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7/22/2019 Engineering Curves and CAGD
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Engineering curves and CAGD
Rubn Dorado Vicente
Department of mechanical & mining engineering
University of Jan
III Engineering Week, Schmalkalden 2013
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Contents
Computer Aided Engineering
Computer Aided Geometric Design CAGD
Mathematical model of a curve
Useful engineering curves
Bzier, B-spline and NURBS representation
Fitting: approximation and interpolation
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CAGD Curve
s
Eng. curves CAD representation Fittin
g
Conclusions
Idea
Concept
& Design
Analysis
Simulation
Prototype
Tests Manufacturing
Product
Computer aided engineering
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CAE= CAxx
,
x = process aided by computer
DesignCADbased on CAGD: computational representation of shapes
Analysis + virtual prototypingCAA
ManufacturingCAM
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CAGD Curve
s
Eng. curves CAD representation Fittin
g
Conclusions
CAE advantages
Design errors lead to high final costs
CAE reduces the need for prototyping: cost & time to take market
Delayslost sales and lock-in
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CAGD Curve
s
Eng. curves CAD representation Fittin
g
Conclusions
Examples of first mover advantages
Operating systems: Microsoft
1981, DOS: Unstable, unsafe, "backward compatibility
CAD software: AutoCAD
First CAD software
QWERTY keyboard
Designed to avoid a mechanical problem
Alternative combustion engine
Contamination, mechanically
complex
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Source: Itedo
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CAGD Curve
s
Eng. curves CAD representation Fittin
g
Conclusions
CAE: disadvantages
There are many CAD format Lost time & errors
Solution for geometry: Neutral formats like IGES, STEP
Data maintenance: Hardware + software, High costs
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CAGD Curve
s
Eng. curves CAD representation Fittin
g
Conclusions
Finite element analysis
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Source: engineering.com
Numerical solution
of mechanical problems
Application in structural,
thermal and dynamic studies
Idea: divide and conquer
Discretization in elements Analytical CAD models.
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CAGD Curve
s
Eng. curves CAD representation Fittin
g
Conclusions
Finite Element Analysis software
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Software Company Analysis
ANSYS INC Fluids, structural, thermal, dynamic.
Dassault Systemes Fluids, structural, thermal, dynamic.
Siemens Fluids, structural, thermal, dynamic.
Open source Fluids, structural, thermal
Open source Dynamic
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CAGD Curve
s
Eng. curves CAD representation Fittin
g
Conclusions
Computer aided manufacturing
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Help to generate ISO code from CAD models
Manufacturing automatization
Control and inspection
CAD software admits CAM plugins
Example DELCAM within SolidWorks
Tool-path generation based on CAGD
An adequate tool-path can reduce time and forces.
Source: delcam.com
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CAGD Curve
s
Eng. curves CAD representation Fittin
g
Conclusions
CAM software
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Software Company Analysis
CATIA Mold &
Tooling Design
Dassault Systemes Machining, mold design, plastic
injection
NX CAM 9 Siemens Machining, inspection
DELCAM Delcam Machining and inspection
Blendef Open source Milling
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CAE Curve
s
Eng. curves CAD representation Fittin
g
Conclusions
Computer Aided Geometric Design
It is a discipline dealing with computational aspects ofgeometric objects G. Farin (2002)
How model engineering curves and surfaces in a computer
It became a scientific & engineering discipline in 1974
Try to reduce errors and time by the use of a pc
Main CAGD advances:
Theory of Bzier curves and B-spline techniques
Other CAGD applications:
Solid modelling, Geographic information systems, Medicalimaging, Computer gaming, Scientific visualization
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CAE Curve
s
Eng. curves CAD representation Fittin
g
Conclusions
Traditional vs. computer design
CAD reduces time and cost
it allows to represent real 3D models
No material and size limitations as paper planes
Lot of different computer graphic solutions
2D-Design
Technical illustrations
Metal sheetSolid modelling
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CAE Curve
s
Eng. curves CAD representation Fittin
g
Conclusions
Design software 2D
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Software Company web
AutoCAD AutoDesk AutoDesk.com
Corel Designer Corel Corp. Corel.com
PTC Creo Schematic PTC Corp. Ptc.com
MicroStation
Powerdraft
Bentley Bentley.com
LibreCAD Open source Librecad.org
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CAE Curve
s
Eng. curves CAD representation Fittin
g
Conclusions
Technical illustrations 2D & 3D
Software Company web
Adobe Illustrator Adobe adobe.com
Corel Draw Corel Corp. Corel.com
PTC Creo Illustrate PTC Corp. Ptc.com
InkScape Open source inkscape.org
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Source: Ptc corp.
