commit d49ebe2fbc20345b35358f6d402cfb11160f87aa · hauleth.dev/advent-of-code

+42 -40
2021/day19.livemd
···

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       ```elixir

     

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       # File.read!("day19.txt")

     

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       input =

     

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         File.read!("day19.txt")

     

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         File.read!("2021/day19.txt")

     

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         |> String.split("\n\n")

     

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         |> Enum.map(fn scanner ->

     

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           scanner

     
···

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       34

     

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       ```

     

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       To find rotation of point $b_1$ and $b_2$ around shift point $b_0$

     

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       +
       

     

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       $$

     

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       \left\{

     

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       \begin{alignat*}{3}

     

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       b_{x_1} r_x + b_{x_0} = a_{x_1} \\

     

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       b_{x_2} r_x + b_{x_0} = a_{x_2}

     

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       {b_1}_x r_x + {b_0}_x = {a_1}_x \\

     

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       {b_2}_x r_x + {b_0}_x = {a_2}_x

     

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       \end{alignat*}

     

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       \right.

     

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       \left\{

     

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       \begin{alignat*}{3}

     

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       b_{y_1} r_y + b_{y_0} = a_{y_1} \\

     

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       b_{y_2} r_y + b_{y_0} = a_{y_2} \\

     

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       {b_1}_y r_y + {b_0}_y = {a_1}_y \\

     

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       {b_2}_y r_y + {b_0}_y = {a_2}_y \\

     

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       \end{alignat*}

     

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       \right.

     

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       \left\{

     

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       \begin{alignat*}{3}

     

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       b_{z_1} r_z + b_{z_0} = a_{z_1} \\

     

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       b_{z_2} r_z + b_{z_0} = a_{z_2} \\

     

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       {b_1}_z r_z + {b_0}_z = {a_1}_z \\

     

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       {b_2}_z r_z + {b_0}_z = {a_2}_z \\

     

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       \end{alignat*}

     

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       \right.

     

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       $$

     
···

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       This mean that we need to solve:

     

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       $$

     

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       B_x x + b_0 = a_x \\

     

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       B_y y + b_0 = a_y \\

     

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       B_z z + b_0 = a_z

     

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       B_x r_x + b_0 = a_x \\

     

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       B_y r_y + b_0 = a_y \\

     

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       B_z r_z + b_0 = a_z

     

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       $$

     

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       Where:

     
···

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       {b_1}_d & 1 \\

     

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       {b_2}_d & 1 \\

     

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       \end{bmatrix}

     

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       d = \begin{bmatrix}

     

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       r_d \\

     

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       b_{d_0}

     

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       b_0 = \begin{bmatrix}

     

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       {b_0}_d \\

     

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       {b_0}_d

     

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       \end{bmatrix}

     

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       a_d = \begin{bmatrix}

     

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       {a_1}_d \\

     
···

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       $$

     

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       By applying [Cramer's Rule](https://en.wikipedia.org/wiki/Cramer%27s_rule) we can solve these

     

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       linear equations with:

     

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       linear equations with (we are looking for $b_0$ and $r$):

     

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       $$

     

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       e_d = \frac{

     

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       r_d = \frac{

     

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       \begin{vmatrix}

     

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       {b_1}_d & 1 \\

     

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       {b_2}_d & 1

     

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       {a_1}_d & 1 \\

     

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       {a_2}_d & 1

     

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       \end{vmatrix}

     

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       }{

     

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       \begin{vmatrix}

     

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       {a_1}_d & 1 \\

     

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       {a_2}_d & 1

     

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       {b_1}_d & 1 \\

     

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       {b_2}_d & 1

     

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       \end{vmatrix}

     

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       } = \frac{{b_1}_d - {b_2}_d}{{a_1}_d - {a_2}_d} \\

     

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       } = \frac{{a_1}_d - {a_2}_d}{{b_1}_d - {b_2}_d} \\

     

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       b_{d_0} = \frac{

     

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         \begin{vmatrix}

     

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       {a_1}_d & {b_1}_d \\

     

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       {a_2}_d & {b_2}_d

     

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       {b_1}_d & {a_1}_d \\

     

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       {b_2}_d & {a_2}_d

     

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       \end{vmatrix}

     

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       }{

     

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         \begin{vmatrix}

     

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       {a_1}_d & 1 \\

     

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       {a_2}_d & 1

     

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       {b_1}_d & 1 \\

     

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       {b_2}_d & 1

     

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       \end{vmatrix}

     

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       } = \frac{{a_1}_d {b_2}_d - {b_1}_d {a_2}_d}{{a_1}_d - {a_2}_d}

     

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       } = \frac{{b_1}_d {a_2}_d - {a_1}_d {b_2}_d}{{b_1}_d - {b_2}_d}

     

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       $$

     

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       ```elixir

     
···

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           b_2 = reaxe(select_common(p0, p2), axes)

     

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           transform =

     

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             for i <- ~w[x y z]a, into: %{} do

     

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               a_1 = a_1[i]

     

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               a_2 = a_2[i]

     

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               b_1 = b_1[i]

     

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               b_2 = b_2[i]

     

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             for d <- ~w[x y z]a, into: %{} do

     

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               a_1 = a_1[d]

     

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               a_2 = a_2[d]

     

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               b_1 = b_1[d]

     

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               b_2 = b_2[d]

     

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               det_b = b_1 - b_2

     

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               r = div(a_1 - a_2, det_b)

     

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               b_0 = div(b_1 * a_2 - a_1 * b_2, det_b)

     

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               {i, {r, b_0}}

     

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               {d, {r, b_0}}

     

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             end

     

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           new_points =

     
···

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       <!-- livebook:{"output":true} -->

     

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       ```

     

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       {:module, Day19, <<70, 79, 82, 49, 0, 0, 26, ...>>, {:pairs, 1}}

     

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       {:module, Day19, <<70, 79, 82, 49, 0, 0, 29, ...>>, {:pairs, 1}}

     

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       ```

     

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       ```elixir

     
···

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       <!-- livebook:{"output":true} -->

     

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       ```

     

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       0.002822

     

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       0.005503

     

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       1: {5, 25}

     

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       0.017052

     

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       0.029354

     

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       2: {5, 20}

     

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       0.0355

     

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       0.067284

     

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       3: {7, 13}

     

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       0.134655

     

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       0.195126

     

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       4: {6, 7}

     

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       0.229952

     

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       0.292686

     

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       5: {5, 2}

     

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       0.288351

     

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       0.374403

     

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       6: {1, 1}

     

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       0.076665

     

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       0.074645

     

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       7: {1, 0}

     

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       0.071942

     

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       0.090199

     

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       ```

     

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       <!-- livebook:{"output":true} -->