
false PATredef
|
||||
PATDict begin
|
||||
CColor PATsc
|
||||
end
|
||||
} bind def
|
||||
|
||||
% left45
|
||||
11 dict begin
|
||||
/PaintType 1 def
|
||||
/PatternType 1 def
|
||||
/TilingType 1 def
|
||||
/BBox [0 0 1 1] def
|
||||
/XStep 1 def
|
||||
/YStep 1 def
|
||||
/PatWidth 1 def
|
||||
/PatHeight 1 def
|
||||
/Multi 2 def
|
||||
/PaintData [
|
||||
{ clippath } bind
|
||||
{ 20 20 true [ 20 0 0 -20 0 20 ]
|
||||
{<8020004010002008001004000802000401000200
|
||||
8001004000802000401080200040100020080010
|
||||
0400080200040100020080010040008020004010>}
|
||||
imagemask } bind
|
||||
] def
|
||||
/PaintProc {
|
||||
pop
|
||||
exec fill
|
||||
} def
|
||||
currentdict
|
||||
end
|
||||
/P4 exch def
|
||||
|
||||
% vertical lines
|
||||
11 dict begin
|
||||
/PaintType 1 def
|
||||
/PatternType 1 def
|
||||
/TilingType 1 def
|
||||
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|
||||
[60] 0 sd
|
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n 5700 6270 300 300 0 360 DrawEllipse gs col6 1.00 shd ef gr gs col8 s gr
|
||||
[] 0 sd
|
||||
% Ellipse
|
||||
[60] 0 sd
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||||
n 3075 5175 300 300 0 360 DrawEllipse gs col6 1.00 shd ef gr gs col8 s gr
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[] 0 sd
|
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% Ellipse
|
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[60] 0 sd
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n 8250 5175 300 300 0 360 DrawEllipse gs col6 1.00 shd ef gr gs col8 s gr
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[] 0 sd
|
||||
% Ellipse
|
||||
[60] 0 sd
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n 9825 2595 300 300 0 360 DrawEllipse gs col6 1.00 shd ef gr gs col8 s gr
|
||||
[] 0 sd
|
||||
% here ends figure;
|
||||
%
|
||||
% here starts figure with depth 25
|
||||
/Times-Bold ff 400.00 scf sf
|
||||
5580 3975 m
|
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gs 1 -1 sc (0) col0 sh gr
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5610 6405 m
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gs 1 -1 sc (1) col0 sh gr
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/Times-Bold ff 400.00 scf sf
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2985 5310 m
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gs 1 -1 sc (2) col0 sh gr
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8160 5310 m
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gs 1 -1 sc (2) col0 sh gr
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/Times-Bold ff 366.67 scf sf
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5550 2430 m
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gs 1 -1 sc (2) col0 sh gr
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/Times-Bold ff 400.00 scf sf
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9735 2730 m
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gs 1 -1 sc (0) col0 sh gr
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% here ends figure;
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$F2psEnd
|
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rs
|
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showpage
|
||||
%%Trailer
|
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%EOF
|
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|
@ -3,25 +3,25 @@ Landscape
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Center
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6 5370 3540 5970 4140
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4 0 0 25 -1 2 24 0.0000 0 270 210 5580 3975 0\001
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4 0 0 25 -1 2 24 0.0000 0 270 195 5580 3975 0\001
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4 0 0 25 -1 2 24 0.0000 0 270 195 9735 2730 0\001
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-6
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2 3 0 1 4 7 60 0 20 0.000 0 0 -1 0 0 4
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5625 2750 4312 4500 6937 4500 5625 2750
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Binary file not shown.
|
|
@ -48,8 +48,8 @@ in $P$ and in $Q$ and performing ``merge sort'' on the edges.
|
|||
\begin{ccTexOnly}
|
||||
\begin{center}
|
||||
\begin{tabular}{c c}
|
||||
\psfig{figure=Minkowski_sum_2/fig/onecyc_in.eps,width=2.5in,silent=} ~&~
|
||||
\psfig{figure=Minkowski_sum_2/fig/onecyc_out.eps,width=2.5in,silent=}
|
||||
\includegraphics{Minkowski_sum_2/fig/onecyc_in} ~&~
|
||||
\includegraphics{Minkowski_sum_2/fig/onecyc_out} \\
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\end{ccTexOnly}
|
||||
|
|
@ -122,6 +122,10 @@ generates a smaller intermediate arrangement, hence it is faster and
|
|||
consumes less space.
