4 - Geometric Modeling, Curve Entities

8/10/2019 4 - Geometric Modeling, Curve Entities

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Ken Youssefi Mechanical Engineering Dept. 1

Curve Enti t ies

Curve entities are divided into two categories,

Analyt ic

Points, lines, arcs, fillets, chamfers, and conics

(ellipses, parabolas, and hyperbolas)

Synthetic (freeform )

Includes various types of spline; Cubic spline, B-spline

and Bezier curve

All CAD/PLM software provide users with curve entities


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Curve Enti t ies

Methods u ti l ized by CAD/CAM systems to create cu rve

Definingpo in ts Geometric modifiers


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Curve Enti t ies

Methods of def in ing po in tsGeometric modifiers


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Curve Enti t ies

Methods of def in ing l ines


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Curve Enti t ies

Methods of d ef in ing l ines


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Curve Enti t iesMethods of d ef in ing circles


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AutoCAD

SW

Creo


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Curve Enti t iesMethods of def in ing ell ipses and parabolas


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SW

AutoCAD

Creo


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Conic Curves - Parabolas

Conic curves or conics are the curves formed by the intersection of a plane

with a right circular cone (parabola, hyperbola and sphere).

A parabolais the curve created when a plane intersects a right circularcone parallel to the side (elements) of the cone

Cutting plane

Parallel

Elements


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Con ic Curv es - Parabo las

Directrix

Parabola

Focus

Axis

Parabola is defined as the set of points in a

plane that are equidistant from a point

(focus, F) and a fixed line (directrix, 1).

PP = PF

Constructing a parabola using the

Tangent method

P

P

FV

A

AA = AF

A


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Con ic Curv es - Parabo las

Engineering applications of parabola

Light source

Searchlight mirror

Light rays

Telescope mirror

Eye piece

Light rays

A parabola revolved about its

axis creates a surface called

paraboloid.

Satell i te dish antenna


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An auditorium ceiling in shape of paraboloid

reduces reverberations if the speaker standsnear the focus


Beam of uniform strength Weightless flight trajectory

Parabola

Zero g

Zero g

Zero g

Load

Parabola

= Mc/I = M / Z


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Parabolic Solar Mirror

designed by MIT.

Perfect mirror with

zero distortion, soundand light waves



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Odeillo Font-Romeux, France, location of the world's largest solar furnace, a parabolic reflector

that focuses solar radiation at a point to generate extremely high temperatures. Sixty-three flat

mirrors, installed on eight terraces, reflect the solar radiation on the eight-story high parabolic

reflector. Every position is calculated so that the reflected light is parallel to the symmetry axis of

the paraboloid. The reflector then concentrates the energy in the focal zone about 18 meters in

front of the paraboloid, The typical range of available temperature is from 800to 2500C (1475to 4500F), with a maximum reachable temperature of approximately 3800C (6850F). These

temperatures correspond to a maximum thermal power of about 1000 kW.


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Conic Cu rves - Hyperbo las

A hyperbolais the curve created when a

plane parallel to the axis and perpendicularto the base intersects a right circular cone.

Hyperbola is defined as the set ofpoints in a plane whose distances

from two fixed points (foci, B1, B2) in

the plane have constant differences.

d2d1 = constant = 2a

P1

a


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Dulles Airport, designed by Eero Saarinen, is

in the shape of a hyperbolic paraboloid

Cooling Towers of Nuclear Reactors

The hyperboloid is the design standard for all nuclear

cooling towers. It is structurally sound and can be

built with straight steel beams.

For a given diameter and height of a tower and a

given strength, this shape requires less material than

any other form.

Conic Curves - Hyperbo las


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Calgry skyline and Pengrowth

Saddledome, July 23, 2005

Munich, Olympia buildings


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Con ic Curves - El lipse

An el l ipseis the curve created when a

plane cuts all the elements (sides) of the

cone but its not perpendicular to the axis.

Ellipse is defined as the set of points in a

plane for which the sum of the distancesfrom two fixed points (foci) in the plane is

constant

AD + DC = AB + BC


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The Statuary Hall in the Rotunda (Capitol

Building in Washington D.C.) has a ceilingcurved as an ellipse. It has been suggested

that after John Quincy Adams left presidency

and became a member of the House, he

would sit in one focus point of the ellipsoid

and listen to the other party located near the

other focus point. The place is labeled in the

floor by a brass name tag.

