From 7304de171e8b31f144565284ebb0aefdf860066d Mon Sep 17 00:00:00 2001
From: baluteshih <brian369999@gmail.com>
Date: Sat, 21 Oct 2023 02:50:21 +0800
Subject: [PATCH] Add Nearest points of two skew lines

---
 codebook/6_Math/Theorem.tex | 11 +++++++++++
 1 file changed, 11 insertions(+)

diff --git a/codebook/6_Math/Theorem.tex b/codebook/6_Math/Theorem.tex
index 7c84641a..b4d98ee5 100644
--- a/codebook/6_Math/Theorem.tex
+++ b/codebook/6_Math/Theorem.tex
@@ -75,5 +75,16 @@
   \item The solution corresponding to the original constrained optimization is always a saddle point of the Lagrangian function.
 \end{itemize}
 
+\item Nearest points of two skew lines
+
+\begin{itemize}
+\item $\text{Line 1}: \boldsymbol{v}_1 = \boldsymbol{p} + t_1\boldsymbol{d}_1$
+\item $\text{Line 2}: \boldsymbol{v}_2 = \boldsymbol{p} + t_2\boldsymbol{d}_2$
+\item $\boldsymbol{n} = \boldsymbol{d}_1\times \boldsymbol{d}_2$
+\item $\boldsymbol{n}_1 = \boldsymbol{d}_1 \times \boldsymbol{n}$
+\item $\boldsymbol{n}_2 = \boldsymbol{d}_2 \times \boldsymbol{n}$
+\item $\boldsymbol{c}_1 = \boldsymbol{p}_1 + \frac{(\boldsymbol{p}_2 - \boldsymbol{p}_1)\cdot\boldsymbol{n}_2}{\boldsymbol{d}_1\cdot\boldsymbol{n}_2}\boldsymbol{d}_1$
+\item $\boldsymbol{c}_2 = \boldsymbol{p}_2 + \frac{(\boldsymbol{p}_1 - \boldsymbol{p}_2)\cdot\boldsymbol{n}_1}{\boldsymbol{d}_2\cdot\boldsymbol{n}_1}\boldsymbol{d}_2$
+\end{itemize}
 
 \end{itemize}