Envelopes and tubular splines
Section snippets
Preliminaries
A particularly intuitive definition of envelope of a monoparametric family of plane curves is provided in [9]. Given a family of curves , for , they consider the surface . The intersection of S with each horizontal plane is a member of the family of curves and on each curve there are points such that their tangent planes to S are orthogonal to the xy plane. These are the contour points and their orthogonal projections onto the xy plane form the envelope of the family
Construction of tubular splines
Consider a sequence of spheres , for such that for each , is a real circle. We will show how to construct (i.e. the tangent plane varies continuously) tubular splines that interpolate the circles . See [7], [6] for a treatment of the problem in terms of Möbius geometry and the projective space.
For each circle we choose a sphere that passes through and two additional spheres and , which meet the spheres and , at the circles
Connectivity of the envelope segment
Consider three spheres , and such that and are real circles. Connectivity of the segment of the envelope given by , which is contained in , depends on the weigth . Fig. 8 ilustrates the connectivity change as the weight varies.
We are interested in finding the interval of variation of so that the envelope segment stays connected. Let be the center of and denote by the plane containing these points. Let C be the circle on
Prescribing angles at joints
It is easy to join envelopes with prescribed dihedral angles. If the spheres and meet at a prescribed angle along the circle , then the envelopes:andalso meet at the same angle, where and pass through , and are arbitrary spheres and , are positive constants. Fig. 9 illustrates the case of two envelopes joining at an angle along a circle.
Tubular splines and four-dimensional space
In [4] we establish a 1:1 relationship between the space of spheres and points in that lie outside a paraboloid of revolution. Paluszny and Boehm [6], study the relationship between the space of spheres and planes in and the exterior of a quadric (of signature ) in projective four-space. The latter is analogous to the former, the difference being the choice of the point at infinity. The monoparametric family of spheres given by (1) corresponds to a Bézier conic in four-dimensional
Applications
Tubular splines are generalizations of classical curve analogs: a spline that follows the path of an object.
Consider an extensible rod whose base moves along a path and its retractible tip is allowed to move on a sphere of varying radius. Then the volume (i.e. the set of points) that can be reached by the tip is the interior of a canal surface: the envelope of the moving sphere.
A canal surface can be a complicated object: non-rational or have high algebraic degree, but it can always be
Acknowledgements
The authors thank J. Yerena and E. Roa for their help with the illustrations. We also appreciate the funding of FONACIT and CDCH-UCV. We are grateful to the referee for some stimulating observations that were helpful to improve the paper.
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