In New York's Grand Central Station, underneath the

main concourse theres a special place known as The

Whispering Gallerywhere the faintest murmur can be

heard 40 feet away across the busy passageway.Look for a place where two walkways intersect, and a

vaulted roof forms a shallow dome. Take a friend and

pick diagonal corners. Turn your faces to the wall and

start talking. It's a popular spot for marriage proposals.

Conic Curves - El lipse

Other famous examples are found in Mormon Tabernacle

in Salt Lake, St Paul's Cathedral in London and St Peter's

Basilica in Rome.


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Pool with elliptical roof

Ellipse wings, gives upto 30% increase in

power compared to the

traditional planes


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Conic Curves - El lipse

Some tanks are in fact elliptical (not circular) in cross section. This gives them

a high capacity, but with a lower center-of-gravity. They're shorter, so that they

can pass under a low bridge. You might see these tanks transporting heating

oil or gasoline on the highway

Ellipses (or half-ellipses) are sometimes used as fins, or airfoils in

structures that move through the air. The elliptical shape reduces drag .

On a bicycle, you might find a chainwheel (the gear that is connected to the

pedal cranks) that is approximately elliptical in shape. Here the difference

between the major and minor axes of the ellipse is used to account fordifferences in the speed and force applied

Elliptical gears are used for certain applications


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Conic Curves


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Curve Enti t ies Synthet ic Curves

Analytical curves are usually not sufficient to meet the design requirements

of complex mechanical parts, car bodies, ship hulls, airplane fuselages and

wings, shoe insoles, propeller blades, bottles, plastic enclosures for

household appliances and power tools, .


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Radio Thermos

Coffee Press


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Mathematically, synthetic curves represent the problem ofconstructing a smooth curve given a set of data points. There

are two methods of curve fitting;Polynomialsand Spl ines

Polynomialgiven a set of data points find a function of

order nthat best presents the curve passing through all the

data points.

Spl ines this method of curve fitting works with the basic

assumption that a cubic function can be passed between

any two points. And curve segments can be connected

using smoothing constraints.


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Order of continuity of curves

A complex curve is molded by several curve segments pieced together

end-to-end. Several continuity requirements can be specified at the data

points to impose various smoothness on the resulting curve; zeroordercontinuity yields position-continuous curve, first-order continuity implies

slope, second-order continuity imposes curvature-continuous curve

A cubical polynomial is

the minimum order

polynomial that can

guarantee the

generation of the curve.

P(x)=Cixi

i= 0

3


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Curve Ent i t ies - Synthet ic Curves

Methods of def in ing

synth etic curves


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Synthet ic Curves Freeform Curves

If the curve is created by smoothly connecting the control points,

the process is called interpolat ion.

If the curve is created by drawing a smooth curve passing through

some control points, but not all of the control points, the process is

called approximat ion (Extrapolat ion).

Control point


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Synthet ic Curves Freeform Curves

For CAD systems, three types of freeform curves have been developed,

B-spline curve

Bezier curve

Cubic spline


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Synthetic Curves Cubic Spl ine

Hermite Cubic Spline

Cubic splines use cubic polynomials (3rdorder polynomials). The

polynomial has four coefficients and needs four conditions to evaluatethe coefficients. The Hermit cubic spline uses two data points at its

ends and two tangent vectors at these points.

The parametric equation of a cubic spline in an expanded vector form

can be written as;

P0

and P1

are the end points

and P0and P1are the

tangent vectors. For planar

spline tangent vectors can

be replaced by slope.


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Synthetic Curves Cubic Spl ine

The control of the curve is not very obvious from the input data. Changing

the data points (end points) and the slope, changes the entire shape of the

spline. This does not provide an intuitive feel required for design, not very

popular.


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Synthet ic Curves Bezier Curve

The Bezier curve is defined by a set of data points. The curve could be

created using interpolation (passing thru the points) or extrapolation.

Some CAD system provide both option, others offer only interpolation.

The slope and shape of the Bezier curve is controlled by itsdata

points. Unlike the cubic curve that the Tangent vector controls the

shape. This provides the designer with a much better feel for the

relationship between the input points and the output curve.