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CAE Curve
s
Eng. curves CAD representation Fittin
g
Conclusions
Surface & Solid Modelling
Software Application Company web
RhinoCeros NURBS curves & surfaces Robert McNeel& Associates
rhino3D.com
SolidWorks, Catia Surfaces & Solid Modelling Dassault Systemes 3ds.com
Creo Elements Surfaces & Solid Modelling PTC Corp. Ptc.com
FreeCAD Solid Modelling Open source freecadweb.org
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Source: linuxaideddesign.
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CAE CAGD Eng. curves CAD representation Fittin
g
Conclusions
Curve models
Curve: A continuous map from one dimension to n-dimensionspace.
They have useful applications:
CAD Modelling uses:Define contours of orthographic projections
Define wireframe models in CAD: rotations and translations ofa curve profile generate revolution and swept surfaces.
Engineering curves: conics, Cycloid, Spirals, Helix & so on
Scientific visualization via approximation or interpolation
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Source: wikimedia
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CAE CAGD Eng. curves CAD representation Fittin
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Conclusions
Single valued curves
Definition: Graphic of a function
a vertical line cuts the curve once
Examples:
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CAE CAGD Eng. curves CAD representation Fittin
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Conclusions
Single valued curves: Problems
This representation is not valid after a rotation
Simple and useful curves are not single valued
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CAE CAGD Eng. curves CAD representation Fittin
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Conclusions
Parametric curves
Definition:
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CAE CAGD Eng. curves CAD representation Fittin
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Conclusions
Parametric curves
Advantages:
Solve the aforementioned drawbacks of single valued curves
Single valued (case x=t)parametric curve
Simple modelling and visualization:
1. give values (t)points c(t)
2. Join c(t) (linear segments)
Computer graphic cards draw 100 Msegments / s
Disadvantages: It is complicated to define areas
Therefore it is difficult to distinguish inside and outside
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CAE CAGD Eng. curves CAD representation Fittin
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Conclusions
Polynomial parametric curves
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c(t)= {x(t),y(t)}, x(t), y(t)are degreenpolynomials.
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CAE CAGD Eng. curves CAD representation Fittin
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Conclusions
Rational parametric curves
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c(t)= {x(t),y(t)}, x(t), y(t)are degreenrational functions.
c(t)= p(t)
w(t),q(t)
w(t)
, p(t), q(t), w(t)are degreenpolynomials.
Advantages of Polynomial-Rational Parametric Curves
Extremely fast point evaluation:
Additions, products (and division for rational)
Operations implemented within microprocessor
GFLOPS (G Floating Point Operations per Second)
Parametric Rational Polynomial
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CAE CAGD Eng. curves CAD representation Fittin
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Conclusions
Rational para. curves: Examples
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CAE CAGD Eng. curves CAD representation Fittin
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Conclusions
Implicit curves
Those plane curves / f(x,y)=0 Algebraic curvesf is a degree n polynomial
Advantages: inside-outside classification. Disadvantages:
It is difficult to model free form curves.
It is also complex to extend this representation to 3D.
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CAE CAGD Eng. curves CAD representation Fittin
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Conclusions
Definition: non algebraic curves
Main drawback: they do not admit an exact computerrepresentation
Polynomial-rational parametric curvesstandard CAD
representation
Solution:polynomial-rational fitting
Detection:intersection between a line and a transcendental
curveinfinite points
Examples:Logarithmic and Archimedes spiral, helix, catenary
Transcendental curves
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CAE CAGD Eng. curves CAD representation Fittin
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Conclusions
Complex shapes Piecewise parametric curves
Spline curves
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CAE CAGD Eng. curves CAD representation Fittin
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Conclusions
Spline continuity type.Mechanical applications: G0or G1tool paths (conventionalmachines), G2vehicles and High speed machining
Geometric continuity Gr
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CAE CAGD Eng. curves CAD representation Fittin
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Conclusions
Evaluation via Curvature k(s), s=Arc length parameterization
Quality assessment
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Smooth curve if k(s):
is a continuous function
has few and monotonic segments
ends at specified points
i i i l i
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CAE CAGD Curve
s
CAD representation Fittin
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Conclusions
Engineering curves
We have two main types of Engineering curves:
Curves with CAD representation:
Polynomial and rationalApplicationsDesign
Curves without CAD representation:
Transcendental curves (examples in the following slides)
CAD incorporation via interpolation or approximation
Lot of engineering applications
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CAE CAGD C CAD i Fi i C l i
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CAE CAGD Curve
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CAD representation Fittin
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Conclusions
Engineering curves: Trochoids
Curve described by a fixed point as a circle rolls
outside a circle (Epitrochoid)
inside a circle (Hypotrochoid)
They admit a rational representation
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CAE CAGD C CAD t ti Fitti C l i
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CAE CAGD Curve
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CAD representation Fittin
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Conclusions
Trochoids applications
Design of roots blower rotors.
