mirror of https://github.com/CGAL/cgal
updated figures
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Efi Fogel
parent
7d0dc2ddcc
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@ -13,17 +13,32 @@ curves, their {\em lower envelope} is defined as the point-wise minimum of
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all curves. Namely, the lower envelope of the set $\calC$ can be
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all curves. Namely, the lower envelope of the set $\calC$ can be
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defined as the following function:
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defined as the following function:
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\begin{eqnarray*}
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\begin{eqnarray*}
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\calL_{\calC} (x) = \min_{1 \leq k \leq n}{\overline{C}_k (x)} \ ,
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\calL_{\calC} (x) = \min_{1 \leq k \leq n}{
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\begin{ccTexOnly}\overline{C}\end{ccTexOnly}
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\begin{ccHtmlOnly}<span style="text-decoration:overline">C</span>\end{ccHtmlOnly}
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_k (x)} \ ,
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\end{eqnarray*}
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\end{eqnarray*}
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where we define $\overline{C}_k(x) = C_k(x)$ for $x \in R_{C_k}$,
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where we define $
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and $\overline{C}_k(x) = \infty$ otherwise.
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\begin{ccTexOnly}\overline{C}\end{ccTexOnly}
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\begin{ccHtmlOnly}<span style="text-decoration:overline">C</span>\end{ccHtmlOnly}
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_k(x) = C_k(x)$ for $x \in R_{C_k}$,
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and $
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\begin{ccTexOnly}\overline{C}\end{ccTexOnly}
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\begin{ccHtmlOnly}<span style="text-decoration:overline">C</span>\end{ccHtmlOnly}
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_k(x) = \infty$ otherwise.
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Similarly, the {\em upper envelope} of $\calC$ is the point-wise maximum of
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Similarly, the {\em upper envelope} of $\calC$ is the point-wise maximum of
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the $x$-monotone curves in the set:
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the $x$-monotone curves in the set:
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\begin{eqnarray*}
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\begin{eqnarray*}
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\calU_{\calC} (x) = \max_{1 \leq k \leq n}{\underline{C}_k (x)} \ ,
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\calU_{\calC} (x) = \max_{1 \leq k \leq n}{
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\begin{ccTexOnly}\underline{C}\end{ccTexOnly}
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\begin{ccHtmlOnly}<span style="text-decoration:underline">C</span>\end{ccHtmlOnly}
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_k (x)} \ ,
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\end{eqnarray*}
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\end{eqnarray*}
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where in this case $\underline{C}_k(x) = -\infty$ for $x
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where in this case $
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\begin{ccTexOnly}\underline{C}\end{ccTexOnly}
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\begin{ccHtmlOnly}<span style="text-decoration:underline">C</span>\end{ccHtmlOnly}
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_k(x) = -\infty$ for $x
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\not\in R_{C_k}$.
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\not\in R_{C_k}$.
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Given a set of $x$-monotone curves $\calC$, the {\em minimization
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Given a set of $x$-monotone curves $\calC$, the {\em minimization
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@ -71,7 +86,7 @@ diagram these functions output.
