diff --git a/HW2/HW2.html b/HW2/HW2.html
index 8236202..70001ac 100644
--- a/HW2/HW2.html
+++ b/HW2/HW2.html
@@ -987,6 +987,16 @@ hue-6-2: #e6c07b
 <li>QuickHull(p5 , p3 , {p6 , p7, p1})</li>
 <li>QuickHull(p7 , p3 , {p1})</li>
 </ul>
+<p>recursion stack:</p>
+<blockquote>
+<ul>
+<li>QuickHull(p9 , p3 , {p0 , p1, p2 , p4, p5 , p6, p7 , p8, p10 , p 11, p12 })</li>
+<li>QuickHull((QuickHull(p9 , p5 , {p4 , p11, p8 , p10}), 5, QuickHull(p5 , p3 , {p6 , p7, p1}))</li>
+<li>QuickHull(QuickHull(p9 , p11 , {p4}), 11, QuickHull(p11 , p5 , {p10}), 5, 7, QuickHull(p7 , p3 , {p1})))</li>
+<li>QuickHull(QuickHull(p9 , p4 , {}}), 4, QuickHull(p4 , p11 , {}}), 11, QuickHull(p11 , p10 , {}), 10, QuickHull(p10 , p5 , {}), 5, 7, QuickHull(p7 , p1 , {}), 1, QuickHull(p1 , p3 , {})))</li>
+<li>QuickHull(4, 11, 10, 5, 7, 1)</li>
+</ul>
+</blockquote>
 </li>
 <li>
 <ol start="2">
diff --git a/HW2/HW2.md b/HW2/HW2.md
index f7ed383..0ae1dd3 100644
--- a/HW2/HW2.md
+++ b/HW2/HW2.md
@@ -63,6 +63,13 @@
    - QuickHull(p11 , p5 , {p10})
    - QuickHull(p5 , p3 , {p6 , p7, p1})
    - QuickHull(p7 , p3 , {p1})
+
+    recursion stack:
+    >  - QuickHull(p9 , p3 , {p0 , p1, p2 , p4, p5 , p6, p7 , p8, p10 , p 11, p12 })
+    >  - QuickHull((QuickHull(p9 , p5 , {p4 , p11, p8 , p10}), 5, QuickHull(p5 , p3 , {p6 , p7, p1}))
+    >  - QuickHull(QuickHull(p9 , p11 , {p4}), 11, QuickHull(p11 , p5 , {p10}), 5, 7, QuickHull(p7 , p3 , {p1})))
+    >  - QuickHull(QuickHull(p9 , p4 , {}}), 4, QuickHull(p4 , p11 , {}}), 11, QuickHull(p11 , p10 , {}), 10, QuickHull(p10 , p5 , {}), 5, 7, QuickHull(p7 , p1 , {}), 1, QuickHull(p1 , p3 , {})))
+    >  - QuickHull(4, 11, 10, 5, 7, 1)
   - 2. 
     - 1. Correct, this is basically the same as Graham scan.
     - 2. Wrong, counter example, c is the smallest angle but c is not a part of convex hull. 
diff --git a/HW2/HW2.pdf b/HW2/HW2.pdf
index b88c57f..99efe2d 100644
Binary files a/HW2/HW2.pdf and b/HW2/HW2.pdf differ
diff --git a/HW2/hw2-quickhull.ipe b/HW2/hw2-quickhull.ipe
index 058ceac..6320f74 100644
--- a/HW2/hw2-quickhull.ipe
+++ b/HW2/hw2-quickhull.ipe
@@ -1,7 +1,7 @@
 <?xml version="1.0"?>
 <!DOCTYPE ipe SYSTEM "ipe.dtd">
 <ipe version="70218" creator="Ipe 7.2.26">
-<info created="D:20210912132529" modified="D:20220918194029"/>
+<info created="D:20210912132529" modified="D:20220920155646"/>
 <ipestyle name="basic">
 <symbol name="arrow/arc(spx)">
 <path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
@@ -301,5 +301,45 @@ h
 144 720 l
 </path>
 <text transformations="translations" pos="144 720" stroke="black" type="label" width="7.61" height="6.808" depth="0" valign="baseline">D</text>
+<path stroke="black">
+80 400 m
+496 464 l
+</path>
+<path stroke="black">
+80 400 m
+276 536 l
+</path>
+<path stroke="black">
+276 536 m
+496 464 l
+</path>
+<path stroke="black">
+80 400 m
+140 476 l
+276 536 l
+</path>
+<path stroke="black">
+80 400 m
+100 436 l
+140 476 l
+</path>
+<path stroke="black">
+140 476 m
+212 520 l
+276 536 l
+</path>
+<path stroke="black">
+276 536 m
+352 536 l
+</path>
+<path stroke="black">
+352 536 m
+496 464 l
+</path>
+<path stroke="black">
+352 536 m
+468 492 l
+496 464 l
+</path>
 </page>
 </ipe>
diff --git a/HW2/hw2-quickhull.ipe.autosave.ipe b/HW2/hw2-quickhull.ipe.autosave.ipe
index 615aad0..3c37e49 100644
--- a/HW2/hw2-quickhull.ipe.autosave.ipe
+++ b/HW2/hw2-quickhull.ipe.autosave.ipe
@@ -1,7 +1,7 @@
 <?xml version="1.0"?>
 <!DOCTYPE ipe SYSTEM "ipe.dtd">
-<ipe version="70218" creator="Ipe 7.2.13">
-<info created="D:20210912132529" modified="D:20220909153350"/>
+<ipe version="70218" creator="Ipe 7.2.26">
+<info created="D:20210912132529" modified="D:20220918194029"/>
 <ipestyle name="basic">
 <symbol name="arrow/arc(spx)">
 <path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
@@ -301,5 +301,45 @@ h
 144 720 l
 </path>
 <text transformations="translations" pos="144 720" stroke="black" type="label" width="7.61" height="6.808" depth="0" valign="baseline">D</text>
+<path stroke="black">
+80 400 m
+496 464 l
+</path>
+<path stroke="black">
+80 400 m
+276 536 l
+</path>
+<path stroke="black">
+276 536 m
+496 464 l
+</path>
+<path stroke="black">
+80 400 m
+140 476 l
+276 536 l
+</path>
+<path stroke="black">
+80 400 m
+100 436 l
+140 476 l
+</path>
+<path stroke="black">
+140 476 m
+212 520 l
+276 536 l
+</path>
+<path stroke="black">
+276 536 m
+352 536 l
+</path>
+<path stroke="black">
+352 536 m
+496 464 l
+</path>
+<path stroke="black">
+352 536 m
+468 492 l
+496 464 l
+</path>
 </page>
 </ipe>