From 8ce7f12fbbde081929413816fabac7c730c3116b Mon Sep 17 00:00:00 2001
From: utensil <utensilcandel@gmail.com>
Date: Thu, 19 Sep 2024 22:48:07 +0800
Subject: [PATCH] Add: Ray marching (naive)

---
 trees/ag-0001.tree |  2 +-
 trees/ag-000N.tree |  2 +-
 trees/ag-000V.tree |  2 +-
 trees/ag-000W.tree |  2 +-
 trees/ag-0018.tree | 55 ++++++++++++++++++++++++++++++++++++++++++++++
 5 files changed, 59 insertions(+), 4 deletions(-)
 create mode 100644 trees/ag-0018.tree

diff --git a/trees/ag-0001.tree b/trees/ag-0001.tree
index f293e27..b7c83a7 100644
--- a/trees/ag-0001.tree
+++ b/trees/ag-0001.tree
@@ -8,7 +8,7 @@
 \author{utensil}
 \date{2024-08-13}
 
-\meta{pdf}{true}
+% \meta{pdf}{true}
 
 % definition theorem lemma construction observation
 % convention corollary axiom example exercise proof
diff --git a/trees/ag-000N.tree b/trees/ag-000N.tree
index 4598874..1aa949e 100644
--- a/trees/ag-000N.tree
+++ b/trees/ag-000N.tree
@@ -16,5 +16,5 @@
 
 \transclude{ag-000V}
 \transclude{ag-000W}
-
+\transclude{ag-0018}
 }
diff --git a/trees/ag-000V.tree b/trees/ag-000V.tree
index a829a9c..df560ac 100644
--- a/trees/ag-000V.tree
+++ b/trees/ag-000V.tree
@@ -14,7 +14,7 @@
 
 \refdeft{ray equation}{sec. 1.1 eq. 3}{gillespie2024ray}{
 \p{
-A \newvocab{ray} anchored at #{\boldsymbol{r}_o} in the direction of the unit vector #{\boldsymbol{r}_d} can be parametrically defined as
+A \newvocab{ray} anchored at origin #{\boldsymbol{r}_o} in the direction of the unit vector #{\boldsymbol{r}_d} can be parametrically defined as
 
 
 ##{
diff --git a/trees/ag-000W.tree b/trees/ag-000W.tree
index 0ec4d85..8adc44f 100644
--- a/trees/ag-000W.tree
+++ b/trees/ag-000W.tree
@@ -21,7 +21,7 @@ F(t)=0
 
 correspond to \newvocab{ray intersection}s with the implicit surface.}
 
-\p{Implicit surface \newvocab{ray-tracing} algorithms simply apply one of the multitude of numerical root finding methods to solve this equation.
+\p{Implicit surface \newvocab{ray-casting}/\newvocab{ray-tracing}/\newvocab{ray-marching} algorithms simply apply one of the multitude of numerical root finding methods to solve this equation.
 }
 
 }
diff --git a/trees/ag-0018.tree b/trees/ag-0018.tree
new file mode 100644
index 0000000..bf3473c
--- /dev/null
+++ b/trees/ag-0018.tree
@@ -0,0 +1,55 @@
+\import{cg-macros}
+% clifford hopf spin tt ag math draft cg
+\tag{cg}
+
+% definition theorem lemma construction observation
+% convention corollary axiom example exercise proof
+% discussion remark notation
+% \taxon{}
+% \refcardt{lemma}{}{}{}{
+
+% kostecki2011introduction leinster2016basic nakahira2023diagrammatic rosiak2022sheaf
+
+% cox1997ideals gathmann2013commutative hart1996sphere gillespie2024ray winchenbach2024lipschitz
+
+\def\alg/input[x]{\textbf{Input:} \x \\ }
+\def\alg/inputc[x]{\phantom{\textbf{Input:}} \x \\ }
+\def\alg/param[x]{\textbf{Parameters:} \x \\ }
+\def\alg/paramc[x]{\phantom{\textbf{Parameters:}} \x \\ }
+\def\alg/output[x]{\textbf{Output:} \x \\ }
+\def\alg/outputc[x]{\phantom{\textbf{Output:}} \x \\ }
+
+\def\ro{\mathbf{r}_o}
+\def\rd{\mathbf{r}_d}
+\def\rt{\mathbf{r}_t}
+\def\dt{\Delta t}
+\def\eps{\varepsilon}
+\def\tmax{t_{\max}}
+
+\card{algorithm}{ray marching (naïve)}{
+\minialg{
+% \begin{algorithm*}{Ray marching (naive)}{}
+\alg/input{$\ro \in \RR^3$, ray origin}
+\alg/inputc{$\rd \in \RR^3$, unit ray direction}
+\alg/inputc{$f: \RR^3 \to \RR$, implicit surface function}
+\alg/param{$\dt$, step size}
+\alg/paramc{$\eps>0$, stopping tolerance}
+\alg/paramc{$\tmax>0$, maximum ray time}
+\alg/output{$t^*$, the time of the first intersection, $\tmax$ if no intersection occurs}
+\begin{algorithmic}
+    \Procedure{RayMarch}{$\ro,\rd,f;\dt, \eps,\tmax$}
+        \State $t \gets 0$
+        \While{$t < \tmax $}
+            \State $\rt \gets \ro + t \rd $  \Comment{current point along ray}
+            \State $f_t \gets f(\rt)$        \Comment{current function value}
+            \If{$f_t < \eps$}                \Comment{stopping condition}
+                \State \textbf{return} $t$
+            \EndIf
+            \State $t \gets t + \dt$         \Comment{take step}
+        \EndWhile
+        \State \textbf{return} $\tmax$       \Comment{ray does not hit surface}
+    \EndProcedure
+\end{algorithmic}
+% \end{algorithm*}
+}
+}