Procedural elements for computer graphics
Procedural elements for computer graphics
Algorithms for Graphics and Imag
Algorithms for Graphics and Imag
Computational Geometry for Design and Manufacture
Computational Geometry for Design and Manufacture
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Certain applications in CAD/CAM require the design of tubular surfaces. The axis curve is usually limited to a straight line (as in the case of a cylinder) or a circular curve (as in the case of a torus). In these cases the surface has circular planar cross sections orthogonal to the axis curve. The tubing used to wind through various compartments of an aircraft is an example where the axis curve is an arbitrary curve instead of a straight line or a circle. There are other applications such as a braided bundle of fiber optic lines where planar cross sections are complex closed curve strings. The mathematical formulation developed here will allow the generation of a tubular surface with arbitrarily shaped planar cross sections along an arbitrarily shaped axis curve. Traditional surfaces where the axis curve is a straight line or a circle become special cases of this representation.The mathematical formulation of the tubular surface is developed in terms of parametric geometry. Let S(u,v) be the tubular surface. The inputs needed to generate this surface are given as follows:Let R(u) be a G1 axis curve and C(u) be the simple continuous curve in the u-parametric direction at v - v0 such that c(u) - S(u,v0). If for v0 ≤ v ≤ v1, r(u,v) is a continuous shape design function in a plane orthogonal to C(u) and | C(u) - R(u) | = r(u,v0) and [ C(u) - R(u) ] . R'(u) = 0, then the tubular surface function S(u,v) is defined by S(u,v) = C(u) + [C(u) - R(u)] M(v,R(u),r(u,v)) where M is the path matrix used to define the closed path r(u,v) in the instantaneous plane through R(u) with normal R'(u).Suppose the shape design function is independent of u. In particular, if the shape design function is a circle or an ellipse, then the resultant surface will be:a (circular or elliptical cross section) torus, if the axis curve R(u) and the initial curve C(u) are both concentric circles;a (circular or elliptical cross section) cylinder, if the axis curve R(u) and the initial curve C(u) are two distinct parallel lines;a (circular or elliptical cross section) cone, if the axis curve R(u) and the initial curve C(u) are two crossing lines.