Choosing nodes in parametric curve interpolation
Computer-Aided Design
Knot selection for parametric spline interpolation
Mathematical methods in computer aided geometric design
A survey of applications of an affine invariant norm
Mathematical methods in computer aided geometric design
Detecting cusps and inflection points in curves
Computer Aided Geometric Design
A recursive evaluation algorithm for a class of Catmull-Rom splines
SIGGRAPH '88 Proceedings of the 15th annual conference on Computer graphics and interactive techniques
On the deviation of a parametric cubic spline interpolant from its data polygon
Computer Aided Geometric Design
Four-point curve subdivision based on iterated chordal and centripetal parameterizations
Computer Aided Geometric Design
Proceedings of the 4th ACM Multimedia Systems Conference
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The behavior of Catmull-Rom curves heavily depends on the choice of parameter values at the control points. We analyze a class of parameterizations ranging from uniform to chordal parameterization and show that, within this class, curves with centripetal parameterization contain properties that no other curves in this family possess. Researchers have previously indicated that centripetal parameterization produces visually favorable curves compared to uniform and chordal parameterizations. However, the mathematical reasons behind this behavior have been ambiguous. In this paper we prove that, for cubic Catmull-Rom curves, centripetal parameterization is the only parameterization in this family that guarantees that the curves do not form cusps or self-intersections within curve segments. Furthermore, we provide a formulation that bounds the distance of the curve to the control polygon and explain how globally intersection-free Catmull-Rom curves can be generated using these properties. Finally, we discuss two example applications of Catmull-Rom curves and show how the choice of parameterization makes a significant difference in each of these applications.