From 4b3fafc1d96222c46dd9aeb810ec02aed90e0241 Mon Sep 17 00:00:00 2001 From: Renge Date: Sun, 18 Sep 2022 19:10:30 -0400 Subject: [PATCH] feat: finish hw2 --- HW2/1_1_1.png | Bin 0 -> 46058 bytes HW2/1_2_1.png | Bin 0 -> 10200 bytes HW2/2_2_2.png | Bin 0 -> 11083 bytes HW2/HW2.html | 1029 +++++++++++++++++++++++++++++++++++++++ HW2/HW2.md | 72 +++ HW2/HW2.pdf | Bin 0 -> 163684 bytes HW2/hw2-graham-scan.ipe | 368 ++++++++++++++ HW2/hw2-quickhull.ipe | 284 +++++++++++ 8 files changed, 1753 insertions(+) create mode 100644 HW2/1_1_1.png create mode 100644 HW2/1_2_1.png create mode 100644 HW2/2_2_2.png create mode 100644 HW2/HW2.html create mode 100644 HW2/HW2.md create mode 100644 HW2/HW2.pdf create mode 100644 HW2/hw2-graham-scan.ipe create mode 100644 HW2/hw2-quickhull.ipe diff --git a/HW2/1_1_1.png b/HW2/1_1_1.png new file mode 100644 index 0000000000000000000000000000000000000000..8efed33400323c725624d95fa9375b2d96b0e472 GIT binary patch literal 46058 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HW2 + + + + + + + + + + +

HW2

+

Problem 1

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    +
  • +
      +
    1. +

      Points:

      + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
      indexposition
      0(3, 1)
      1(4, 4)
      2(5, -1)
      3(2, -1)
      4(2, 2)
      5(2, 1)
      6(4, -1)
      7(6, 1)
      8(8, 1)
      9(8, 5)
      10(7, 4)
      11(3, 3)
      +

      Afer sorting:

      + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
      indexpositiondeleted
      2(5, -1)False
      8(8, 1)False
      7(6, 1)True
      9(8, 5)False
      10(7, 4)False
      1(4, 4)False
      11(3, 3)False
      0(3, 1)True
      4(2, 2)False
      5(2, 1)False
      6(4, -1)True
      3(2, -1)False
      +

      Stack history:

      + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
      istack
      28,2
      39,8,2
      410,9,8,2
      49,8,2
      51,9,8,2
      611,1,9,8,2
      61,9,8,2
      74,1,9,8,2
      85,4,1,9,8,2
      84,1,9,8,2
      93,4,1,9,8,2
      +

      image: +1_1_1

      +
    2. +
    +
  • +
  • +
      +
    1. +
    +
      +
    • +
        +
      1. Because the stack only iterate through points once and exactly once, one point either push in or push in then pop out which take constant time k. So the total time is k*n which is O(n).
      2. +
      +
    • +
    • +
        +
      1. Consider a square ABDC, p0p_0 and p1p_1 is A and B. Then we go to C and push it. Then D, however B-C-D makes a right turn so we pop C then push D. Then we form a convex polygon with points A,B,D, but C is not inside the polygon. +1_2_1
      2. +
      +
    • +
    +
  • +
+

