ken youssefimechanical engineering dept. 1 curve entities curve entities are divided into two...

Ken Youssefi Mechanical Engineering Dept. 1

Curve Entities

Curve entities are divided into two categories,

AnalyticPoints, lines, arcs, fillets, chamfers, and conics (ellipses, parabolas, and hyperbolas)

Synthetic

Includes various types of spline; Cubic spline, B-spline and Bezier curve

All CAD/CAM systems provide users with curve entities


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 parabola is the curve created when a plane intersects a right circular cone parallel to the side (elements) of the cone

Cutting plane

Parallel


Conic Curves - Parabolas

Applications of parabola

A parabola revolved about its axis creates a surface called paraboloid. An auditorium ceiling in shape of paraboloid reduces reverberations if the speaker stands near the focus

Light source

Searchlight mirror

Light rays

Telescope mirror

Eye piece

Light rays

Beam of uniform strength Weightless flight trajectory

Parabola

Zero g

Zero g

Zero g

Load

Parabola


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


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 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.

Conic Curves - Hyperbolas


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.


The Statuary Hall in the Rotunda (Capitol Building in Washington D.C.) has a ceiling curved 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 there’s 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.

Conic Curves - Ellipse

Other famous examples are found in Mormon Tabernacle in Salt Lake, St Paul's Cathedral in London and St Peter's Basilica in Rome.


Conic Curves - EllipseSome 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

Elliptical gears are used for certain applications


Conic Curves


Curve Entities – Synthetic CurvesAnalytical 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, ….


Radio Thermos

Coffee Press


Synthetic Curves – Freeform Curves

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

B-spline curve

Bezier curve

Cubic spline


Synthetic 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 points define the shape of the curve.

Characteristic polygon



Modifying the curve by changing one 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)


Synthetic Curves – Bezier CurveA 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.



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

Cubic curve Bezier curve


Synthetic Curves – Bezier CurveThe designer should be able to predict the shape of the curve once its control points are given.


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.



Local control of B-spline curve


Synthetic Curves – B-Spline CurveEffect of the degree of B-spline curve on the shape

Tangent to the curve at the midpoints of all the internal polygon segments

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

7 degree 5 degree 3 degree



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)


SolidWorks Commands – Parabola and Spline


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 SWB-Spline Curve using interpolation – SolidWorks generates a smooth curve passing through all data points

Point #2 modified

X & Y coordinates of the point

Data point #2

Point #


Spline Command in SWThe spline shape can be modified by manipulating the tangent vector for each point.

Data point #3 is selected

Size (weight)

angle


Spline Toolbar in SW


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


Spline 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 Spline



Change Tangent Direction

Change Curvature

Change Tangent Magnitude

Manipulating the spline curve


Spline in NX5 (Unigraphics)B-Spline Curve, extrapolation method (does not pass thru points)

Closed option

Open option



• 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


Example - Spline in NX5 (Unigraphics)Five data points using 3rd order polynomial to fit

A Fit Spline created on a faceted Body

Five data points using 4th order polynomial to fit


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


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.

Hyperbola Command in NX


Example - Hyperbola Command in NX

Revolved feature

Hyperbola


General Conic Curve Command in NX

The General Conic option 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 a hyperbola, depending on the mathematical results of the input data.

Overview of ConicsConics 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).


Example - Hyperbola Command in NXThis 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

ken youssefimechanical engineering dept. 1 curve entities curve entities are divided into two...

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