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\begin{figure}[t]
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\begin{figure}[t]
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\begin{ccTexOnly}
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\begin{ccTexOnly}
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\begin{center}
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\begin{center}
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\epsfig{figure=Envelope_2/fig/min_diag.eps,width=5in,silent=}
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\includegraphics[width=5in]{Envelope_2/fig/min_diag}
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\end{center}
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\end{center}
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\end{ccTexOnly}
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\end{ccTexOnly}
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\begin{ccHtmlOnly}
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\begin{ccHtmlOnly}
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@ -169,7 +184,7 @@ diagrams:
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\begin{figure}[t]
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\begin{figure}[t]
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\begin{ccTexOnly}
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\begin{ccTexOnly}
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\begin{center}
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\begin{center}
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\epsfig{figure=Envelope_2/fig/ex_circle.eps,silent=}
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\includegraphics{Envelope_2/fig/ex_circle}
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\end{center}
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\end{center}
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\end{ccTexOnly}
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\end{ccTexOnly}
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\begin{ccHtmlOnly}
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\begin{ccHtmlOnly}
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@ -1,9 +1,10 @@
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%!PS-Adobe-2.0 EPSF-2.0
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%!PS-Adobe-2.0 EPSF-2.0
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%%Title: ex_circle.fig
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%%Title: ex_circle.fig
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%%Creator: fig2dev Version 3.2 Patchlevel 5-alpha5
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%%Creator: fig2dev Version 3.2 Patchlevel 5-alpha5
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%%CreationDate: Thu Aug 3 16:04:45 2006
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%%CreationDate: Sun Aug 13 17:26:49 2006
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%%For: wein@cgal (Ron Wein,,09-576985,)
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%%For: efif@lap-eye (Efraim Fogel,,,)
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%%BoundingBox: 0 0 185 177
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%%BoundingBox: 0 0 185 177
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%%HiResBoundingBox: 0 0 185 177
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%Magnification: 0.5000
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%Magnification: 0.5000
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%%EndComments
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%%EndComments
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/$F2psDict 200 dict def
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/$F2psDict 200 dict def
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@ -117,17 +118,17 @@ n 7639 7425 101 101 0 360 DrawEllipse gs col7 s gr
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% here starts figure with depth 50
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% here starts figure with depth 50
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% Ellipse
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% Ellipse
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15.000 slw
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15.000 slw
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n 5850 3600 900 900 0 360 DrawEllipse gs col0 s gr
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% Ellipse
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n 3600 5400 900 900 0 360 DrawEllipse gs col0 s gr
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% Ellipse
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n 5400 6300 450 450 0 360 DrawEllipse gs col0 s gr
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n 5400 6300 450 450 0 360 DrawEllipse gs col0 s gr
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% Ellipse
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% Ellipse
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n 4950 4500 1800 1800 0 360 DrawEllipse gs col0 s gr
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n 4950 4500 1800 1800 0 360 DrawEllipse gs col0 s gr
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% Ellipse
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n 5850 3600 900 900 0 360 DrawEllipse gs col0 s gr
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% Ellipse
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n 3600 5400 900 900 0 360 DrawEllipse gs col0 s gr
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% Polyline
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% Polyline
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0 slj
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0 slj
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0 slc
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0 slc
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@ -210,15 +211,10 @@ n 3150 2700 m
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%
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%
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% here starts figure with depth 40
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% here starts figure with depth 40
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% Arc
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% Arc
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30.000 slw
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90.000 slw
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0 slc
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0 slc
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[120] 0 sd
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[120] 0 sd
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n 3600.0 5400.0 900.0 180.0000 53.1301 arcn
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n 5119.2 4570.4 1632.3 63.4042 -2.4732 arcn
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gs col4 s gr
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[] 0 sd
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% Arc
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[120] 0 sd
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n 4965.0 4320.0 1980.1 114.6236 90.4341 arcn
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gs col4 s gr
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gs col4 s gr
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[] 0 sd
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[] 0 sd
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% Arc
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% Arc
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@ -228,23 +224,28 @@ gs col4 s gr
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[] 0 sd
