this repo has no description
fix: math
+42
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2021/day19.livemd
+42
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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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```
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To find rotation of point $b_1$ and $b_2$ around shift point $b_0$
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$$
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\left\{
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\begin{alignat*}{3}
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b_{x_1} r_x + b_{x_0} = a_{x_1} \\
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b_{x_2} r_x + b_{x_0} = a_{x_2}
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{b_1}_x r_x + {b_0}_x = {a_1}_x \\
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{b_2}_x r_x + {b_0}_x = {a_2}_x
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\end{alignat*}
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\right.
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\left\{
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\begin{alignat*}{3}
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b_{y_1} r_y + b_{y_0} = a_{y_1} \\
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b_{y_2} r_y + b_{y_0} = a_{y_2} \\
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{b_1}_y r_y + {b_0}_y = {a_1}_y \\
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{b_2}_y r_y + {b_0}_y = {a_2}_y \\
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\end{alignat*}
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\right.
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\left\{
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\begin{alignat*}{3}
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b_{z_1} r_z + b_{z_0} = a_{z_1} \\
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b_{z_2} r_z + b_{z_0} = a_{z_2} \\
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{b_1}_z r_z + {b_0}_z = {a_1}_z \\
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{b_2}_z r_z + {b_0}_z = {a_2}_z \\
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\end{alignat*}
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\right.
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$$
···
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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} -->