They handles large quantities of air at a small pressure diff.
Wankel engine: few vibration & mechanical stress at high rpm
Design of gear tooth profiles
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Rotary blower
Mazdas Wankel engine
CAE CAGD C CAD t ti Fitti C l i
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CAE CAGD Curve
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CAD representation Fittin
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Conclusions
Catenary
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CAE CAGD C CAD t ti Fitti C l i
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CAE CAGD Curve
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CAD representation Fittin
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Conclusions
Clothoid spiral
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Curvature is proportional to distance.
Parameterization: Fresnel integrals (Transcendental curve)
Application: road design, roller coasters
Speed = Constant
c(t)=C(t)
S(t)
,C(t)= cos(
2 2 )dz
0
t
S(t)= sin( 2 2 )dz0t
.
CAE CAGD Curve CAD representation Fittin Conclusions
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CAE CAGD Curve
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CAD representation Fittin
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Conclusions
Involute or evolvent
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Definition:Path follow by the end of a string which is woundtangent to a profile.
It is a transcendental curve.
Application:design of gear teeth
CAE CAGD Curve CAD representation Fittin Conclusions
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CAE CAGD Curve
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CAD representation Fittin
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Conclusions
Offset
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Definition:parallel curve cd(t)
In general, it is a non polynomial/rational curve
cd(t)= c(t) dn(t),
n(t)normal
+ up, - down
.
CAE CAGD Curve CAD representation Fittin Conclusions
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CAE CAGD Curve
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CAD representation Fittin
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Conclusions
Offset: Applications
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Tool path and road design
Cam design
CAE CAGD Curve CAD representation Fittin Conclusions
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CAE CAGD Curve
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CAD representation Fittin
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Conclusions
Offset: representation problems
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Self-intersections
CAE CAGD Curve Eng curves Fittin Conclusions
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CAE CAGD Curve
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Eng. curves Fittin
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Conclusions
Standard CAD representation
We can incorporate polynomial curves intoCAD but there are different ways to describe a
polynomial.
Our first choice is the monomial basis, but is it
the best representation for CAD?
Next slides try to answer this question
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CAE CAGD Curve Eng curves Fittin Conclusions
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CAE CAGD Curve
s
Eng. curves Fittin
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Conclusions
Monomial basis
Definition:A way to describe a polynomial using a linearcombination of monomials
Main advantages:
Simple algebraic manipulation: additions, products,
derivation
OK to approximate functions around a point (Taylor, t = 0)
Efficient computing algorithm (Horner)
Everybody learned this representation in high school
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Monomialbasis: 1,t,t2 ,...,tn{ },
Polynomialrepresentation: a(t)= aiti
i=0
n
.
CAE CAGD Curve Eng curves Fittin Conclusions
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CAE CAGD Curve
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Eng. curves Fittin
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Conclusions
Monomial basis
Disadvantages:
Coefficients have not geometric meaning
Poor numerical properties: small errors in aigaps
Gaps and non geometric meaning are unacceptable for CAD
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CAE CAGD Curve Eng curves Fittin Conclusions
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CAE CAGD Curve
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Eng. curves Fittin
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Conclusions
Bzier curves
Provide a polynomial representation: where coefficients have geometric meaningControl
points
without numerical problemsno gaps
End control points = curve end points.
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CAE CAGD Curve Eng curves Fittin Conclusions
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CAE CAGD Curve
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Eng. curves Fittin
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Conclusions
Examples of Bzier curves
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CAE CAGD Curve Eng curves Fittin Conclusions
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CAE CAGD Curve
s
Eng. curves Fittin
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Conclusions
Bernstein polynomials
Definition:
Bzier: combination of control pts via Bernstein polynomials
biControl points
effect of biin the curve shape.
bipulls the curve(u=i/nMaximum )
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Bi
n(u) = ni
1 u( )n i ui .
b(u) = biBi
n
(u)i=0
n
.
Bi
n(u)
Bi
n(u)
CAE CAGD Curve Eng. curves Fittin Conclusions
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CAE CAGD Curve
s
Eng. curves Fittin
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Conclusions
Bzier curves: geometric properties
Affine invariance. Reposition-scale-rotation so ontransformation of control points
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Other Application
Typography:
S S S
CAE CAGD Curve Eng. curves Fittin Conclusions
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CAE CAGD Curve
s
Eng. curves Fittin
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Conclusions
End point interpolation. The curve pass through the end controlpointsno gaps
Linear precision. If bi are alignedline
Convex hull property. B(t) lies in the convex hull of the controlpolygon. Application: Interference checking.
