4 - geometric modeling, curve entities
TRANSCRIPT
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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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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
Engineering applications of parabola
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
Engineering applications of parabola
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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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Curve Enti t ies Synthet ic Curves
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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Curve Enti t ies Synthet ic Curves
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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Synthet ic Curves Bezier Curve
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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Synthet ic Curves Bezier Curve
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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Synthet ic Curves Bezier Curve
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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Synthet ic Curves Bezier Curve
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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Synthet ic Curves Bezier Curve
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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Synthet ic Curves Bezier Curve
The designer should be able to predict the shape of the curve once its
control points are given.
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Synthet ic Curves Bezier Curve
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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
Synthet ic Curves B-Spl ine Curve
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Synthet ic Curves B-Spl ine Curve
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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Synthet ic Curves B-Spl ine Curve
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.
Synthet ic Curves B-Spl ine Curve
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
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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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Spline Command in SW
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
Spline Command in SW
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Spline Command in SW
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
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Mathematical
Equation
x +sin(x)
Spline in Creo
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Spline in Creo
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Double click any
point to changethe type of spline
S li i C
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Spline in Creo
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Interpolation
Creo
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Creo
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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
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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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Spline in NX5 (Unigraphics)
Change Tangent Direction
Change Curvature
Change Tangent Magnitude
Manipulating the spline curve
Spline in NX5 (Unigraphics)
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Spline in NX5 (Unigraphics)
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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Spline in NX5 (Unigraphics)
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
Spline in NX5 (Unigraphics)
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Spline in NX5 (Unigraphics)
Spline
You can create splines using one of several methods.
There are four creation methods for splines:
Splin e in NX5 (Unigraphics)
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Splin e in NX5 (Unigraphics)
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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Splin e in NX5 (Unigraphics)
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
Ken Youssefi Mechanical Engineering Dept. 76
1select the two
end points start
2select the Apex
3Select the Rho () value
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