The cubic spline is a third order curve, whereas the order ofthe Bezier curve is defined by the number of data points and

is variable. n+ 1 data points define nthdegree curve , whichpermits higher order continuity. CAD systems limit the

degree of the curve.

S th t i C B i C


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Mathematically, for n+ 1 control points, the Bezier curve is defined by thefollowing polynomial of degree n:

Point on the

curve

Control point Bernstein polynomials

The Bernstein polynomial serves as the blending function, C(n, i) is

the binomial coefficient.


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The curve is always tangent to the first and the last polygon

segment. The curve shape tends to follow the polygon shape.

The data points of the Bezier curve are called control points. Only

the first and the last control points lie on the curve. The other pointsdefine the shape of the curve.

Characteristic polygon


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Modifying the curve by changingone or more vertices of its

polygon (control points).

Modifying the curve by keeping

the polygon fixed and specifying

multiple coincident points at a

vertex (control point)

S th t i C B i C


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A desired feature of the Bezier curve or any curve defined by a polygon is

the Convex hul l property. This property guarantees that curve lies in the

convex hull regardless of changes made in control points.

The curve never oscillates wildly away from its defining control

points

The size of the convex hull is the upper bound on the size of the

curve itself.


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Disadvantages of Bezier curve over the cubic spline curve

The curve lacks local control, if one control point is changed,

the whole curve changes (global control)

The curve degree depends on the number of data points,

most CAD software limit the number of points used to define

a Bezier curve

Cubic curve Bezier curve


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The designer should be able to predict the shape of the curve once its

control points are given.


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Synthetic Curves B -Spl ine Curve

B-spline curves are powerful generalization of Bezier curve.

The curves have the same characteristics as Beziercurves

They provide local control as opposed to the global control

of the curve by using blending functions which provides

local influence.

The B-spline curves also provide the ability to separate the

curve degree from the number of data points.

S th t i C B S l i C


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Synthet ic Curves B-Spl ine Curve

Local control of B-spline curve



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Effect of the degree of B-spline curve on the shape

As the degree decreases, the generated B-spline curve moves closer to its

control polyline.

7 degree 5 degree 3 degree

Tangent to the curve at the midpoints of

all the internal polygon segments

Midpoint


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Effect of point multiplicity of B-spline curve on the shape

Multiple control points induce regions of high curvature, increase the number of

multiplicity to pull the curve towards the control point (3 points at P3)

S th t i C B S l i C


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B-spline curve property allows us to design complex shapes with lower degree

polynomials. For example, the right figure below shows a Bezier curve with the

same set of control points. It still cannot follow the control polyline nicely eventhough its degree is 10.


Bezier curveB-spline curve


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SolidWorks Commands Parabo la and Spline

2010/11 version same as 2012/13 version

2013/14 version


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SW

2012/2013

Select Tools

and then

Sketch Entities

SolidWorks Commands Spl ine on Surface


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SolidWorks Commands Spl ine on Surface


The only option to sketch

on a curved surface isSpline on Surface

Parabola Command in SW


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Parabola Command in SW

1 - Select the Focus point

2 - Select the Apex

3 - Select the Start point, and drag

to the End point

StartEnd

Parabola

Start

End

Focus

Vertex

Spline Command in SW


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Cubic Spl in e Curve SolidWorks generates a smooth curve passing through all

data points. The shape can be manipulated by control points and tangent vectors.

Point #2 modified

from (1,1) to (1,2)

X & Y coordinates of

the point, Y changed

from 1 to 2.

Data point #2

Point #

Tangent Driving box is checked off



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The spline shape can be modified by

manipulating the tangent vector for each

point.Data point #3 is selected

Size (weight)

angle

Spl ine Too lbar in SW


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Spl ine Too lbar in SW

Curves def ined by equat ions


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Curves def ined by equat ions


Mathematical

Equation

x +sin(x)

Spline in Creo


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Spline in Creo


Double click any

point to changethe type of spline

S li i C


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Spline in Creo


Interpolation

Creo


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Creo


Modification is done by dragging

a control point, cubic spline (local

control)

User can control the

slope at the end.

5 control points (data points)

S C


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Spline in Creo


B-Spline

Spline in NX5 (Unigraphics)


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Spline in NX5 (Unigraphics)All splines created in NX are Non Uniform Rational B-splines (NURBS). In NX

the terms "B-spline" and "spline" are used interchangeably.

