A survey of curve and surface methods in CAGD
Computer Aided Geometric Design
Curvature continuous curves and surfaces
Computer Aided Geometric Design
Geometric continuity with interpolating Be´zier curves
Proceedings of Graphics Interface '85 on Computer-generated images: the state of the art
An intuitive approach to geometric continuity for parametric curves and surfaces
Proceedings of Graphics Interface '85 on Computer-generated images: the state of the art
An introduction to splines for use in computer graphics & geometric modeling
An introduction to splines for use in computer graphics & geometric modeling
Computer graphics and geometric modeling using Beta-splines
Computer graphics and geometric modeling using Beta-splines
Local Control of Bias and Tension in Beta-splines
ACM Transactions on Graphics (TOG)
Scan line methods for displaying parametrically defined surfaces
Communications of the ACM
Scalar- and planar-valued curve fitting using splines under tension
Communications of the ACM
Geometric Continuity: A Parameterization Independent Measure of
Geometric Continuity: A Parameterization Independent Measure of
The Beta2-spline: a Special Case of the Beta-spline Curve and
The Beta2-spline: a Special Case of the Beta-spline Curve and
Geometric Continuity of Parametric Curves
Geometric Continuity of Parametric Curves
The beta-spline: a local representation based on shape parameters and fundamental geometric measures
The beta-spline: a local representation based on shape parameters and fundamental geometric measures
Incremental computation of planar maps
SIGGRAPH '89 Proceedings of the 16th annual conference on Computer graphics and interactive techniques
Polar forms for geometrically continuous spline curves of arbitrary degree
ACM Transactions on Graphics (TOG)
Mathematical Methods for Curves and Surfaces
IEEE Computer Graphics and Applications
On the parameterization of Catmull-Rom curves
2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling
ICIRA'11 Proceedings of the 4th international conference on Intelligent Robotics and Applications - Volume Part II
Collision-free and smooth trajectory computation in cluttered environments
International Journal of Robotics Research
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Catmull-Rom splines have local control, can be either approximating or interpolating, and are efficiently computable. Experience with Beta-splines has shown that it is useful to endow a spline with shape parameters, used to modify the shape of the curve or surface independently of the defining control vertices. Thus it is desirable to construct a subclass of the Catmull-Rom splines that has shape parameters.We present such a class, some members of which are interpolating and others approximating. As was done for the Beta-spline, shape parameters are introduced by requiring geometric rather than parametric continuity. Splines in this class are defined by a set of control vertices and a set of shape parameter values. The shape parameters may be applied globally, affecting the entire curve, or they may be modified locally, affecting only a portion of the curve near the corresponding joint. We show that this class results from combining geometrically continuous (Beta-spline) blending functions with a new set of geometrically continuous interpolating functions related to the classical Lagrange curves.We demonstrate the practicality of several members of the class by developing efficient computational algorithms. These algorithms are based on geometric constructions that take as input a control polygon and a set of shape parameter values and produce as output a sequence of Bézier control polygons that exactly describes the original curve. A specific example of shape design using a low-degree member of the class is given.