Methods Used

Lines uses 2-D and 3D-space curves to define a hullform, of which the mathematical basis is B-Splines.

The mathematical spline is a close analog of the draughtsman's spline, a long narrow strip of wood or plastic shaped into the required curve form by lead weights called ducks.

If the draughtsman's spline is considered as a long thin elastic beam, then Euler's equation yields:

where M(x) is the bending moment, E is Young's modulus, I is the moment of inertia and R(x) is the radius of curvature. For small deflections we can assume:

where y denotes the second derivative of the deflection y with respect to x. If the ducks are assumed as simple supports:

(a linear function)

where A and B are constant, therefore

Integrating the above equation twice shows that the physical spline can be described by cubic polynomials between supports:

where a, b, c and d are coefficients of the polynomial.

B-spline curves are splines in which the vertices of an open polygon together with an appropriate knot vector and the B-spline Basis functions of the required order uniquely define the curve shape (see Figure 1:2 in B-spline Properties). The prime user functions of the B-Spline polynomial used in the Lines system are:

• Display of polygon vertices on the graphics screen and interactive modification of the local curve shape by redefining the polygon vertices.

• Provision for creating a knuckle: that means, a discontinuity in the first derivative of the curve. The B-spline definition for the entire curve span is unaffected by the introduction of a knuckle.

• Provide a stable fit through uneven or irregularly spaced data points.

• The B-spline can be easily interrogated for properties of the curve such as coordinates of a point, or the slope and curvature at a point and the area enclosed by the spline.