Catmull-Rom curves are an important and frequently used tool in computer graphics and geometric modeling. These curves are visually smooth, interpolate the given control points, and have local support. In this work, they are defined through Lagrange basis polynomials and B-splines, and we also present the Barry–Goldman recursive algorithm for efficient point computation on the curve. The focus is primarily on cubic Catmull–Rom splines, with a detailed investigation of their properties. We also analyze how different parameterizations (uniform, centripetal, chordal) influence the shape of the curves. By converting the spline segments into Bézier form, we demonstrate that with centripetal parametrization, self-intersections and cusps cannot occur within an individual segment of the spline.
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