Envelopes and tubular splines

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Abstract

Envelopes of monoparametric families of spheres determine canal surfaces. In the particular case of a quadratic family of spheres the envelope is an algebraic surface of degree four that is composed of circles. We are interested in the construction of smooth tubular splines with pieces of envelopes of quadratic families of spheres. We present a scheme for the construction of a tubular spline that interpolates a sequence of circles in 3D. We control the shape near each circle by prescribing a sphere that contains it and is tangent to the spline. We offer further shape handles for local control through weights that are assigned to the controlling spheres.

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 ft=0, for tR, they consider the surface S={(x,y,z):fz(x,y)=0}. 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 E2i+1, for i=0,,n such that for each i=1,,n, C2i=E2i1E2i+1 is a real circle. We will show how to construct G1 (i.e. the tangent plane varies continuously) tubular splines that interpolate the circles C2i. See [7], [6] for a treatment of the problem in terms of Möbius geometry and the projective space.

For each circle C2i we choose a sphere E2i that passes through C2i and two additional spheres E0 and E2n+2, which meet the spheres E1 and E2n+1, at the circles C0

Connectivity of the envelope segment

Consider three spheres E0, E1 and E2 such that E0E1 and E1E2 are real circles. Connectivity of the segment of the envelope given by E0(x,y,z)E2(x,y,z)w2E1(x,y,z)2=0, which is contained in E1, depends on the weigth w. Fig. 8 ilustrates the connectivity change as the weight w varies.

We are interested in finding the interval of variation of w so that the envelope segment stays connected. Let (xi,yi,zi) be the center of Ei and denote by π 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 E2i and Ē2i meet at a prescribed angle along the circle C2i, then the envelopes:E2i2(x,y,z)E2i(x,y,z)w2i12E2i1(x,y,z)2=0andĒ2i(x,y,z)E2i+2(x,y,z)w2i+12E2i+1(x,y,z)2=0also meet at the same angle, where E2i1 and E2i+1 pass through C2i, E2i2 and E2i+2 are arbitrary spheres and w2i1, w2i+1 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 R4 that lie outside a paraboloid of revolution. Paluszny and Boehm [6], study the relationship between the space of spheres and planes in 3D 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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