Centripetal Catmull–Rom spline
Centripetal Catmull–Rom spline
Definition
![{\mathbf {P}}_{i}=[x_{i}\quad y_{i}]^{T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94b787f14d85118f4669426adc26edc700fc97e7)
where
and
![t_{{i+1}}=\left[{\sqrt {(x_{{i+1}}-x_{i})^{2}+(y_{{i+1}}-y_{i})^{2}}}\right]^{{\alpha }}+t_{i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3fbc45aa0c46308c07c445d0e1359cafca90a17)
Advantages
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.
Other uses
In computer vision, centripetal Catmull-Rom spline has been used to formulate an active model for segmentation. The method is termed active spline model.[4] 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.
Code example in Python
The following is an implementation of the Catmull–Rom spline in Python.
Code example in Unity C#
For an implementation in 3D space, after converting Vector2 to Vector3 points, the float could be extended a in function GetT to this : Mathf.Pow((p1.x-p0.x), 2.0f) + Mathf.Pow((p1.y-p0.y), 2.0f) + Mathf.Pow((p1.z-p0.z), 2.0f).
See also
Cubic Hermite splines