add Hilbert Curve

fractals/hilbert_curve.py

@@ -0,0 +1,103 @@
+"""
+Hilbert Curve
+
+The Hilbert curve is a space-filling curve that recursively fills a square
+with an intricate pattern. It is used in computer science, graphics, and various fields.
+
+Reference:
+https://en.wikipedia.org/wiki/Hilbert_curve
+
+Requirements:
+    - matplotlib
+    - numpy
+"""
+
+import doctest
+
+import matplotlib.pyplot as plt
+
+
+def hilbert_curve(
+    order: int,
+    x0: float = 0,
+    y0: float = 0,
+    xi: float = 1,
+    xj: float = 0,
+    yi: float = 0,
+    yj: float = 1,
+) -> list[tuple[float, float]]:
+    """
+    Generates the points for the Hilbert curve of the specified order.
+
+    The Hilbert curve is built using recursive rules. This function returns
+    a list of (x, y) points.
+
+    Parameters:
+    - order: the recursion depth or order of the curve
+    - x0, y0: the starting coordinates
+    - xi, xj: the transformation matrix for x coordinates
+    - yi, yj: the transformation matrix for y coordinates
+
+    Returns:
+    - A list of (x, y) coordinates representing the Hilbert curve.
+
+    >>> len(hilbert_curve(1))
+    4
+    >>> len(hilbert_curve(2))
+    16
+    >>> len(hilbert_curve(3))
+    64
+    """
+    if order == 0:
+        return [(x0 + (xi + yi) / 2, y0 + (xj + yj) / 2)]
+
+    points = []
+    points += hilbert_curve(order - 1, x0, y0, yi / 2, yj / 2, xi / 2, xj / 2)
+    points += hilbert_curve(
+        order - 1, x0 + xi / 2, y0 + xj / 2, xi / 2, xj / 2, yi / 2, yj / 2
+    )
+    points += hilbert_curve(
+        order - 1,
+        x0 + xi / 2 + yi / 2,
+        y0 + xj / 2 + yj / 2,
+        xi / 2,
+        xj / 2,
+        yi / 2,
+        yj / 2,
+    )
+    points += hilbert_curve(
+        order - 1,
+        x0 + xi / 2 + yi,
+        y0 + xj / 2 + yj,
+        -yi / 2,
+        -yj / 2,
+        -xi / 2,
+        -xj / 2,
+    )
+
+    return points
+
+
+def plot_hilbert_curve(points: list[tuple[float, float]]) -> None:
+    """
+    Plots the Hilbert curve using matplotlib.
+
+    Parameters:
+    - points: A list of (x, y) coordinates representing the Hilbert curve.
+
+    Returns:
+    - None
+    """
+    x, y = zip(*points)
+    plt.plot(x, y)
+    plt.title("Hilbert Curve")
+    plt.gca().set_aspect("equal", adjustable="box")
+    plt.show()
+
+
+if __name__ == "__main__":
+    doctest.testmod()
+
+    order = 5  # Order of the Hilbert curve
+    curve_points = hilbert_curve(order)
+    plot_hilbert_curve(curve_points)