|
||||
\end{description}
|
||||
|
||||
\subsection{Computing Minkowski Sum using Convolutions}
|
||||
\label{mink_ssec:sum_conv}
|
||||
%------------------------------------------------------
|
||||
|
||||
The function \ccc{minkowski_sum (P, Q)} accepts two simple polygons $P$
|
||||
and $Q$, represented using the \ccc{Polygon_2<Kernel,Container>}
|
||||
class-template and uses the convolution method in order to compute and
|
||||
|
|
@ -154,11 +158,39 @@ in the example program \ccc{ex_sum_triangles.C}.}
|
|||
|
||||
The following example program constructs the Minkwoski sum of two triangles,
|
||||
as depicted in Figure~\ref{mink_fig:sum_tri}. The result in this case is
|
||||
a convex hexagon:
|
||||
a convex hexagon. This program, as other example porgrams in this chapter,
|
||||
includes the auxiliary header file \ccc{ms_rational_nt.h} which defines
|
||||
\ccc{Number_type} as either \ccc{Gmpq} or \ccc{Quotient<MP_Float>},
|
||||
depending on whther the {\sc Gmp} library is installed or not.
|
||||
The file \ccc{print_util.h} includes auxiliary functions for printing polygons.
|
||||
|
||||
\ccIncludeExampleCode{../examples/Minkowski_sum_2/ex_sum_triangles.C}
|
||||
|
||||
% RWRW : Here should come another example, with non-convex polygons.
|
||||
\begin{figure}[t]
|
||||
\begin{ccTexOnly}
|
||||
\begin{center}
|
||||
\input{Minkowski_sum_2/fig/tight.pstex_t}
|
||||
\end{center}
|
||||
\end{ccTexOnly}
|
||||
\begin{ccHtmlOnly}
|
||||
<p><center>
|
||||
<img src="./fig/tight.gif" border=0 alt="Minkowski sum of two non-convex polygon that contains holes">
|
||||
</center>
|
||||
\end{ccHtmlOnly}
|
||||
\caption{Computing the Minkowski sum of two non-convex polygons $P$ and $Q$,
|
||||
as done in the example programs \ccc{ex_sum_with_holes.C} and
|
||||
\ccc{ex_sum_by_decomposition.C}.}
|
||||
\label{mink_fig:sum_holes}
|
||||
\end{figure}
|
||||
|
||||
In the following program we compute the Minkowski sum of two polygons
|
||||
that are read from an input file. In this case, the sum is not simple
|
||||
and contains four holes, as illustrated in Figure~\ref{mink_fig:sum_holes}.
|
||||
Note that this example uses the predefined \cgal\ kernel with exact
|
||||
constrcutions. In general, instantiating polygons with this kernel yield
|
||||
the fastest running times for Minkowski-sum computations.
|
||||
|
||||
\ccIncludeExampleCode{../examples/Minkowski_sum_2/ex_sum_with_holes.C}
|
||||
|
||||
\subsection{Decomposition Strategies}
|
||||
\label{mink_ssec:decomp}
|
||||
|
|
@ -179,14 +211,15 @@ introduced in~\cite{???}. The convex decompositions that it creates
|
|||
usually yield efficient running times for Minkowski sum computations:
|
||||
\begin{itemize}
|
||||
\item
|
||||
The class \ccc{Optimal_convex_decomposition} uses the dynamic-programming
|
||||
algorithm of Greene~\cite{g-dpcp-83} for computing an optimal decomposition
|
||||
of a polygon into a minimal number of convex sub-polygons. The main drawback
|
||||
of this strategy is that it runs in $O(n^4)$ time and $O(n^3)$ in the worst
|
||||
case,where $n$ is the number of vertices in the input polygon.
|
||||
The class \ccc{Optimal_convex_decomposition<Kernel>} uses the
|
||||
dynamic-programming algorithm of Greene~\cite{g-dpcp-83} for computing an
|
||||
optimal decomposition of a polygon into a minimal number of convex
|
||||
sub-polygons. The main drawback of this strategy is that it runs in $O(n^4)$
|
||||
time and $O(n^3)$ in the worst case,where $n$ is the number of vertices in
|
||||
the input polygon.
