Centripetal Catmull–Rom spline

Barry and Goldman's pyramidal formulation

Knot parameterization for the Catmull–Rom algorithm

Let ${\displaystyle \mathbf {P} _{i}=[x_{i}\quad y_{i}]^{T}}$ denote a point. For a curve segment ${\displaystyle \mathbf {C} }$ defined by points ${\displaystyle \mathbf {P} _{0},\mathbf {P} _{1},\mathbf {P} _{2},\mathbf {P} _{3}}$ and knot sequence ${\displaystyle t_{0},t_{1},t_{2},t_{3}}$, the centripetal Catmull–Rom spline can be produced by:

${\displaystyle \mathbf {C} ={\frac {t_{2}-t}{t_{2}-t_{1}}}\mathbf {B} _{1}+{\frac {t-t_{1}}{t_{2}-t_{1}}}\mathbf {B} _{2}}$

where

${\displaystyle \mathbf {B} _{1}={\frac {t_{2}-t}{t_{2}-t_{0}}}\mathbf {A} _{1}+{\frac {t-t_{0}}{t_{2}-t_{0}}}\mathbf {A} _{2}}$

${\displaystyle \mathbf {B} _{2}={\frac {t_{3}-t}{t_{3}-t_{1}}}\mathbf {A} _{2}+{\frac {t-t_{1}}{t_{3}-t_{1}}}\mathbf {A} _{3}}$

${\displaystyle \mathbf {A} _{1}={\frac {t_{1}-t}{t_{1}-t_{0}}}\mathbf {P} _{0}+{\frac {t-t_{0}}{t_{1}-t_{0}}}\mathbf {P} _{1}}$

${\displaystyle \mathbf {A} _{2}={\frac {t_{2}-t}{t_{2}-t_{1}}}\mathbf {P} _{1}+{\frac {t-t_{1}}{t_{2}-t_{1}}}\mathbf {P} _{2}}$

${\displaystyle \mathbf {A} _{3}={\frac {t_{3}-t}{t_{3}-t_{2}}}\mathbf {P} _{2}+{\frac {t-t_{2}}{t_{3}-t_{2}}}\mathbf {P} _{3}}$

and

${\displaystyle t_{i+1}=\left[{\sqrt {(x_{i+1}-x_{i})^{2}+(y_{i+1}-y_{i})^{2}}}\right]^{\alpha }+t_{i}}$

in which ${\displaystyle \alpha }$ ranges from 0 to 1 for knot parameterization, and ${\displaystyle i=0,1,2,3}$ with ${\displaystyle t_{0}=0}$. For centripetal Catmull–Rom spline, the value of ${\displaystyle \alpha }$ is ${\displaystyle 0.5}$. When ${\displaystyle \alpha =0}$, the resulting curve is the standard uniform Catmull–Rom spline; when ${\displaystyle \alpha =1}$, the result is a chordal Catmull–Rom spline.

Gif animation for uniform, centripetal and chordal parameterization of Catmull–Rom spline depending on the α value

Plugging ${\displaystyle t=t_{1}}$ into the spline equations ${\displaystyle \mathbf {A} _{1},\mathbf {A} _{2},\mathbf {A} _{3},\mathbf {B} _{1},\mathbf {B} _{2},}$ and ${\displaystyle \mathbf {C} }$ shows that the value of the spline curve at ${\displaystyle t_{1}}$ is ${\displaystyle \mathbf {C} =\mathbf {P} _{1}}$. Similarly, substituting ${\displaystyle t=t_{2}}$ into the spline equations shows that ${\displaystyle \mathbf {C} =\mathbf {P} _{2}}$ at ${\displaystyle t_{2}}$. This is true independent of the value of ${\displaystyle \alpha }$ since the equation for ${\displaystyle t_{i+1}}$ is not needed to calculate the value of ${\displaystyle \mathbf {C} }$ at points ${\displaystyle t_{1}}$ and ${\displaystyle t_{2}}$.

3D centripetal Catmull-Rom spline segment.

The extension to 3D points is simply achieved by considering ${\displaystyle \mathbf {P} _{i}=[x_{i}\quad y_{i}\quad z_{i}]^{T}}$a generic 3D point ${\displaystyle \mathbf {P} _{i}}$ and

${\displaystyle t_{i+1}=\left[{\sqrt {(x_{i+1}-x_{i})^{2}+(y_{i+1}-y_{i})^{2}+(z_{i+1}-z_{i})^{2}}}\right]^{\alpha }+t_{i}}$

Centripetal Catmull–Rom spline has several desirable mathematical properties compared to the original and the other types of Catmull-Rom formulation.[3] First, it will not form loop or self-intersection within a curve segment. Second, cusp will never occur within a curve segment. Third, it follows the control points more tightly.[4]

In this figure, there is a self-intersection/loop on the uniform Catmull-Rom spline (green), whereas for chordal Catmull-Rom spline (red), the curve does not follow tightly through the control points.

