introduction to virtual engineering Óbuda university john von neumann faculty of informatics...

Introduction to virtual engineering

Óbuda UniversityJohn von Neumann Faculty of Informatics

Institute of Intelligent Engineering Systems

Lecture 3.

Description of geometry

László Horváth university professor

http://nik.uni-obuda.hu/lhorvath/

Parametric representation of curves

Global and local characteristics of a curve

Functions for the description of curves

Methods for interpolation

Bezier curves: characteristics, convex hull

Curve: one-piece of piecewise (segmented)?

Segments in a B-spline curve

Characteristics of B-spline curve

Control of a B-spline curve

Contents

László Horváth ÓU-JNFI-IIES http://nik.uni-obuda.hu/lhorvath/

P (x,y,z)

(u)umin

umax

Z

X

Y

P

General form of the parametric equation of curve:

P(u)=[x(u) y(u) z(u)]

where umin <= u <= umax

The x, y, and z coordinates of model space point P in the function of the parameter u:

x=x(u), y=y(u) és z=z(u)

It defines points along a curve in the function of parameter u.

It gives x,y, and z coordinate values at the point P for the parameter value uP of that point.

Pu is the position vector to point P.

Parametric equation of a three dimensional curve

Cartesian space

Parametric representation of curves


Local characteristics at a point with parameter value u are:

Tangent (t),Normal (n)Binormal (b)Curvature (r)

tn

br

Local characteristics t, n, and b define the accompanying trieder.

Global characteristics:ControlDegreeclass

Global and local characteristics of a curve


Simple functions: analytical curves

Polynomial: arbitrary analytical and free form shape

Suitable for determination of tangents, normals, and curvatures. Differentiation of the function is easy. This class of functions is widely applied in description of geometry.

General form of a polynomial of degree n is

in

ii xaxp

0

=

011

1 axaxaxaxp nn

nn

… =

Functions for the description of curves


Creating curve through points defined by experiments or calculations.

Linear interpolation: straight segment between two points

Arc through three points (quadratic analytical curve)

Four points defne cubic curve.

A Hermite interpolation: definition of a curve using two points and the demanded tangents at those points. Method by Hermite was applied by Ferguson and Coons.

Interpolation polynomials were developed as solutions for the interpolation task. The most widely known method is the simple Lagrange polynomial for fitting curve through points.

Methods for interpolation


First applications of mathematical descriptions of curves and surfaces were in design of cars and aircraft.

Paul Bezier (Renault): introduced the control polygon, that controls the shape of curve by the position of control points. He applied Bernstein polynomials as blending (base) function.

Concurently, de Casteljau (Citroen) developed similar method for the same purpose.

Characteristics of the Bezier curve:•Global control.•Degree depends on the number or control points.•Curve goes through the first and last control points. •First and last segments of the control polygon are tangents of the curve.•It is within the convex hull (see below).

P0

P1

P2

P3

P0

P1

P2

P3

P0

P1

P2

P3

Bezier curves: characteristics, convex hull


One-piece curve Bezier curve: global control. Change of position at a control point modifies the entire curve.

Chain of Bezier curves.

B-spline curve consists of segments. Pi -1

PiPi +1

i -1 segment

i segment

u0 u1 u2 u3

Curve: one-piece of piecewise (segmented)?


u= u=

u0

u1u2

u3

u4

u1 u2

Knot Segment

Parameter range of a segment

Segments in a B-spline curve


Consists of segments.Continuity at segment borders.Local control.Spline base functions.Degree of the curve is same as degree of the base function. Different degree of segments is allowed.Curve goes through of the first and last control points only in case of special parameterization.

Spline: Flexible steel ribbon in ship building. It was modeled.

Characteristics of B-spline curve


The B-spline base (blending) function is defined within a given parameter range.

Example: Closed curve is controlled by six control points (vertices). It consists of six segments.

Segment 1 is controlled by V0-V2.Segment 2 is controlled by V1-V3.

And so on.

V0

V1

V2

V3

V4

V5

Segment 1 Segment 2

Control of a B-spline curve


introduction to virtual engineering Óbuda university john von neumann faculty of informatics...

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