|
||||
%
|
||||
\item
|
||||
The class \ccc{Hertel_Mehlhorn_convex_decomposition} implements the
|
||||
The class \ccc{Hertel_Mehlhorn_convex_decomposition<Kernel>} implements the
|
||||
approximation algorithm suggested by Hertel and Mehlhorn~\cite{hm-ftsp-83},
|
||||
which triangulates the input polygon and proceeds by throwing away
|
||||
unnecessary triangulation edges.
|
||||
|
|
@ -195,7 +228,7 @@ the number of sub-polygons it generates is not more than four
|
|||
times the optimum.
|
||||
%
|
||||
\item
|
||||
The class \ccc{Greene_convex_decomposition} is an implementation of
|
||||
The class \ccc{Greene_convex_decomposition<Kernel>} is an implementation of
|
||||
Greene's approximation algorithm~\cite{g-dpcp-83}, which computes a
|
||||
convex decomposition of the polygon based on its partitioning into
|
||||
$y$-monotone polygons. This algorithm runs in $O(n \log n)$ time
|
||||
|
|
@ -203,7 +236,7 @@ and $O(n)$ space, and has the same approximation guarantee as
|
|||
Hertel and Mehlhorn's algorithm.
|
||||
%
|
||||
\item
|
||||
The class \ccc{short_side_angle_bisecttor_convex_decomposition} uses
|
||||
The class \ccc{Small_side_angle_bisecttor_convex_decomposition<Kernel>} uses
|
||||
a heuristic improvement to the angle-bisector decomposition method
|
||||
suggested by Chazelle and Dobkin~\cite{cd-ocd-85}, which runs in
|
||||
$O(n^2)$ time. It starts by examining each pair of reflex vertices
|
||||
|
|
@ -219,7 +252,12 @@ diagonal that is closest to the angle bisector emanating from this
|
|||
vertex.
|
||||
\end{itemize}
|
||||
|
||||
% RWRW : Here comes an example.
|
||||
The following example demonstrates the computation of the Minkowski sum
|
||||
of the same input polygons as used in \ccc{ex_sum_with_holes.C} (as
|
||||
depicted in Figure~\ref{mink_fig:sum_holes}), using the small-side
|
||||
angle-bisector decomposition strategy:
|
||||
|
||||
\ccIncludeExampleCode{../examples/Minkowski_sum_2/ex_sum_by_decomposition.C}
|
||||
|
||||
\section{Offsetting a Polygon}
|
||||
\label{mink_sec:offset}
|
||||
|
|
@ -233,9 +271,9 @@ is widely known as \emph{offsetting} the polygon $P$ by a radius $r$.
|
|||
\begin{ccTexOnly}
|
||||
\begin{center}
|
||||
\begin{tabular}{ccc}
|
||||
\psfig{figure=Minkowski_sum_2/fig/convex_offset.eps,width=1.8in,silent=} &
|
||||
\psfig{figure=Minkowski_sum_2/fig/offset_decomp.eps,width=1.8in,silent=} &
|
||||
\psfig{figure=Minkowski_sum_2/fig/offset_conv.eps,width=1.8in,silent=} \\
|
||||
\includegraphics{Minkowski_sum_2/fig/convex_offset} &
|
||||
\includegraphics{Minkowski_sum_2/fig/offset_decomp} &
|
||||
\includegraphics{Minkowski_sum_2/fig/offset_conv} \\
|
||||
{\small (a)} & {\small (b)} & {\small (c)}
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
|
|
@ -285,43 +323,44 @@ illustration. We finally compute the winding numbers of the faces of the
|
|||
arrangement induced by the convolution cycle to obtain the offset
|
||||
polygon.
|
||||
|
||||
%RWRW : From here ...
