def inflection_points(points, axis, span):
'''
Find the list of vertices that preceed inflection points in a curve. The curve is differentiated
with respect to the coordinate system defined by axis and span.
axis: A vector representing the vertical axis of the coordinate system.
span: A vector representing the the horiztonal axis of the coordinate system.
returns: a list of points in space corresponding to the vertices that
immediately preceed inflection points in the curve
'''
coords_on_span = points.dot(span)
dx = np.gradient(coords_on_span)
coords_on_axis = points.dot(axis)
# Take the second order finite difference of the curve with respect to the
# defined coordinate system
finite_difference_2 = np.gradient(np.gradient(coords_on_axis, dx), dx)
# Compare the product of all neighboring pairs of points in the second derivative
# If a pair of points has a negative product, then the second derivative changes sign
# at one of those points, signalling an inflection point
is_inflection_point = [finite_difference_2[i] * finite_difference_2[i + 1] <= 0 for i in range(len(finite_difference_2) - 1)]
inflection_point_indices = [i for i, b in enumerate(is_inflection_point) if b]
if len(inflection_point_indices) == 0: # pylint: disable=len-as-condition
return []
return points[inflection_point_indices]
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