In computer vision, centripetal Catmull-Rom spline has been used to formulate an active model for segmentation. The method is termed active spline model.[5] The model is devised on the basis of active shape model, but uses centripetal Catmull-Rom spline to join two successive points (active shape model uses simple straight line), so that the total number of points necessary to depict a shape is less. The use of centripetal Catmull-Rom spline makes the training of a shape model much simpler, and it enables a better way to edit a contour after segmentation.

The following is an implementation of the Catmull–Rom spline in Python that produces the plot shown beneath.

import numpy
import matplotlib.pyplot as plt


QUADRUPLE_SIZE: int = 4


def num_segments(point_chain: tuple) -> int:
    # There is 1 segment per 4 points, so we must subtract 3 from the number of points  
    return len(point_chain) - (QUADRUPLE_SIZE - 1)


def flatten(list_of_lists) -> list:
    # E.g. mapping [[1, 2], [3], [4, 5]] to  [1, 2, 3, 4, 5] 
    return [elem for lst in list_of_lists for elem in lst]


def catmull_rom_spline(
    P0: tuple,
    P1: tuple,
    P2: tuple,
    P3: tuple,
    num_points: int,
    alpha: float = 0.5,
):
    """
    Compute the points in the spline segment
    :param P0, P1, P2, and P3: The (x,y) point pairs that define the Catmull-Rom spline
    :param num_points: The number of points to include in the resulting curve segment
    :param alpha: 0.5 for the centripetal spline, 0.0 for the uniform spline, 1.0 for the chordal spline.
    :return: The points
    """

    # Calculate t0 to t4. Then only calculate points between P1 and P2.
    # Reshape linspace so that we can multiply by the points P0 to P3
    # and get a point for each value of t.
    def tj(ti: float, pi: tuple, pj: tuple) -> float:
        xi, yi = pi
        xj, yj = pj
        dx, dy = xj - xi, yj - yi
        l = (dx ** 2 + dy ** 2) ** 0.5
        return ti + l ** alpha

    t0: float = 0.0
    t1: float = tj(t0, P0, P1)
    t2: float = tj(t1, P1, P2)
    t3: float = tj(t2, P2, P3)
    t = numpy.linspace(t1, t2, num_points).reshape(num_points, 1)

    A1 = (t1 - t) / (t1 - t0) * P0 + (t - t0) / (t1 - t0) * P1
    A2 = (t2 - t) / (t2 - t1) * P1 + (t - t1) / (t2 - t1) * P2
    A3 = (t3 - t) / (t3 - t2) * P2 + (t - t2) / (t3 - t2) * P3
    B1 = (t2 - t) / (t2 - t0) * A1 + (t - t0) / (t2 - t0) * A2
    B2 = (t3 - t) / (t3 - t1) * A2 + (t - t1) / (t3 - t1) * A3
    points = (t2 - t) / (t2 - t1) * B1 + (t - t1) / (t2 - t1) * B2
    return points


def catmull_rom_chain(points: tuple, num_points: int) -> list:
    """
    Calculate Catmull-Rom for a sequence of initial points and return the combined curve.
    :param points: Base points from which the quadruples for the algorithm are taken
    :param num_points: The number of points to include in each curve segment
    :return: The chain of all points (points of all segments)
    """
    point_quadruples = (  # Prepare function inputs
        (points[idx_segment_start + d] for d in range(QUADRUPLE_SIZE))
        for idx_segment_start in range(num_segments(points))
    )
    all_splines = (catmull_rom_spline(*pq, num_points) for pq in point_quadruples)
    return flatten(all_splines)


if __name__ == "__main__":
    POINTS: tuple = ((0, 1.5), (2, 2), (3, 1), (4, 0.5), (5, 1), (6, 2), (7, 3))  # Red points
    NUM_POINTS: int = 100  # Density of blue chain points between two red points

    chain_points: list = catmull_rom_chain(POINTS, NUM_POINTS)
    assert len(chain_points) == num_segments(POINTS) * NUM_POINTS  # 400 blue points for this example

    plt.plot(*zip(*chain_points), c="blue")
    plt.plot(*zip(*POINTS), c="red", linestyle="none", marker="o")
    plt.show()

Plot obtained by the Python example code given above

• Cubic Hermite splines

• Catmull-Rom curve with no cusps and no self-intersections – implementation in Java
• Catmull-Rom curve with no cusps and no self-intersections – simplified implementation in C++
• Catmull-Rom splines – interactive generation via Python, in a Jupyter notebook
• Smooth Paths Using Catmull-Rom Splines – another versatile implementation in C++ including centripetal CR splines