1. Ship Computer Aided Design
MR 422

2. Geometry of Surfaces Introduction
Mathematical Surface Definitions: Parametric vs. Explicit vs. Implicit.
Analytic Properties of Parametric Surfaces
Surface Curvature
Continuity Between Surfaces
Fairness of Surfaces
Spline Surfaces

3. 1. Introduction
A surface is a 2-D continuous point set embedded in a 2D or 3-D space.
Surfaces may be used as:
Explicit design elements, such as the hull or weather deck surfaces
Construction elements, such as a horizontal rectangular surface locating an interior deck
Boundaries for solids.

4. 2. Mathematical Surface Definitions
Defining surfaces mathematically ways are:
Implicit surface definition
Explicit surface definition
Parametric surface definition

5. f(x, y, z) = (x _ a)2 + (y _ b)2 + (z _ c)2 - r2 = 0 .
Implicit surface definition: A surface is defined in 3-D as the set of points that satisfy an implicit equation in the three coordinates:
f(x, y, z) = 0
Homogeneous, free of the coordinate singularities but Lacks any natural surface coordinate system
Example: a complete spherical surface is defined as the set of points at a given distance r from a given center point {a, b, c}:
f(x, y, z) = (x _ a)2 + (y _ b)2 + (z _ c)2 - r2 = 0 .

6. Explicit surface definition: In 3-D, one coordinate is expressed as an explicit function of the other two, for example:
z = f(x, y)
have seen some use in ship form definitions, but usually have problems with some shapes which restrict the range of shapes that can be accommodated without encountering mathematical singularities.  
Example of explicit definition of ship hull forms is the series of algebraic shapes, commonly called the “Wigley parabolic hull”.

7. The waterlines (z= constant) and underwater sections (x= constant) are families of parabolas.
These simple explicit surface equations allowed the computation of Michell’s integral analytically, allowing an early comparison of this influential theory with towing-tank results.

8. x = f(u, v), y = g(u, v), z = h(u, v)
Parametric surface definition: In either 2-D or 3-D, each coordinate is expressed as an explicit function of two common dimensionless parameters:
x = f(u, v), y = g(u, v), z = h(u, v)
The parametric surface can be described as a locus in three different ways:
The locus of a moving point {x, y, z} as the parameters u, v vary continuously over a specified domain such as [0, 1] X [0, 1], or
The locus of a moving parametric curve (parameter u or v) as the other parameter (v or u) varies continuously over a domain such as [0, 1].
Subdivision surfaces.
Parametric surface definitions avoid the limitations of implicit and explicit definitions and are widely employed in 3-D CAD systems today

10. 3. Analytic Properties of Parametric Surfaces
Definitions:
x(u, v) : a parametric surface
A bold face letter signifies a vector of three components.
Assume the range of each parameter u, v is [0, 1]
The bounded surface patch corresponds to the nominal parameter range.
The parameter space of the surface: is the 2-D space of u and v.
The 3-D surface is a mapping of the parameter-space points into three-space points, moderated by the surface equations x(u, v).

11. The unit normal vector varies with u and v unless the surface is flat.
The tangent plane is the plane passing through a surface point, normal to the unit normal vector at that point.
The direction of the unit normal on, for example, one of the wetted surfaces of a ship may be inward (into the hull interior) or outward (into the water), depending on the orientation chosen for the parameters u, v.
For many purposes the normal orientation will not matter; however, for other purposes it is of critical importance.

12. For hydrostatic or hydrodynamic analysis, it is usually necessary to create panels having a consistent orientation of corner points, e.g., counterclockwise when viewed from the water; this may well require that the surface normal have a prescribed orientation.
The bevel angles: the angles of the unit normal with respect to the coordinate planes, are sometimes required during construction. The angle between n and the unit vector in the x direction is most often used

13. 4. Surface Curvature
At a point P on a surface S, where S is sufficiently smooth (i.e., a unique normal line N and tangent plane T exist), several measures of surface curvature can be defined.
Each plane that cuts the surface S in a plane curve C, known as a normal section.
The curvature of C at P is called a normal curvature (kn) of S (dimensions 1/length) at this location.

14. Normal curvature depends on α
Normal curvature depends on α. As α varies, kn varies sinusoidally with respect to α, and in general goes through maximum and minimum values k1,k2 (the principal curvatures).
The directions of the two principal curvatures are orthogonal, and are called the principal directions. 
The product k1 k2 of the two principal curvatures is called Gaussian curvature K (dimensions 1/length2).
The average (k1 +k2)/2 of the two principal curvatures is called mean curvature H (dimensions 1/length).

15. 5. Continuity Between Surfaces
Levels of geometric continuity are defined as follows:
G0: Surfaces that join with an angle or knuckle (different normal directions) at the junction.
G1: Surfaces that join with the same normal direction at the junction.
G2: Surfaces that join with the same normal direction and the same normal curvatures in any direction that crosses the junction.

16. G0 continuity
easy to achieve, used in “industrial” contexts when a sharp corner does not interfere with function, for example, the longitudinal chines of a typical metal workboat.
G1 continuity
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.
G2 continuity
still more difficult to attain, is required for the highest levels of visual design, as in automobile and yacht exteriors.

17. 6. Fairness of Surfaces
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 G2 continuity)
unnecessary inflection points

18. 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 planning hull, where the flow deliberately separates from the surface).
To reveal unfairness of physical surfaces:
 Reflection lines: Viewing the reflections that occur at low angles (assuming a polished, reflective surface) of a regular grid, can be computed and presented in computer displays to simulate this process using the visualization technology known as ray tracing.
 Highlight lines: contours of equal “slope” s on the surface.

19. 7. Spline Surfaces
A spline surface is typically divided along certain parameter lines (its knotlines) into sub-surfaces or spans, each of which is a parametric polynomial (or rational polynomial) surface in u and v.
Within each span, the surface is analytic (continuous derivatives of all orders)
At the knotlines, the spans join with levels of continuity depending on the spline degree.
Cubic spline surfaces have C2 continuity across their knotlines, which is generally considered adequate continuity for all practical visual and hydrodynamic purposes.
Splines of lower order than cubic (i.e., linear and quadratic) are simpler to apply and provide adequate continuity (C0 and C1, respectively) for many less demanding applications.

20. Interpolating Spline Lofted Surface
Interpolating spline curves passes through an arbitrary set of data points.
A lofted surface interpolates an arbitrary set of parent curves, known as master curves or control curves.