1. Ship Computer Aided Design MR 422

2. Geometry of Curves 1.Introduction 2.Mathematical Curve Definitions 3.Analytic Properties of Curves 4.Fairness of Curves. 5.Spline Curves. 6.Interpolating Splines 7.Approximating Splines and Smoothness 8.B- spline Curves 9.NURBS Curves 10.Re-parameterization of Parametric Curves 11.Continuity of Curves 12.Projections and Intersections. 13.Relational Curves 14.Points Embedded in Curves 2

3. 1. Introduction A curve is a 1-D continuous point set embedded in a 2-D or 3-D space. Curves are used in: as explicit design elements, such as the sheer line, chines, or stem profile of a ship as components of a wireframe representation of surfaces. as control curves for generating surfaces by various constructions. 3

4. 2. Mathematical Curve Definitions Implicit, Explicit, and Parametric Implicit curve definition: A curve is implicitly defined as the set of points that satisfy an implicit equation f(x, y)=0 or f(x, y, z) =0 Explicit curve definition: one coordinate is expressed as an explicit function of the other: y= f(x) or y=f(x), z= g(x) Parametric curve definition: In either 2-D or 3-D each coordinate is expressed as an explicit function of a common dimensionless parameter: x=f(t), y = g(t), z = h(t) 4

5. 3. Analytic Properties of Curves x(t) signifying a vector of two or three components ({x, y} for 2-D curves and {x,y, z} for 3-D curves). Differential geometry Tangent vector Parametric velocity Arc length The tangent vector 5

6. 4. Fairness of Curves Ships and boats of all types are visual as will as functional objects. Fairness is a visual rather than mathematical property of a curve. Many aspects of fairness can be directly related to analytic properties of a curve. It is not possible to give an exact mathematical definition that every one can agree on The vessel may be viewed or photographed from widely varying viewpoints, it is valuable to check these properties in 3-D as well as in 2D orthographic views. 6

7. 4. Fairness of Curves Features that contrary to fairness : Unnecessarily hard turns ( local high curvature ). Flat spots (local low curvature). Abrupt change of curvature, as in the transition from a straight line to a tangent circular arc Unnecessary inflection points (reversals of curvature). If a curve is planar and is free of inflection in any particular perspective or orthographic view, from a view point not in the plane, then it is free of inflection in all perspective and orthographic views. 7

8. 5. Spline Curves Spline curves originated as mathematical models of the flexible curves used for drafting and lofting of freeform curves in ship design. Splines are composite function generated by splicing together spans of relatively simple function usually low order polynomials. The location where spans join called knots. 8

9. 5. Spline Curves The spline function and its first two derivatives (i.e., slope and curvature) are continuous across a typical knot. The cubic spline is a model of a drafting spline, arising very naturally from the small-deflection theory for a thin uniform beam subject to concentrated shear loads at the points of support. Spline curves used in geometric design can be explicit or parametric. – Explicit Spline curves: The most basic definition of a curve in two dimensions is y=f(x). ( limited if slope is infinite) – Parametric Spline curve: can be used to generate curves that are more general the explicit equations of the form y= f(x). 9

10. 6. Interpolating Splines A common form of spline curve is the cubic interpolation spline. This is a parametric spline in 2D or 3D that passes through a sequence of N 2D,3D data points Xi=1,….., N N-1 spans of such a spline is a parametric cubic curve, and at the knots the individual spans join with continuous slop and curvature. Issues need to be resolved to specify a cubic spline uniquely: Parameters values at the knots( uniformly spaces data points, chord-length parameterization) End conditions

11. 7. Approximating /Smoothing Splines not pass through all its data points, but rather is adjusted to pass optimally “close to” its data points in some defined sense such as least squares or minmax deviation.

12. 8. B-Spline Curves A B-spline basis function (“B-spline”) is a continuous curve x(t) defined in relation to a sequence of control points. the B-splines can be viewed as variable weights applied to the control points to generate or sweep out the curve. The parametric B-spline curve imitates in shape the usually open) control polygon or polyline joining its control points in sequence. Another interpretation of B-spline curves is that they act as if they are attracted to their control points, or attached to the interior control points by springs.

13. 9. NURBS Curves NURBS Curves = Non Uniform Rational B- Splines – Non uniform means : non uniform knots – Rational reflects to representation of a NURBS curve as a fraction involving non negative weights. The NURBS curve with uniform weights is just a B-spline curve Advantages of NURBS 1.All advantages of B-Splines. 2.Specific choices of weights and knots exist which will make a NURRBS curve take the exact shape of any choice section 3.Provides a single unified representation 4.Used to approximate any other curve 5.Widely adopted for communication of curves between CAD system.