Create hilbert_curve.py

fractals/hilbert_curve.py

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+"""
+Author Atharva Date | [email protected] | git/Atharva9621
+
+The Hilbert Curve (also known as the Hilbert Space Filling Curve) is a continuous
+fractal space-filling curve and is a variant of the space-filling Peano curves.
+
+Because it is space-filling, its Hausdorff dimension is 2. Precisely, its image
+is the unit square, whose dimension is 2 in any definition of dimension. Its
+graph is a compact set homeomorphic to the closed unit interval, with Hausdorff
+dimension 1.
+
+Credits:
+    description adapted from
+    https://en.wikipedia.org/wiki/Hilbert_curve
+
+    also see
+    https://youtu.be/3s7h2MHQtxc?si=_qIusAJFHYfXIOKn
+        (3b1b - Hilbert's Curve: Is infinite math useful?)
+    https://dl.acm.org/doi/pdf/10.1145/290200.290219
+
+Requirements (pip):
+    - matplotlib
+"""
+
+import matplotlib.pyplot as plt
+
+
+def rotate_pnts(
+    pnts: list[tuple[float, float]], angle: int
+) -> list[tuple[float, float]]:
+    """
+    Rotates a list of points by a given angle (in multiples of 90 degrees).
+
+    Since the rotation is limited to multiples of 90 degrees (90, 180, 270, 360),
+    this function simply reorders the list of points accordingly. The rotation in
+    each quadrant is achieved  by adjusting the starting index of the list
+    and wrapping around.
+
+    Parameters:
+    -----------
+    pnts : List[Tuple[float, float]]
+        A list of tuples, where each tuple represents a point (x, y).
+    angle : int
+        The angle of rotation, should be a multiple of 90 degrees
+        (e.g., 90, 180, 270, 360).
+
+    Returns:
+    --------
+    List[Tuple[float, float]]
+        A reordered list of points, rotated by the specified angle.
+
+    Example:
+    --------
+    >>> rotate_pnts([(1, 1), (0, 1), (0, 0), (1, 0)], 90)
+    [(0, 1), (0, 0), (1, 0), (1, 1)]
+    """
+    start_index = angle // 90 % 4
+    return pnts[start_index:] + pnts[:start_index]
+
+
+def hilbert_curve(
+    center: tuple[float, float], level: int, side: float = 1, angle: int = 90
+) -> list[tuple[float, float]]:
+    """
+    Params:
+    ------
+        center: Tuple[float, float]- The (x, y) center coordinate of the subsection.
+        level: int- The recursion depth or subdivision level of the Hilbert curve.
+        side : float, optional
+                The length of the side of the square region in which the curve is drawn.
+        angle : int, optional
+                The initial rotation angle of the curve in degrees.
+                It should be a multiple of 90 degrees. (default=90)
+
+    Returns:
+    ------
+        pts: List[Tuple[float, float]] -
+            A list of points (x, y) representing the Hilbert curve for the given level.
+
+    Example:
+    --------
+    >>> hilbert_curve((0, 0), 1, angle=0)
+    [(0.25, 0.25), (-0.25, 0.25), (-0.25, -0.25), (0.25, -0.25)]
+    """
+    x, y = center
+    angle = angle % 360
+    pnts = [
+        (x + side / 4, y + side / 4),
+        (x - side / 4, y + side / 4),
+        (x - side / 4, y - side / 4),
+        (x + side / 4, y - side / 4),
+    ]
+    pnts = rotate_pnts(pnts, angle)
+
+    if level == 1:
+        return pnts
+    else:
+        return (
+            hilbert_curve(pnts[0], level - 1, side / 2, angle=angle + 90)[::-1]
+            + hilbert_curve(pnts[1], level - 1, side / 2, angle=angle)
+            + hilbert_curve(pnts[2], level - 1, side / 2, angle=angle)
+            + hilbert_curve(pnts[3], level - 1, side / 2, angle=angle - 90)[::-1]
+        )
+
+
+def plot_hilbert_curve(points: list[tuple[float, float]]) -> None:
+    """
+    Plots the hilbert curve using mtplotlib
+
+    Example
+    --------
+    >>> plot_hilbert_curve([(-0.25, 0.25), (-0.25, -0.25), (0.25, -0.25), (0.25, 0.25)])
+    """
+    x_coords = [p[0] for p in points]
+    y_coords = [p[1] for p in points]
+
+    plt.plot(x_coords, y_coords, marker="o", linestyle="-")
+    plt.gca().set_aspect("equal", adjustable="box")  # Make the plot square
+    plt.title("Hilbert Curve")
+    plt.show()
+
+
+if __name__ == "__main__":
+    import doctest
+
+    # Run doctests
+    doctest.testmod()
+
+    # Plotting Hilbert Curve
+    plot_hilbert_curve(hilbert_curve((0, 0), 4))