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[] 0 sd
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% Arc
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% Arc
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[120] 0 sd
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[120] 0 sd
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n 5119.2 4570.4 1632.3 63.4042 -2.4732 arcn
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n 4965.0 4320.0 1980.1 114.6236 90.4341 arcn
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gs col4 s gr
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gs col4 s gr
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[] 0 sd
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[] 0 sd
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% Arc
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% Arc
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[120] 0 sd
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[120] 0 sd
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n 5896.7 3542.4 855.2 -118.3937 3.8642 arc
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n 3600.0 5400.0 900.0 180.0000 53.1301 arcn
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gs col11 s gr
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gs col4 s gr
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[] 0 sd
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[] 0 sd
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% Arc
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% Arc
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[120] 0 sd
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[120] 0 sd
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n 3735.0 5445.0 1036.0 -177.5104 -124.3803 arc
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n 5876.5 3573.7 873.9 -116.2512 1.7217 arc
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gs col11 s gr
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gs col16 s gr
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[] 0 sd
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[] 0 sd
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% Arc
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% Arc
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[120] 0 sd
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[120] 0 sd
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n 4918.5 4464.0 1768.9 178.8338 -71.1502 arc
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n 4918.5 4464.0 1768.9 178.8338 -71.1502 arc
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gs col11 s gr
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gs col16 s gr
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[] 0 sd
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% Arc
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[120] 0 sd
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n 3735.0 5445.0 1036.0 -177.5104 -124.3803 arc
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gs col16 s gr
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[] 0 sd
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[] 0 sd
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% here ends figure;
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% here ends figure;
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$F2psEnd
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$F2psEnd
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@ -7,19 +7,19 @@ A4
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Single
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Single
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-2
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-2
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1200 2
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1200 2
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5 1 1 3 4 7 40 -1 -1 8.000 0 1 0 0 3600.000 5400.000 2700 5400 3600 6300 4140 6120
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5 1 1 7 4 7 40 -1 -1 8.000 0 1 0 0 5119.239 4570.435 5850 6030 6525 5400 6750 4500
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5 1 1 3 4 7 40 -1 -1 8.000 0 1 0 0 4965.000 4320.000 4140 6120 4545 6255 4950 6300
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5 1 1 7 4 7 40 -1 -1 8.000 0 1 0 0 5400.000 6300.000 4950 6300 5400 6750 5850 6300
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5 1 1 3 4 7 40 -1 -1 8.000 0 1 0 0 5400.000 6300.000 4950 6300 5400 6750 5850 6300
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5 1 1 7 4 7 40 -1 -1 8.000 0 1 0 0 4965.000 4320.000 4140 6120 4545 6255 4950 6300
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5 1 1 3 4 7 40 -1 -1 8.000 0 1 0 0 5119.239 4570.435 5850 6030 6525 5400 6750 4500
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5 1 1 7 4 7 40 -1 -1 8.000 0 1 0 0 3600.000 5400.000 2700 5400 3600 6300 4140 6120
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5 1 1 3 11 7 40 -1 -1 8.000 0 0 0 0 5896.695 3542.363 5490 2790 6570 3015 6750 3600
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5 1 1 7 16 7 40 -1 -1 8.000 0 0 0 0 5876.521 3573.744 5490 2790 6462 2925 6750 3600
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5 1 1 3 11 7 40 -1 -1 8.000 0 0 0 0 3735.000 5445.000 2700 5400 2880 4860 3150 4590
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5 1 1 7 16 7 40 -1 -1 8.000 0 0 0 0 4918.500 4464.000 3150 4500 3375 3600 5490 2790
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5 1 1 3 11 7 40 -1 -1 8.000 0 0 0 0 4918.500 4464.000 3150 4500 3375 3600 5490 2790
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5 1 1 7 16 7 40 -1 -1 8.000 0 0 0 0 3735.000 5445.000 2700 5400 2880 4860 3150 4590
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1 3 0 2 0 7 50 -1 -1 0.000 1 0.0000 5850 3600 900 900 5850 3600 6750 3600
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1 3 0 2 0 7 50 -1 -1 0.000 1 0.0000 3600 5400 900 900 3600 5400 4500 5400
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1 3 0 2 0 7 50 -1 -1 0.000 1 0.0000 5400 6300 450 450 5400 6300 5850 6300
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1 3 0 2 0 7 50 -1 -1 0.000 1 0.0000 5400 6300 450 450 5400 6300 5850 6300
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1 3 0 1 7 7 55 -1 -1 0.000 1 0.0000 2014 2025 101 101 2014 2025 2115 2025
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1 3 0 1 7 7 55 -1 -1 0.000 1 0.0000 2014 2025 101 101 2014 2025 2115 2025
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1 3 0 1 7 7 55 -1 -1 0.000 1 0.0000 7639 7425 101 101 7639 7425 7740 7425
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1 3 0 1 7 7 55 -1 -1 0.000 1 0.0000 7639 7425 101 101 7639 7425 7740 7425
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1 3 0 2 0 7 50 -1 -1 0.000 1 0.0000 4950 4500 1800 1800 4950 4500 6750 4500
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1 3 0 2 0 7 50 -1 -1 0.000 1 0.0000 4950 4500 1800 1800 4950 4500 6750 4500
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1 3 0 2 0 7 50 -1 -1 0.000 1 0.0000 5850 3600 900 900 5850 3600 6750 3600
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1 3 0 2 0 7 50 -1 -1 0.000 1 0.0000 3600 5400 900 900 3600 5400 4500 5400
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2 1 0 1 15 7 50 -1 -1 0.000 0 0 -1 1 0 2
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2 1 0 1 15 7 50 -1 -1 0.000 0 0 -1 1 0 2
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0 0 1.00 90.00 120.00
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0 0 1.00 90.00 120.00
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3150 7200 3150 2250
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3150 7200 3150 2250
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