The Geometry of Ships

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THE GEOMETRY OF SHIPS 19 Fig. 13 Patches with positive, zero, and negative Gaussian curvature. • The directions of the two principal curvatures are orthogonal, and are called the principal directions. • The product 1 2 of the two principal curvatures is called Gaussian curvature K (dimensions 1/length 2 ). • The average ( 1 2 )/2 of the two principal curvatures is called mean curvature H (dimensions 1/length). Normal curvature has important applications in the fairing of free-form surfaces. Gaussian curvature is a quantitative measure of the degree of compound curvature or double curvature of a surface and has important relevance to forming curved plates from flat material. Color displays of Gaussian curvature are sometimes used as an indication of surface fairness. Mean curvature displays are useful for judging fairness of developable surfaces, for which K 0 identically. Figure 13 shows example surface patches having positive, zero, and negative Gaussian curvature. 4.4 Continuity Between Surfaces. A major consideration in assembling different surface entities to build a composite surface model is the degree of continuity required between the various surfaces. Levels of geometric continuity are defined as follows: G 0 : Surfaces that join with an angle or knuckle (different normal directions) at the junction have G 0 continuity. G 1 : Surfaces that join with the same normal direction at the junction have G 1 continuity. G 2 : Surfaces that join with the same normal direction and the same normal curvatures in any direction that crosses the junction have G 2 continuity. The higher the degree of continuity, the smoother the junction will appear. G 0 continuity is relatively easy to achieve and is often used in “industrial” contexts when a sharp corner does not interfere with function (for example, the longitudinal chines of a typical metal workboat). G 1 continuity is more trouble to achieve, and is widely used in industrial design when rounded corners and fillets are functionally required (for example, a rounding between two perpendicular planes achieved by welding in a quarter-section of cylindrical pipe). G 2 continuity, still more difficult to attain, is required for the highest levels of aesthetic design, as in automobile and yacht exteriors. 4.5 Fairness of Surfaces. As with curves, the concept of fairness of surfaces is a subjective one. It is closely related to the fairness of normal sections as curves. Fairness is best described as the absence of certain kinds of features that would be considered unfair: • surface slope discontinuities (creases, knuckles) • local regions of high curvature (e.g., bumps and dimples) • flat spots (local low curvature) • abrupt change of curvature (adjoining regions with less than G 2 continuity) • unnecessary inflection points On a vessel, because of the principally longitudinal flow of water, fairness in the longitudinal direction receives more emphasis than in the transverse direction. Thus, for example, longitudinal chines are tolerated for ease of construction, but transverse chines are very much avoided (except as steps in a high speed planing hull, where the flow deliberately separates from the surface). Most surface modeling design programs provide forms of color-coded rendered display in which each region of the surface has a color indicating its curvature. This can include displays of Gaussian, mean, and normal curvature. A sensitive way to reveal unfairness of physical surfaces is to view the reflections that occur at low or grazing angles (assuming a polished, reflective surface). Reflection lines, e.g., of a regular grid, can be computed and presented in computer displays to simulate this process using the visualization technology known as ray tracing. A simpler and somewhat less sensitive alternative is to display so-called “highlight lines,” i.e., contours of equal “slope” s on the surface; for example, s ŵ n(u, v), where ŵ is a selectable constant unit vector and n is the unit normal vector. 4.6 Spline Surfaces. Various methods are known to generate parametric surfaces based on piecewise polynomials. These include the dominant surface representations used in most CAD programs today. Some may be viewed as a composition of splines in the two parametric directions (u and v), others as an extension of spline curve concepts to a higher level of dimensionality. From their roots in spline curves, spline surfaces inherit the advantages of being made up of relatively simple functions (polynomials) which are easy to evaluate, differentiate, and integrate. A spline surface is typically divided along certain parameter lines (its knotlines) into subsurfaces or spans, each of which is a parametric polynomial (or rational polynomial) surface in u and v. Within each span, the surface is analytic, i.e., it has
