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
Geometric continuity, shape parameters, and geometric constructions for Catmull-Rom splines
ACM Transactions on Graphics (TOG)
ACM Transactions on Graphics (TOG)
Mathematical methods in computer aided geometric design
Algebraic aspects of geometric continuity
Mathematical methods in computer aided geometric design
Mathematical methods in computer aided geometric design
Geometric Continuity of Parametric Curves: Three Equivalent Characterizations
IEEE Computer Graphics and Applications
Geometric Continuity of Parametric Curves
Geometric Continuity of Parametric Curves
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In the previous Oslo conference held in June 1988, we observed that a sufficient condition for the affine combination of two geometrically continuous curves to be a geometrically continuous curve as that the two curves satisfy that Beta-constraints for the same B values [11]. This result was motivated by John Gregory's observation at the same conference(see page 361 of [15]) that an affine combination of two geometrically continuous curves does not necessarily yield a geometrically continuous curve. This anomaly causes havoc in various curve and surface constructions. For example. A ruled. Lofted, or Boolean sum surface constructed groom geometrically continuous curves need not be a geometrically continuous surface. Similarly, Catmull-From splines and rational curves constructed from geometrically continuous curves or geometrically continuous blending functions need not be geometrically continuous curves. Here, we examine geometrically continuous curves described in piecewise Bézier form that do not satisfy the Beta-constraints for the sameB values. From our previous work, it follows that after reparametrization, an affine combination of the two curves will necessarily be geometrically continuous. We show how to determine the location of the Bézier control vertices of the same curves will be geometrically continuous, thereby permitting classical smooth curve and surface centurions such as ruled and lofted surfaces and Catmull-Rom splines.