|
||||
\begin{figure}[t]
|
||||
\begin{ccTexOnly}
|
||||
\begin{center}
|
||||
\includegraphics{Minkowski_sum_2/fig/ex_offset}
|
||||
\end{center}
|
||||
\end{ccTexOnly}
|
||||
\begin{ccHtmlOnly}
|
||||
<p><center>
|
||||
<img src="./fig/ex_offset.gif" border=0 alt="Offsetting a polygon">
|
||||
</center>
|
||||
\end{ccHtmlOnly}
|
||||
\caption{The offset computation performed by the example programs
|
||||
\ccc{ex_exact_offset.C} and \ccc{ex_approx_offset.C}. The input polygon
|
||||
is shaded and the boundary of its offset is drawn in a thick black line.}
|
||||
\label{mink_fig:ex_offset}
|
||||
\end{figure}
|
||||
|
||||
\ccIncludeExampleCode{../examples/Minkowski_sum_2/ex_exact_offset.C}
|
||||
|
||||
\subsection{Approximating the Offset with a Guaranteed Error-Bound}
|
||||
\label{mink_ssec:approx_offset}
|
||||
%------------------------------------------------------------------
|
||||
|
||||
\begin{figure}[t]
|
||||
\begin{ccTexOnly}
|
||||
\begin{center}
|
||||
\input{Minkowski_sum_2/fig/approx_offset.pstex_t}
|
||||
\end{center}
|
||||
\end{ccTexOnly}
|
||||
\begin{ccHtmlOnly}
|
||||
<p><center>
|
||||
<img src="./fig/tight.gif" border=0 alt="Approximating an offset edge">
|
||||
</center>
|
||||
\end{ccHtmlOnly}
|
||||
\caption{Approximating the offset edge $o_1 o_2$ induced by the polygon
|
||||
edge $p_1 p_2$ by two line segments $o'_1 q'$ and $q' o'_2$.}
|
||||
\label{mink_fig:approx_offset}
|
||||
\end{figure}
|
||||
|
||||
\ccIncludeExampleCode{../examples/Minkowski_sum_2/ex_approx_offset.C}
|
||||
|
||||
The circular arcs of the convolution cycle are clearly supported
|
||||
by rational circles. Let us examine how the offset edges look
|
||||
like. We consider the polygon edge directed from $v_1 = (x_1,
|
||||
y_1)$ to $v_2 = (x_2, y_2)$, we denote by $\theta$ the angle
|
||||
$\vec{v_1 v_2}$ forms with the $x$ axis. Let $\ell = \sqrt{(x_2 -
|
||||
x_1)^2 + (y_2 - y_1)^2}$ be the edge length, so we have
|
||||
$\cos\theta = \frac{1}{\ell}(x_2 - x_1)$ and $\sin\theta =
|
||||
\frac{1}{\ell}(y_2 - y_1)$. As we traverse the polygon edges in a
|
||||
counterclockwise orientation, we construct the offset edge $p_1
|
||||
p_2$ that corresponds to $v_1 v_2$ by shifting either polygon
|
||||
vertex by a vector whose length is $r$ and which forms an angle of
|
||||
$\phi = \theta - \frac{\pi}{2}$ with the $x$-axis. It is easy to
|
||||
see that:
|
||||
\begin{eqnarray}
|
||||
\sin\phi &=& \sin\theta\cdot\cos\frac{\pi}{2} -
|
||||
\cos\theta\cdot\sin\frac{\pi}{2}
|
||||
= - \cos\theta = \frac{1}{\ell}(x_1 - x_2) \ , \\
|
||||
\cos\phi &=& \cos\theta\cdot\cos\frac{\pi}{2} +
|
||||
\sin\theta\cdot\sin\frac{\pi}{2}
|
||||
= \sin\theta = \frac{1}{\ell}(y_2 - y_1) \ .
|
||||
\end{eqnarray}
|
||||
Thus, $p_j = \left( x_j + \frac{r}{\ell}(y_2 - y_1), y_j +
|
||||
\frac{r}{\ell}(x_1 - x_2) \right)$ for $j = 1,2$. Indeed, the
|
||||
coordinates of these points are one-root numbers, but the segment
|
||||
$p_1 p_2$ is supported by a line whose coefficient are
|
||||
\emph{irrational}: it is easy to show that if the supporting line
|
||||
of $v_1 v_2$ is $a x + b y + c = 0$ (where $a,b,c \in {\mathbb Q}$),
|
||||
then the line supporting $p_1 p_2$ is $a x + b y + (c + \ell r) =
|
||||
0$, where $\ell$ is usually an irrational number. The coordinates
|
||||
of the intersection points between two offset edges, or between an
|
||||
offset edge and a circular arc that represents an offset vertex,
|
||||
are therefore \emph{not} one-root numbers. It should be mentioned
|
||||
that the locus of all points lying at distance $r$ from the line
|
||||
$a x + b y + c = 0$ is given by $\frac{(a x + b y + c)^2}{a^2 +
|
||||
b^2} = r^2$, which is a degenerate conic curve (a pair of parallel
|
||||
lines) with rational coefficients. Offset polygons can therefore
|
||||
by constructed in an exact manner using the conic-arc
|
||||
traits-class.
|
||||
|
|
|
|||
Loading…
Reference in New Issue