20 THE PRINCIPLES OF NAVAL ARCHITECTURE SERIES continuous derivatives of all orders. At the knotlines, the spans join with levels of continuity depending on the spline degree. Cubic spline surfaces have C 2 continuity across their knotlines, which is generally considered adequate continuity for all practical aesthetic and hydrodynamic purposes. Splines of lower order than cubic (i.e., linear and quadratic) are simpler to apply and provide adequate continuity (C 0 and C 1 , respectively) for many less demanding applications. 4.7 Interpolating Spline Lofted Surface. In Section 3.5, we described interpolating spline curves which pass through an arbitrary set of data points. This curve construction can be the basis of a lofted surface which interpolates an arbitrary set of parent curves, known as master curves or control curves. Suppose we have defined a set of curves X i (t), i 1, . . . N, e.g., the stem curve and some stations of a hull. The following rule produces a parametric surface definition which interpolates these master curves: Given u and v, • Evaluate each master curve at t u, resulting in the points X i (u), i 1, . . . N • Construct an interpolating spline S(t) passing through the points X i in sequence • Evaluate X(u, v) S(v). A little more has to be specified to make this construction definite: the order k of the interpolating spline, how its knots are determined (knots at the master curves are common), and the end conditions to be applied (Fig. 14). If the master curves X i are interpolating splines, this surface passes exactly through all its data points. Note that there do not have to be the same number of data points along each master curve, but the data points do have to be organized into rows or columns; they can’t just be scattered points. The smoothness of the resulting surface may not be acceptable unless the data itself is of very high quality, e.g., sampled from a smooth surface, with a very low level of measurement error. 4.8 B-spline Lofted Surface. In a similar construction, B-spline curves can be used instead of interpolating splines to create another form of lofted surface. Again, we start with N master curves, but the construction is as follows: Given u and v, • Evaluate each master curve at t u, resulting in the points X i (u), i 1, . . . N • Construct a B-spline curve S(t) using the points X i in sequence as control points • Evaluate X(u, v) S(v). To be definite, we have to specify the order k of the B- spline, and its knots (which might just be uniform). The B-spline lofted surface interpolates its first and last master curves but in general not the others (unless k 2). It behaves instead as if it is attracted to the interior master curves. It has the following additional useful properties, analogous to those of B-spline curves: • End tangency: At v 0, X(u, v) is tangent to the ruled surface between X 1 and X 2 ; likewise at v 1, X(u, v) is tangent to the ruled surface between the last two master curves. This property makes it easy to control the slopes in the v direction at the ends. • Straight section: If k or more consecutive master curves lie on a general cylinder (i.e., their projections on a plane normal to the cylinder generators are identical), a portion of the surface will lie accurately on that same cylinder. • Mesh velocity: The parametric velocity in the v-direction reflects the spacing of master curves, i.e., the velocity will be relatively low where master curves are close together. • Local control: When N k the influence of any one master curve will extend over a limited part of the surface in the v-direction (less than k knot spans). Figure 15 shows lines of a ship hull with bow and stern rounding based on property (1) and parallel middle body based on property (2). Fig. 14 A parametric hull surface lofted through five B-spline master curves. Fig. 15 A ship hull defined as a B-spline lofted surface with eight master curves.
Page 2 and 3: The Principles of Naval Architectur
Page 4 and 5: Nomenclature m A A
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Page 8 and 9: Section 1 Geometric Modeling for Ma
Page 10 and 11: THE GEOMETRY OF SHIPS 3 Although fi
Page 12 and 13: THE GEOMETRY OF SHIPS 5 and their f
Page 14 and 15: THE GEOMETRY OF SHIPS 7 The concept
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Page 26 and 27: THE GEOMETRY OF SHIPS 17 plicit sur
Page 30 and 31: THE GEOMETRY OF SHIPS 21 4.9 B-spli
Page 32 and 33: THE GEOMETRY OF SHIPS 23 Fig. 19 A
Page 34 and 35: THE GEOMETRY OF SHIPS 25 A typical
Page 36 and 37: THE GEOMETRY OF SHIPS 27 Section 5
Page 38 and 39: THE GEOMETRY OF SHIPS 29 A curve ly
Page 46: Fig. 32 The lines plan of a cargo s
Page 49 and 50: THE GEOMETRY OF SHIPS 37 curves, ty
Page 51 and 52: THE GEOMETRY OF SHIPS 39 tration of
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Page 55 and 56: THE GEOMETRY OF SHIPS 43 The genera
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Page 70 and 71: THE GEOMETRY OF SHIPS 55 Fig. 41 A
Page 74 and 75: THE GEOMETRY OF SHIPS 57 surface ab
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Index Terms Links center of buoyanc
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Index Terms Links coordinates, homo
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Index Terms Links displacement 3 4
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Index Terms Links geometry definiti
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Index Terms Links isoparms 17 K kee
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Index Terms Links model (Cont.) sca
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Index Terms Links pole 18 polygon 1
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Index Terms Links seakeeping 2 31 3
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Index Terms Links surfaces (Cont.)
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Index Terms Links triangle mesh 27f
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The Geometry of Ships

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