Curves - Representation: Prepared By R.sakthivel Murugan, Ap/mech, Kcet

CURVES - REPRESENTATION Prepared by R.Sakthivel Murugan, AP/Mech, KCET.

Why designing curves?  Many technological applications  Design of products (e.g. CAD)  Calculation of the path for a robot

 Design of fonts  Large sized fonts must be smooth

 Interpolating measuring data  Approximating measuring data

Criterias for curves Controllability  Changes must be predictable in effect  Intuitive to use for the designer

Locality  Local changes should stay local

Smoothness  No sharp bends

Curve Representations  3 basic representation strategies:  Explicit: y = mx + b  Implicit: ax + by + c = 0  Parametric: P = P0 + t (P1 - P0)

Advantages of parametric forms        

More degrees of freedom Directly transformable Dimension independent No infinite slope problems Separates dependent and independent variables Inherently bounded Easy to express in vector and matrix form Common form for many curves and surfaces

Spline Representations  Spline curve  Convex hull  Control graph  Piecewise cubic splines

Spline curve  Smooth curve that is defined by a sequence of points.

Interpolating spline

Approximating spline

Convex hull  Smallest polygon that encloses all points

Convex hull

Interpolating spline

Approximating spline

Control graph  Polyline through sequence of points

Control graph

Interpolating spline

Approximating spline

Piecewise cubic splines

Segments

Interpolation vs Approximation Interpolation  Very bad locality  Tend to oscillate  Small changes may result in catastrophe

 Bad controllability  All you know is, that it interpolates the points

 High effort to evaluate curve  Imagine a curve with several million given points

Approximation  Unlike interpolation the points are not necessarily interpolated  Points give a means for controlling of where the curve goes  Often used when creating the design of new (i.e. nonexisting) things  No strict shape is given

Continuity in Curves - Representation  Parametric continuity Cx  Only P is continuous: C0  Positional continuity

 P and first derivative dP/du are continuous: C1  Tangential continuity

 P + first + second: C2  Curvature continuity

 Geometric continuity Gx  Only directions have to match

Parametric continuity Cx - Order of continuity Zero-order parametric continuity C0: P(1) = Q(0).

P(u)

Q(v)

P(u)

Q(v)

Endpoint of P(u) coincides with start point Q(v).

First order parametric continuity C1: dP(1)/du = dQ(0)/dv. Direction of P(1) coincides with direction of Q(0).

First order parametric continuity gives a smooth curve. Sometimes good enough, sometimes not.

Contd.. Second order parametric continuity C2: d2P(1)/du2 = d2Q(0)/dv2. Curvatures in P(1) and Q(0) are equal.

P(u)

Q(v)

Geometric continuity Gx  Here the vectors are exactly equal.  It suffices to require that the directions are the same.

First order geometric continuity: G1: dP(1)/du =  dQ(0)/dv with  >0. Direction of P(1) coincides with direction Q(0).

P(u)

Q(v)

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