diff --git a/geometry/ramer_douglas_peucker.py b/geometry/ramer_douglas_peucker.py
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+++ b/geometry/ramer_douglas_peucker.py
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+"""
+Ramer-Douglas-Peucker polyline simplification algorithm.
+
+Given a sequence of 2-D points and a tolerance epsilon, the algorithm
+reduces the number of points while preserving the overall shape of the curve.
+
+Time complexity:  O(n log n) average, O(n²) worst case
+Space complexity: O(n)
+
+References:
+    https://en.wikipedia.org/wiki/Ramer%E2%80%93Douglas%E2%80%93Peucker_algorithm
+"""
+
+from __future__ import annotations
+
+import math
+
+
+def _euclidean_distance(
+    point_a: tuple[float, float],
+    point_b: tuple[float, float],
+) -> float:
+    """Return the Euclidean distance between two 2-D points.
+
+    >>> _euclidean_distance((0.0, 0.0), (3.0, 4.0))
+    5.0
+    >>> _euclidean_distance((1.0, 1.0), (1.0, 1.0))
+    0.0
+    """
+    return math.hypot(point_b[0] - point_a[0], point_b[1] - point_a[1])
+
+
+def _perpendicular_distance(
+    point: tuple[float, float],
+    line_start: tuple[float, float],
+    line_end: tuple[float, float],
+) -> float:
+    """Return the distance from *point* to the line **segment** between
+    *line_start* and *line_end*.
+
+    When the perpendicular projection of *point* onto the infinite line falls
+    within the segment, this equals the perpendicular distance to that line.
+    When the projection falls outside the segment, the distance to the nearest
+    endpoint is returned instead (projection clamped to [0, 1]).
+
+    This is the correct distance measure for the Ramer-Douglas-Peucker
+    algorithm: using the infinite-line distance can incorrectly discard points
+    whose projection lies beyond a segment endpoint.
+
+    >>> _perpendicular_distance((4.0, 0.0), (0.0, 0.0), (0.0, 3.0))
+    4.0
+    >>> # order of line_start and line_end does not affect the result
+    >>> _perpendicular_distance((4.0, 0.0), (0.0, 3.0), (0.0, 0.0))
+    4.0
+    >>> _perpendicular_distance((4.0, 1.0), (0.0, 1.0), (0.0, 4.0))
+    4.0
+    >>> _perpendicular_distance((2.0, 1.0), (-2.0, 1.0), (-2.0, 4.0))
+    4.0
+    >>> # projection falls outside the segment; distance to nearest endpoint
+    >>> round(_perpendicular_distance((0.0, 2.0), (1.0, 0.0), (3.0, 0.0)), 6)
+    2.236068
+    """
+    px, py = point
+    ax, ay = line_start
+    bx, by = line_end
+    dx, dy = bx - ax, by - ay
+    seg_len_sq = dx * dx + dy * dy
+    if seg_len_sq == 0.0:
+        # line_start and line_end coincide; fall back to point-to-point distance
+        return _euclidean_distance(point, line_start)
+    # Project point onto the segment line, then clamp t to [0, 1] so the
+    # nearest point is always on the segment rather than the infinite line.
+    t = max(0.0, min(1.0, ((px - ax) * dx + (py - ay) * dy) / seg_len_sq))
+    nearest_x = ax + t * dx
+    nearest_y = ay + t * dy
+    return math.hypot(px - nearest_x, py - nearest_y)
+
+
+def ramer_douglas_peucker(
+    pts: list[tuple[float, float]],
+    epsilon: float,
+) -> list[tuple[float, float]]:
+    """Simplify a polyline using the Ramer-Douglas-Peucker algorithm.
+
+    Given a sequence of 2-D points and a maximum allowable deviation
+    *epsilon* (>= 0), returns a simplified list of points such that no
+    discarded point is farther than *epsilon* from the simplified polyline.
+
+    Parameters
+    ----------
+    pts:
+        Ordered sequence of ``(x, y)`` points describing the polyline.
+    epsilon:
+        Maximum allowable distance of any discarded point from the
+        simplified polyline.  Must be non-negative.
+
+    Returns
+    -------
+    list[tuple[float, float]]
+        Simplified list of ``(x, y)`` points.  The first and last points of
+        *pts* are always preserved.
+
+    Raises
+    ------
+    ValueError
+        If *epsilon* is negative.
+
+    References
+    ----------
+    https://en.wikipedia.org/wiki/Ramer%E2%80%93Douglas%E2%80%93Peucker_algorithm
+
+    Examples
+    --------
+    >>> ramer_douglas_peucker([], epsilon=1.0)
+    []
+    >>> ramer_douglas_peucker([(0.0, 0.0)], epsilon=1.0)
+    [(0.0, 0.0)]
+    >>> ramer_douglas_peucker([(0.0, 0.0), (1.0, 0.0)], epsilon=1.0)
+    [(0.0, 0.0), (1.0, 0.0)]
+    >>> # middle point is within epsilon - it is discarded
+    >>> ramer_douglas_peucker([(0.0, 0.0), (1.0, 0.1), (2.0, 0.0)], epsilon=0.5)
+    [(0.0, 0.0), (2.0, 0.0)]
+    >>> # middle point exceeds epsilon - it is kept
+    >>> ramer_douglas_peucker([(0.0, 0.0), (1.0, 1.0), (2.0, 0.0)], epsilon=0.5)
+    [(0.0, 0.0), (1.0, 1.0), (2.0, 0.0)]
+    >>> ramer_douglas_peucker([(0.0, 0.0), (1.0, 0.5), (2.0, 0.0)], epsilon=-1.0)
+    Traceback (most recent call last):
+        ...
+    ValueError: epsilon must be non-negative, got -1.0
+    """
+    if epsilon < 0:
+        msg = f"epsilon must be non-negative, got {epsilon!r}"
+        raise ValueError(msg)
+
+    if len(pts) < 3:
+        return list(pts)
+
+    # ---------------------------------------------------------------------------
+    # Iterative, stack-based implementation.
+    #
+    # The naive recursive approach copies sublists at every level via slicing
+    # (pts[:max_index+1] / pts[max_index:]), which is O(n) per call and makes
+    # the overall algorithm O(n²) in memory even for well-balanced splits.  An
+    # explicit stack operating on index ranges avoids all copying and also
+    # eliminates the risk of hitting Python's recursion limit for long polylines.
+    # ---------------------------------------------------------------------------
+    n = len(pts)
+
+    # keep[i] is True when pts[i] must appear in the output.
+    keep: list[bool] = [False] * n
+    keep[0] = True
+    keep[-1] = True
+
+    # Stack of (start_index, end_index) pairs still to be examined.
+    stack: list[tuple[int, int]] = [(0, n - 1)]
+
+    while stack:
+        start, end = stack.pop()
+        if end - start < 2:
+            # Only one interior candidate at most; nothing to split further.
+            continue
+
+        # Find the interior point with the greatest distance to the segment.
+        max_dist = 0.0
+        max_index = start
+        for i in range(start + 1, end):
+            dist = _perpendicular_distance(pts[i], pts[start], pts[end])
+            if dist > max_dist:
+                max_dist = dist
+                max_index = i
+
+        if max_dist > epsilon:
+            keep[max_index] = True
+            stack.append((start, max_index))
+            stack.append((max_index, end))
+        # else: all interior points are within epsilon; discard them all.
+
+    return [pts[i] for i in range(n) if keep[i]]
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()