diff --git a/Envelope_2/doc_tex/Envelope_2/envelope.tex b/Envelope_2/doc_tex/Envelope_2/envelope.tex index c876199287a..22cae24e6ec 100644 --- a/Envelope_2/doc_tex/Envelope_2/envelope.tex +++ b/Envelope_2/doc_tex/Envelope_2/envelope.tex @@ -13,17 +13,32 @@ curves, their {\em lower envelope} is defined as the point-wise minimum of all curves. Namely, the lower envelope of the set $\calC$ can be defined as the following function: \begin{eqnarray*} -\calL_{\calC} (x) = \min_{1 \leq k \leq n}{\overline{C}_k (x)} \ , +\calL_{\calC} (x) = \min_{1 \leq k \leq n}{ +\begin{ccTexOnly}\overline{C}\end{ccTexOnly} +\begin{ccHtmlOnly}C\end{ccHtmlOnly} +_k (x)} \ , \end{eqnarray*} -where we define $\overline{C}_k(x) = C_k(x)$ for $x \in R_{C_k}$, -and $\overline{C}_k(x) = \infty$ otherwise. +where we define $ +\begin{ccTexOnly}\overline{C}\end{ccTexOnly} +\begin{ccHtmlOnly}C\end{ccHtmlOnly} +_k(x) = C_k(x)$ for $x \in R_{C_k}$, +and $ +\begin{ccTexOnly}\overline{C}\end{ccTexOnly} +\begin{ccHtmlOnly}C\end{ccHtmlOnly} +_k(x) = \infty$ otherwise. Similarly, the {\em upper envelope} of $\calC$ is the point-wise maximum of the $x$-monotone curves in the set: \begin{eqnarray*} -\calU_{\calC} (x) = \max_{1 \leq k \leq n}{\underline{C}_k (x)} \ , +\calU_{\calC} (x) = \max_{1 \leq k \leq n}{ +\begin{ccTexOnly}\underline{C}\end{ccTexOnly} +\begin{ccHtmlOnly}C\end{ccHtmlOnly} +_k (x)} \ , \end{eqnarray*} -where in this case $\underline{C}_k(x) = -\infty$ for $x +where in this case $ +\begin{ccTexOnly}\underline{C}\end{ccTexOnly} +\begin{ccHtmlOnly}C\end{ccHtmlOnly} +_k(x) = -\infty$ for $x \not\in R_{C_k}$. Given a set of $x$-monotone curves $\calC$, the {\em minimization @@ -71,7 +86,7 @@ diagram these functions output. \begin{figure}[t] \begin{ccTexOnly} \begin{center} - \epsfig{figure=Envelope_2/fig/min_diag.eps,width=5in,silent=} + \includegraphics[width=5in]{Envelope_2/fig/min_diag} \end{center} \end{ccTexOnly} \begin{ccHtmlOnly} @@ -169,7 +184,7 @@ diagrams: \begin{figure}[t] \begin{ccTexOnly} \begin{center} - \epsfig{figure=Envelope_2/fig/ex_circle.eps,silent=} + \includegraphics{Envelope_2/fig/ex_circle} \end{center} \end{ccTexOnly} \begin{ccHtmlOnly} diff --git a/Envelope_2/doc_tex/Envelope_2/fig/ex_circle.eps b/Envelope_2/doc_tex/Envelope_2/fig/ex_circle.eps index ae0cb8fea57..3d5bbbcc107 100644 --- a/Envelope_2/doc_tex/Envelope_2/fig/ex_circle.eps +++ b/Envelope_2/doc_tex/Envelope_2/fig/ex_circle.eps @@ -1,9 +1,10 @@ %!PS-Adobe-2.0 EPSF-2.0 %%Title: ex_circle.fig %%Creator: fig2dev Version 3.2 Patchlevel 5-alpha5 -%%CreationDate: Thu Aug 3 16:04:45 2006 -%%For: wein@cgal (Ron Wein,,09-576985,) +%%CreationDate: Sun Aug 13 17:26:49 2006 +%%For: efif@lap-eye (Efraim Fogel,,,) %%BoundingBox: 0 0 185 177 +%%HiResBoundingBox: 0 0 185 177 %Magnification: 0.5000 %%EndComments /$F2psDict 200 dict def @@ -117,17 +118,17 @@ n 7639 7425 101 101 0 360 DrawEllipse gs col7 s gr % here starts figure with depth 50 % Ellipse 15.000 slw -n 5850 3600 900 900 0 360 DrawEllipse gs col0 s gr - -% Ellipse -n 3600 5400 900 900 0 360 DrawEllipse gs col0 s gr - -% Ellipse n 5400 6300 450 450 0 360 DrawEllipse gs col0 s gr % Ellipse n 4950 4500 1800 1800 0 360 DrawEllipse gs col0 s gr +% Ellipse +n 5850 3600 900 900 0 360 DrawEllipse gs col0 s gr + +% Ellipse +n 3600 5400 900 900 0 360 DrawEllipse gs col0 s gr + % Polyline 0 slj 0 slc @@ -210,15 +211,10 @@ n 3150 2700 m % % here starts figure with depth 40 % Arc -30.000 slw +90.000 slw 0 slc [120] 0 sd -n 3600.0 5400.0 900.0 180.0000 53.1301 arcn -gs col4 s gr - [] 0 sd -% Arc - [120] 0 sd -n 4965.0 4320.0 1980.1 114.6236 90.4341 arcn +n 5119.2 4570.4 1632.3 63.4042 -2.4732 arcn gs col4 s gr [] 0 sd % Arc @@ -228,23 +224,28 @@ gs col4 s gr [] 0 sd % Arc [120] 0 sd -n 5119.2 4570.4 1632.3 63.4042 -2.4732 arcn +n 4965.0 4320.0 1980.1 114.6236 90.4341 arcn gs col4 s gr [] 0 sd % Arc [120] 0 sd -n 5896.7 3542.4 855.2 -118.3937 3.8642 arc -gs col11 s gr +n 3600.0 5400.0 900.0 180.0000 53.1301 arcn +gs col4 s gr [] 0 sd % Arc [120] 0 sd -n 3735.0 5445.0 1036.0 -177.5104 -124.3803 arc -gs col11 s gr +n 5876.5 3573.7 873.9 -116.2512 1.7217 arc +gs col16 s gr [] 0 sd % Arc [120] 0 sd n 4918.5 4464.0 1768.9 178.8338 -71.1502 arc -gs col11 s gr +gs col16 s gr + [] 0 sd +% Arc + [120] 0 sd +n 3735.0 5445.0 1036.0 -177.5104 -124.3803 arc +gs col16 s gr [] 0 sd % here ends figure; 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