43 lines
1.2 KiB
GDScript3
43 lines
1.2 KiB
GDScript3
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class_name ComplexMath
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func _init() -> void:
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push_error("Not instantiable")
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free()
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static func mul(z: Vector2, w: Vector2) -> Vector2:
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# (a + ib)(c + id) = ac + ibc + iad - bd
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return Vector2(
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z.x*w.x - z.y*w.y,
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z.y*w.x + z.x*w.y
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)
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static func div(z: Vector2, w: Vector2) -> Vector2:
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# (a + ib)(a - ib) = a^2 + aib - aib - (ib)^2 = a^2 + b^2
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# ergo:
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# (a + ib)/(c + id) = a/(c + id) + ib/(c + id)
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# = a*(c - id)/(c^2 + d^2) + ib*(c - id)/(c^2 + d^2)
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# = (ac - iad + ibc + bd)/(c^2 + d^2)
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return Vector2(
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z.x*w.x + z.y*w.y,
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z.y*w.x - z.x*w.y
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)/(w.x*w.x + w.y*w.y)
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static func abs(z: Vector2) -> float:
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return z.length()
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static func arg(z: Vector2) -> float:
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return atan2(z.y, z.x)
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static func exp(z: Vector2) -> Vector2:
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# e^(i*theta) = cos(theta) + i*sin(theta)
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# ergo e^(a + ib) = (e^a)(e^(ib)) = (e^a)(cos(b) + i*sin(b))
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return exp(z.x)*Vector2(cos(z.y), sin(z.y))
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static func pow(z: Vector2, w: Vector2) -> Vector2:
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# Credit to StackExchange user Shobhit for this formula
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# <https://math.stackexchange.com/questions/476968\
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# /complex-power-of-a-complex-number>.
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var r: float = z.length()
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var theta: float = ComplexMath.arg(z)
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return ComplexMath.exp(ComplexMath.mul(Vector2(log(r), theta), w))
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