Problem 2

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  • +
      +
    1. +
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      +
    • QuickHull(p9 , p3 , {p0 , p1, p2 , p4, p5 , p6, p7 , p8, p10 , p 11, p12 })
    • +
    • QuickHull(p9 , p5 , {p4 , p11, p8 , p10})
    • +
    • QuickHull(p9 , p11 , {p4})
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    • QuickHull(p11 , p5 , {p10})
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    • QuickHull(p5 , p3 , {p6 , p7, p1})
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    • QuickHull(p7 , p3 , {p1})
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  • +
  • +
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    1. +
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    • +
        +
      1. Correct, this is basically the same as Graham scan.
      2. +
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    • +
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        +
      1. Wrong, counter example, c is the smallese angle but c is not a part of convex hull. +2_2_2
      2. +
      +
    • +
    • +
        +
      1. Correct, base on Akl–Toussaint heuristic, for a set of points, the one with the highest y coordinate must be a part of the convex hull.
      2. +
      +
    • +
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        +
      1. Correct, this is a Ellipse with same focuses but different axis. The one with longer axis must contain the one with shorter axis but not vise versa. Thus, the point with highest sum must be a part of convex hull.
      2. +
      +
    • +
    • +
        +
      1. Wrong, same graph as question 2, C is the one with minimal distance sum however it is not a part of convex hull.
      2. +
      +
    • +
    +
  • +
+ + + + + \ No newline at end of file diff --git a/HW2/HW2.md b/HW2/HW2.md new file mode 100644 index 0000000..90cd21f --- /dev/null +++ b/HW2/HW2.md @@ -0,0 +1,72 @@ +# HW2 +## Problem 1 + - 1. + Points: + | index | position | + | ----- | -------- | + | 0 | (3, 1) | + | 1 | (4, 4) | + | 2 | (5, -1) | + | 3 | (2, -1) | + | 4 | (2, 2) | + | 5 | (2, 1) | + | 6 | (4, -1) | + | 7 | (6, 1) | + | 8 | (8, 1) | + | 9 | (8, 5) | + | 10 | (7, 4) | + | 11 | (3, 3) | + + Afer sorting: + | index | position | deleted | + | ----- | -------- | ------- | + | 2 | (5, -1) | False | + | 8 | (8, 1) | False | + | 7 | (6, 1) | True | + | 9 | (8, 5) | False | + | 10 | (7, 4) | False | + | 1 | (4, 4) | False | + | 11 | (3, 3) | False | + | 0 | (3, 1) | True | + | 4 | (2, 2) | False | + | 5 | (2, 1) | False | + | 6 | (4, -1) | True | + | 3 | (2, -1) | False | + + Stack history: + | i | stack | + | --- | ----------- | + | 2 | 8,2 | + | 3 | 9,8,2 | + | 4 | 10,9,8,2 | + | 4 | 9,8,2 | + | 5 | 1,9,8,2 | + | 6 | 11,1,9,8,2 | + | 6 | 1,9,8,2 | + | 7 | 4,1,9,8,2 | + | 8 | 5,4,1,9,8,2 | + | 8 | 4,1,9,8,2 | + | 9 | 3,4,1,9,8,2 | + + image: + ![1_1_1](1_1_1.png) +- 2. + - 1. Because the stack only iterate through points once and exactly once, one point either push in or push in then pop out which take constant time k. So the total time is k*n which is O(n). + - 2. Consider a square ABDC, $p_0$ and $p_1$ is A and B. Then we go to C and push it. Then D, however B-C-D makes a right turn so we pop C then push D. Then we form a convex polygon with points A,B,D, but C is not inside the polygon. + ![1_2_1](1_2_1.png) + +## Problem 2 + - 1. + - QuickHull(p9 , p3 , {p0 , p1, p2 , p4, p5 , p6, p7 , p8, p10 , p 11, p12 }) + - QuickHull(p9 , p5 , {p4 , p11, p8 , p10}) + - QuickHull(p9 , p11 , {p4}) + - QuickHull(p11 , p5 , {p10}) + - QuickHull(p5 , p3 , {p6 , p7, p1}) + - QuickHull(p7 , p3 , {p1}) + - 2. + - 1. Correct, this is basically the same as Graham scan. + - 2. Wrong, counter example, c is the smallese angle but c is not a part of convex hull. + ![2_2_2](2_2_2.png) + - 3. Correct, base on Akl–Toussaint heuristic, for a set of points, the one with the highest y coordinate must be a part of the convex hull. + - 4. Correct, this is a Ellipse with same focuses but different axis. The one with longer axis must contain the one with shorter axis but not vise versa. Thus, the point with highest sum must be a part of convex hull. + - 5. 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