Curve Entities

1
Curve Entities
All CAD/CAM systems provide users with curve
entities
Curve entities are divided into two categories,
2
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).
3
Conic Curves - Parabolas
Light rays
Light rays
Applications of parabola
Light source
Eye piece
Searchlight mirror
Telescope mirror
4
Conic Curves - Hyperbolas
A hyperbola is the curve created when a plane
parallel to the axis and perpendicular to the
base intersects a right circular cone.
Element (side)
Hyperbola
Orthographic view
5
Conic Curves - Hyperbolas
Cooling Towers of Nuclear Reactors
The hyperboloid is the design standard for all
nuclearcooling 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.
6
Conic Curves - Ellipse
An ellipse is the curve created when a plane cuts
all the elements (sides) of the cone but its not
perpendicular to the axis.
7
Conic Curves - Ellipse
In New York's Grand Central Station, underneath
the main concourse theres a special place known
as The Whispering Gallery where 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.
Other famous examples are found in Mormon
Tabernacle in Salt Lake, St Paul's Cathedral in
London and St Peter's Basilica in Rome.
8
Conic Curves - Ellipse
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 for differences in
the speed and force applied
9
Conic Curves
10
Curve Entities Synthetic 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, .
11
(No Transcript)
12
Radio
Thermos
13
(No Transcript)
14
Synthetic Curves Freeform Curves
For CAD systems, three types of freeform curves
have been developed,
15
Synthetic Curves Bezier Curve

• 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 points
  define the shape of the curve.

• The curve is always tangent to the first and the
  last polygon segment. The curve shape tends to
  follow the polygon shape.

Characteristic polygon
16
Synthetic Curves Bezier Curve
Modifying the curve by changing one or more
vertices of its polygon (control points).
17
Synthetic Curves Bezier Curve
A desired feature of the Bezier curve or any
curve defined by a polygon is the convex hull
property. This property guarantees that curve
lies in the convex hull regardless of changes
made in control points.

• The curve never oscillates wildly away its
  defining control points

• The size of the convex hull is the upper bound on
  the size of the curve itself.

18
Synthetic Curves Bezier Curve
Disadvantages of Bezier curve over the cubic
spline curve

• The curve lacks local control, if one control 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

19
Synthetic Curves Bezier Curve
The designer should be able to predict the shape
of the curve once its control points are given.
20
Synthetic Curves Bezier Curve
21
Synthetic Curves B-Spline Curve
B-spline curves are powerful generalization of
Bezier curve.

• The curves have the same characteristics as
  Bezier curves

• 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.

22
Synthetic Curves B-Spline Curve
Local control of B-spline curve
23
Synthetic Curves B-Spline Curve
Effect of the degree of B-spline curve on the
shape
Tangent to the curve at the midpoints of all the
internal polygon segments
24
Synthetic Curves B-Spline 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)
25
SolidWorks Commands Parabola and Spline
26
Parabola Command in SW
Parabola
Start
End
Focus
Vertex
27
Spline Command in SW
B-Spline Curve using interpolation SolidWorks
generates a smooth curve passing through all data
points
28
Spline Command in SW
The spline shape can be modified by manipulating
the tangent vector for each point.
Data point 3 is selected
29
Spline Toolbar in SW
30
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
31
Spline in NX5 (Unigraphics)
32
Spline in NX5 (Unigraphics)
Manipulating the spline curve
33
Spline in NX5 (Unigraphics)
B-Spline Curve, extrapolation method (does not
pass thru points)
Closed option
34
Spline in NX5 (Unigraphics)
Fit Spline

• degrees and segments
• degrees and tolerance
• a template curve

35
Example - Spline in NX5 (Unigraphics)
Five data points using 3rd order polynomial to fit
36
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.
37
Example - Parabola Command in NX
38
Hyperbola Command in NX
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 through this
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 to half the length of these axes. The
relationship between these two axes determines
the slope of the curve.
39
Example - Hyperbola Command in NX
Hyperbola
Revolved feature
40
General Conic Curve Command in NX
Overview of Conics Conics are created
mathematically by sectioning cones. 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).
41
Example - Hyperbola Command in NX
This 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.