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Page 1: Ship Computer Aided Design MR 422. Geometry of Curves 1.Introduction 2.Mathematical Curve Definitions 3.Analytic Properties of Curves 4.Fairness of Curves.

Ship Computer Aided Design

MR 422

Page 2: Ship Computer Aided Design MR 422. Geometry of Curves 1.Introduction 2.Mathematical Curve Definitions 3.Analytic Properties of Curves 4.Fairness of Curves.

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Geometry of Curves

1. Introduction2. Mathematical Curve

Definitions3. Analytic Properties of

Curves4. Fairness of Curves.5. Spline Curves. 6. Interpolating Splines7. Approximating Splines

and Smoothness

8. B- spline Curves9. NURBS Curves10. Re-parameterization of

Parametric Curves11. Continuity of Curves12. Projections and

Intersections.13. Relational Curves14. Points Embedded in

Curves

Page 3: Ship Computer Aided Design MR 422. Geometry of Curves 1.Introduction 2.Mathematical Curve Definitions 3.Analytic Properties of Curves 4.Fairness of Curves.

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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.

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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) =0Explicit 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)

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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

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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.

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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.

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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 .

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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).

Page 10: Ship Computer Aided Design MR 422. Geometry of Curves 1.Introduction 2.Mathematical Curve Definitions 3.Analytic Properties of Curves 4.Fairness of Curves.

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

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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.

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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.

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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 curveAdvantages 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 section3. Provides a single unified representation4. Used to approximate any other curve5. Widely adopted for communication of curves between CAD system.

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