The Geometry of Ships

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THE GEOMETRY OF SHIPS 21 4.9 B-spline (Tensor-Product) Surface. A B-spline surface is defined in relation to a N u N v rectangular array (or net) of control points X ij by the surface equation: X(u,v) N u N v i1 j1 X ij B i (u) B j (v) (39) where the B i (u) and B j (v) are B-spline basis functions of specified order k u , k v for the u and v directions, respectively. The total amount of data required to define the surface is then: N u , N v number of control points for u and v directions k u , k v spline order for u and v directions U i , i 1, . . . N u k u , knotlist for u-direction V j , j 1, . . . N v k v , knotlist for v-direction X ij , i 1, . . . N u , j 1, . . . N v , control points. If the knots are uniformly spaced for both directions, the surface is a “uniform” B-spline (UBS) surface, otherwise it is “nonuniform” (NUBS). As in a B-spline curve, the B-spline products B i (u)B j (v) can be viewed as variable weights applied to the control points. The surface imitates the net in shape, but does it with a degree of smoothness depending on the spline orders. Alternatively, you can envisage the surface patch as being attracted to the control points, or connected to them by springs. The following are useful properties of the B-spline surface, analogous to those of B-spline curves: • Corners: The four corners of the patch are at the four corner points of the net. • Edges: The four edges of the surface are the B-spline curves made from control points along the four edges of the net. • Edge tangents: Slopes along edges are controlled by the two rows or columns of control points closest to the edge. • Straight sections: If k or more consecutive columns of control points are copies of one another translated along an axis, a portion of the surface will be a general cylinder. • Local support: If N u k u or N v k v , the effect of any one control point is local, i.e., it only affects a limited portion (at most k u or k v spans) of the surface in the vicinity of the point. • Rigid body: The shape of the surface is invariant under rigid-body transformations of the net. • Affine: The surface scales affinely in response to affine scaling of the net. • Convex hull: The surface does not extend beyond the convex hull of the control points, i.e., the minimal closed convex polyhedron enclosing the control points. The hull surface shown in Fig. 10 is in fact a B-spline surface; its control point net is shown in Fig. 16. 4.10 NURBS Surface. The generalizations from a uniform B-spline surface to a NURBS surface are similar to those for NURBS curves: • Nonuniform indicates nonuniform knots are permitted • Rational reflects the representation of the surface Fig. 16 The same B-spline surface shown in Fig. 10 with its control point net displayed. equation as a quotient (ratio) involving weights applied to the control points: X(u,v) N u i1 N v w ij X ij B i (u) B j (v) j1 N u i1 N v w ij B i (u) B j (v) j1 (40) This adds to the data required, compared with a B- spline surface, only the weights w ij , i 1, . . . N u , j 1, . . . N v . The NURBS surface shares all the properties — corners, edges, edge slopes, local control, affine invariance, etc. — ascribed to B-spline surfaces above. If the weights are all the same, the NURBS surface degenerates to a NUBS (Non-Uniform B-Spline) surface. The NURBS surface behaves as if it is “attracted” to its control points, or attached to the control points with springs. We can interpret the weights roughly as the strength of attraction, or the spring constant (stiffness) of each spring. A high weight on a particular control point causes the surface to be drawn relatively close to that point. A zero weight causes the corresponding control point to be ignored. With appropriate choices of knots and weights, the NURBS surface can produce exact surfaces of revolution and other shapes generated from conic sections (ellipsoids, hyperboloids, etc.) (Piegl & Tiller 1995). These properties in combination with its freeform capabilities and the development of standard data exchange formats (IGES and STEP) have led to the widespread adoption of NURBS surfaces as the de facto standard surface representation in almost all CAD programs today. 4.11 Ruled and Developable Surfaces. A ruled surface is any surface generated by the movement of a straight line. For example, given two 3-D curves X 0 (t) and X 1 (t), each parameterized on the range [0, 1], one can construct a ruled surface by connecting corresponding parametric locations on the two curves with straight lines (Fig. 17). The parametric surface equations are: X(u, v) (1 v)X 0 (u) vX 1 (u) (41)
22 THE PRINCIPLES OF NAVAL ARCHITECTURE SERIES If either or both of the curves is reparameterized, a different ruled surface will be produced by this construction. The straight lines (u constant isoparms) are called the generators or rulings of the surface. A ruled surface has zero or negative Gaussian curvature. A developable surface is one that can be rolled out flat by bending alone, without stretching of any element. Conversely, it is a surface that can be formed from flat sheet material by bending alone, without in-plane strain (Fig. 18). The opposite of “developable” is “compound-curved.” Geometrically, a developable surface is characterized by zero Gaussian curvature. Developable surfaces are profoundly advantageous in ship design because of their relative ease of manufacture, compared with compoundcurved surfaces. A key strategy for “produceability” is to make as many of the surfaces of a vessel as possible from developable surfaces; this can be 100 percent. Cylinders and cones are well-known examples of developable surfaces. A general cylinder is a surface swept by movement of a straight line that remains parallel to a given line. A general cone is a surface swept by movement of a straight line that always passes through a given Fig. 18 Fig. 17 A chine hull constructed from two ruled surfaces. A chine hull made from developable surfaces spanning three longitudinal curves. point (the apex). One design method for developable surfaces, “multiconic development,” pieces together patches from a series of cones, constructed to have G 1 continuity with one another, to produce a developable composite surface. All developable surfaces are ruled. However — and this is a geometric fact that is widely misapprehended in manufacturing and design — not all ruled surfaces are developable. In fact, developable surfaces are a very narrow and specialized subset of ruled surfaces. One way to distinguish developable surfaces is that they are the ruled surfaces with zero Gaussian curvature. Alternatively, a developable surface is a special ruled surface with the property that it has the same tangent plane at all points of each generator. This latter property of developable surfaces is the basis of Kilgore’s method, a valid drafting procedure for construction of developable hulls and other developable surfaces (Kilgore 1967). Nolan (1971) showed how to implement Kilgore’s method in a computer program for the design of developable hull forms. 4.12 Transfinite Surfaces. The B-spline and NURBS surfaces, supported as they are by arrays of points, each have a finite supply of data and, therefore, a finite space of possible configurations. Generally, this is not limiting when designing a single surface in isolation, but many problems arise when surfaces have to join each other in a complex assembly. In order for two NURBS surfaces to join (G 0 continuity) with mathematical precision, they must have (in general): • the same set of control points along the common edge; • the same polynomial degree in this direction; • the same knotlist in this direction; and • proportional weights on the corresponding control points. These are stringent requirements rarely met in practice. Further, if a surface needs to meet an arbitrary (non- NURBS) curve (for example, a parametric curve embedded in another surface), it will have only a finite number of control points along that edge, and therefore can only approximate the true curve to a finite precision. In NURBS-based modeling, therefore, nearly all junctions are approximate, or defined by intersections. This causes a large variety of problems in manufacturing and in transfer of surface and solid models between systems which have different tolerances. Transfinite surfaces are generated from curves rather than points and, consequently, are not subject to the same limitations. Examples of transfinite surfaces already mentioned above are: • Ruled surface: it interpolates its two edge curves exactly • Lofted surfaces: they interpolate their two end master curves • Developable surface: constructed between two curves by Kilgore’s method.
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Index Terms Links coordinates, homo
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The Geometry of Ships

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