Splines


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Splin e in NX5 (Unigraphics)

Use this command to interactively create an associative or non-associative

spline. You can create splines by dragging defining points or poles. You can

assign slope or curvature constraints at given defining points or to end poles.

Making splines associative preserves their creation parameters and links them

parametrically to parent features

Studio Spl ine

Interpolation

Extrapolation

Cubic

polynomial


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Change Tangent Direction

Change Curvature

Change Tangent Magnitude

Manipulating the spline curve



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B-Spl ine Curve, extrapolat ion method(does not pass thru points)

Closed option

Open option

Convex hull

Spline in NX5 (Unigraphics )


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degrees and segments

degrees and tolerance

a template curve

This option lets you create a spline by fitting it to specified data

points. The data points can reside in a set of chained points, or on faceted

bodies, curves, or faces. You can set endpoint and inner continuity

constraints, and you can control the accuracy and shape of the fit by

specifying:

Fit Spline

Examp le - Spl ine in NX5 (Unigraphics)


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Examp le Spline in NX5 (Unigraphics)

Five data points using 3rdorder

polynomial to fit

A Fit Spline created on a faceted Body

Five data points using 4th order

polynomial to fit



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Spline

You can create splines using one of several methods.

There are four creation methods for splines:



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Causes the spline to gravitate towards each data point (that is, pole), but

not pass through it, except at the endpoints.

By Poles

The spline passes through a set of data points.Through

Points

S li i NX5 (U i hi )


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Fit A specified tolerance is used in "fitting" the spline to its data points; the

spline does not necessarily pass through the points.

Perpendicular

to Planes

The spline passes through and is perpendicular to each plane in a set.

Parabola Command in NX


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Parabola Command in NX

A parabola is a set of points equidistant from a point (the focus) and a line

(the directrix), lying in a plane parallel to the work plane. The default parabola

is constructed with its axis of symmetry parallel to the XC axis.

To create a parabola:

Indicate the vertex for the parabola using the Point Constructor.

Define the creation parameters of the parabola.

Example Parabola Command in NX


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Example - Parabola Command in NX

Hyperbo la Command in NX


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This option allows you to create a hyperbola. By definition, a

hyperbola contains two curves - one on either side of its center. In NX,

only one of these curves is constructed. The center lies at the

intersection of the asymptotes and the axis of symmetry passes throughthis intersection. The hyperbola is rotated from the positive XC axis

about the center and lies in a plane parallel to the XC-YC plane.

To create a hyperbola:

Indicate the center of the hyperbola

using Point Constructor.Define the parameters of the

hyperbola.

A hyperbola has two axes: a

transverse axis and a conjugate

axis. The semi-transverse and

semi-conjugate parameters refer tohalf the length of these axes. The

relationship between these two

axes determines the slope of the

curve.

E l H b l C d i NX


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Example - Hyperbo la Command in NX

Revolved feature

Hyperbola

Example General Conic Command in NX


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pThis option lets you create a conic section by defining five coplanar points.

Define the points using the Point Constructor. If the conic section created is an

arc, an ellipse, or a parabola, it will pass through the points starting at the first

point and ending at the fifth.

Revolved

feature

General Con ic Curve Command in NX


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The General Conicoption lets you create conic sections by using

either one of the various loft conic methods or the general conic equation.

The resulting conic is either a circle, an ellipse, a parabola, or ahyperbola, depending on the mathematical results of the input data.

Overview of Conics

Conics are created

mathematically by sectioningcones. The type of curve that

results from the section depends

on the angle at which the section

passes through the cone. A conic

curve is located with its center at

the point you specify, in a plane

parallel to the work plane (the

XC-YC plane).


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2013/14 version

Creating a conic in SolidWorks is very simple. It builds much like a 3-

point-arc, but instead of adjusting a radius value, we adjust a parameter

called Rho (). If you imagine the conic as a rounded corner, then Rho isthe ratio of the distance of the peak of the rounded corner to the sharp

corner (D1/D2). This gives us an intuitive way to adjust the curvature of

the conic without having to delve into which type of conic section it is, or

what its mathematical eccentricity is.

SolidWorks Conic Command


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SolidWorks Conic Command


1select the two

end points start

2select the Apex

3Select the Rho () value


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4 - Geometric Modeling, Curve Entities