Operationslike integration and derivation via additions,products and divisions of control points
Good numerical properties
Bzier curves: geometric properties
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CAE CAGD Curve Eng. curves Fittin Conclusions
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CAE CAGD Curve
s
Eng. curves Fittin
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Conclusions
Bzier curves: smoothness conditions
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CAE CAGD Curve Eng. curves Fittin Conclusions
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s
g
g
Bzier curve: Derivation
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CAE CAGD Curve Eng. curves Fittin Conclusions
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s
g
g
Bzier curves: Integration
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CAE CAGD Curve Eng. curves Fittin Conclusions
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s
g
g
Design Example: cam profile
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Hypothesis. Constant angular speed
Conditions:
Constraint forcesy() is C1
Avoid jerky()is C2
= cte
CAE CAGD Curve Eng. curves Fittin Conclusions
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s
g
g
Rational Bzier curves
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Several conic curves (ellipse, circle, hyperbola) can not be
represented by means of a polynomial Bzier curve.
Solution:Rational Bzier curvea perspective transformation
CAE CAGD Curve Eng. curves Fittin Conclusions
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Rational Bzier curves
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Effect of control points reposition Effect of weight modification
CAE CAGD Curve Eng. curves Fittin Conclusions
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Conic construction via Rational Bzier curves
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C on ic a rc
C en tr e ( ci rc le ), f oc us : C
Angle 2Weights: w0=w2=1,w1
w1
[0,1] E ll ip se (w2= cos( ) , C i r cl e )
1 Parabola
[1,] Hyperbola
CAE CAGD Curve Eng. curves Fittin Conclusions
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Conics applications
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Source: Alternative-energy-tutorials.com
Source: radartutorial.eu
CAE CAGD Curve Eng. curves Fittin Conclusions
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B-spline curves
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Definition: Spline of Bzier curvessame Bzier properties
Applicationcurves with complex shape
d(u) = diN
i
n(u)
i=0
n
. Nin
(u)piecewisebasis
CAE CAGD Curve Eng. curves Fittin Conclusions
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B-spline versus Bzier
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Bzier B-spline
DesignPseudo local shape
controlLocal control
Continuity
guaranteeComplex Simple (polynomial degree - 1)
Evaluation costN control points increase
computational cost
It doesnt dependon number
of control points
Implementation Straightforward Complex
CAE CAGD Curve Eng. curves Fittin Conclusions
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Example: Degree n=3 B-spline
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Continuityn-1=2
Number of control pointsp=7
9 knots, Number of pieces mm=p-n=4
CAE CAGD Curve Eng. curves Fittin Conclusions
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NURBS: Non uniform rational B-spline
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Definition: Spline of Rational Bzier curves with a non uniform
knot sequence.
CAE CAGD Curve Eng. curves CAD representation Conclusions
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s
Fitting
Definition:Construction of a NURBS that approximates a non
polynomial-rational curve Main fitting techniques:
Approximation. NURBS passes close to a set of points
Interpolation. NURBS satisfies a set of conditions
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CAE CAGD Curve Eng. curves CAD representation Conclusions
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s
Lagrange Interpolation
Lagrange interpolation: the interpolation curve passes
through an uniformly distributed set of points Drawbacks:
Runges phenomenon
New datare-compute
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Original curve
Degree 4 Lagrange Interpolation
Degree 8 Lagrange Interpolation
Source: wikipedia
CAE CAGD Curve Eng. curves CAD representation Conclusions
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s
Hermite Interpolation
Definition: the interpolation curve passes through an non uniformly
distributed set of pointsdata: end points and their derivate Applications:
Approximation of boundary problems
Useful to control continuity
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CAE CAGD Curve Eng. curves CAD representation Conclusions
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s
Approximation
The construction passes close to a set of data.
Complex implementation: iterative algorithm.
Basic approximation technique: Least squared method.
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CAE CAGD Curve Eng. curves CAD representation Fittin
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Conclusions
CAGD decrease design errors & contribute toreduce the time to take the market.
CAGD is devoted to Engineering design via curvesand surfaces.
There are different types of curve representations Bzier B-Spline NURBS is the CAD standard
Advantages: local shape control, fast evaluation andstraightforward implementation of differentoperations
To incorporate transcendental curvesInterpolation or approximation (fitting)
Recommended