From eddb050508fe8a0cf8c88189807b8c1131c48d90 Mon Sep 17 00:00:00 2001
From: Juan Marín Noguera <juan.marinn@um.es>
Date: Thu, 14 Jan 2021 13:49:43 +0100
Subject: gcs/a3d

---
 gcs/n3.lyx | 157 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 157 insertions(+)

diff --git a/gcs/n3.lyx b/gcs/n3.lyx
index 03db12a..e49afd8 100644
--- a/gcs/n3.lyx
+++ b/gcs/n3.lyx
@@ -1894,5 +1894,162 @@ curvatura normal
 .
 \end_layout
 
+\begin_layout Standard
+Ejemplos:
+\end_layout
+
+\begin_layout Enumerate
+Un plano tiene curvatura normal 0 en todo punto y dirección.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Como 
+\begin_inset Formula $A_{p}=0$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa_{n}(v,p)={\cal II}_{p}(v)=\langle A_{p}v,v\rangle=0$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+\begin_inset Formula $\mathbb{S}^{2}(r)$
+\end_inset
+
+ tiene curvatura normal constante 
+\begin_inset Formula $-\frac{1}{r}$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Como 
+\begin_inset Formula $N(p)=\frac{1}{r}p$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa_{n}(v,p)=\langle A_{p}v,v\rangle=\langle-\frac{1}{r}v,v\rangle=-\frac{1}{r}|v|^{2}=-\frac{1}{r}$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Dados 
+\begin_inset Formula $p\in S$
+\end_inset
+
+, 
+\begin_inset Formula $v\in T_{p}S$
+\end_inset
+
+ unitario y 
+\begin_inset Formula $\Pi_{v}:=\text{span}\{v,N(p)\}$
+\end_inset
+
+, llamamos 
+\series bold
+sección normal
+\series default
+ 
+\begin_inset Formula $C_{v}$
+\end_inset
+
+ a la curva regular plana resultante de intersecar 
+\begin_inset Formula $S$
+\end_inset
+
+ con 
+\begin_inset Formula $\Pi_{v}$
+\end_inset
+
+.
+ Sea entonces 
+\begin_inset Formula $\alpha:I\to S$
+\end_inset
+
+ una parametrización por arco de 
+\begin_inset Formula $C_{v}$
+\end_inset
+
+ con 
+\begin_inset Formula $\alpha(0)=p$
+\end_inset
+
+ y 
+\begin_inset Formula $\alpha'(0)=v$
+\end_inset
+
+, entonces 
+\begin_inset Formula $\kappa_{n}(v,p)=\kappa(0)$
+\end_inset
+
+, siendo 
+\begin_inset Formula $\kappa$
+\end_inset
+
+ la curvatura de 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ como curva plana.
+ En efecto, como 
+\begin_inset Formula $v\in T_{p}S$
+\end_inset
+
+, 
+\begin_inset Formula $v\bot N(p)$
+\end_inset
+
+ y el vector normal es 
+\begin_inset Formula $\mathbf{n}=J_{\Pi_{v}}v=\pm N(p)$
+\end_inset
+
+, y como todavía no hemos orientado el plano podemos tomar 
+\begin_inset Formula $\mathbf{n}=N(p)$
+\end_inset
+
+, pero entonces 
+\begin_inset Formula $\kappa_{n}(v,p)=\langle\alpha''(0),N(p)\rangle=\langle\kappa(0)\mathbf{n}(0),N(p)\rangle=\kappa(0)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Si 
+\begin_inset Formula $\alpha:I\to S$
+\end_inset
+
+ es una curva parametrizada por arco, 
+\begin_inset Formula $\alpha''(s)=\kappa_{g}(s)J\alpha'(s)+\kappa_{n}(s)N(\alpha(s))$
+\end_inset
+
+, siendo 
+\begin_inset Formula $\kappa_{n}(s):=\kappa_{n}(\alpha'(s),\alpha(s))=\langle\alpha''(s),N(\alpha(s))\rangle$
+\end_inset
+
+, luego 
+\begin_inset Formula 
+\[
+\kappa(s)^{2}=\kappa_{g}(s)^{2}+\kappa_{n}(s)^{2}.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Curvaturas principales
+\end_layout
+
 \end_body
 \end_document
-- 
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