PTT105: Engineering Graphics1 PREPARED BY: NOR HELYA IMAN BT KAMALUDIN

Lecture 5: Curves And Surfaces, and

Geometric Modeling

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PREPARED BY: NOR HELYA IMAN BT KAMALUDIN

INTRODUCTION ABOUT GEOMETRIC MODELING…

CAD tools have been defined as the melting pot of three disciplines: design, geometric modeling, and computer graphic.

A geometric model should be unique and complete to all engineering functions, from documentation to engineering analysis to manufacturing.

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CURVES Geometric description of curves

defining an object can be tackled in several ways.

A curve can be described by arrays of coordinate data or by an analytic equation.

Majority of the curves were circles, but some were free-form.

Those are curves arising from applications such as ship hull design to architecture.PTT105: Engineering

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When they had to be drawn exactly, the most common tool was a set of templates known as French curves.

French curve:

carefully designed wooden curves

consist of pieces of conics and spirals.

A conic section is a curve obtained by intersecting a cone with a plane.

3 types of conic section are ellipse, parabola, hyperbola.PTT105: Engineering

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1. Ellipse An ellipse is a smooth closed curve which is

symmetric about its horizontal and vertical axes.

In geometry, an ellipse is results from the intersection of a cone by a plane in a way that produces a closed curve.

Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis.

An ellipse is also the locus of all points of the plane whose distances to two fixed points add to the same constant.

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Example of an ellipse:

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2. Parabola

Parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface.

For parabola, the locus of points are equidistant from a given point (the focus) and a corresponding line (the directrix) on the plane.

The parabola has many important applications such as in designing automobile headlight reflectors and ballistic missiles.

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Example of a parabola:

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3. Hyperbola

The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a cone.

A hyperbola is an open curve with two branches, the intersection of a plane with both halves of a double cone.

The two pieces of branches formed mirror images of each other and resembling two infinite bows.

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Example of a hyperbola:

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SPLINES

Another mechanical tool of curve, called a spline was also used.

Spline is a flexible strip of wood that are held in place and shape by metal weights, known as ducks.

A spline “tries" to bend as little as possible, resulting in shapes which are both aesthetically pleasing and physically optimal.

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Spline Curve Description:

Spline curve is one of the most fundamental parametric curve forms. It is a mathematical counterpart to a mechanical spline.

Features:

Curve that can minimize certain functionality

Act as piecewise polynomial (or rational polynomial) curves with certain smoothness properties.

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Example of Splines

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SURFACES Surface models have an infinitely thin

computer-calculated surface between their edges.

Although they appear to be solid, they are an empty shell.

The model on the left in the figure shows the previous object as a surface model.

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They appear to be a round hole through the model.

The hole is actually a tube simulating the surface of a hole, as can be seen in the center model, in which a surface panel has been removed.

Surface model often use wireframe models as a frame for their surfaces.

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Rectangular Surfaces

Rectangular surfaces are a map of a rectangular domain into 3D parametric surface.

Mapping the rectangular domain to a 2D parametric surface, resulting in a distortion of that rectangle.

For example, if we embed a curve in this domain rectangle, we will obtain a deformed curve.

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Application of rectangular surfaces

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SOLID GEOMETRIC MODELING

• Solid models have both edge and surfaces, plus computer-calculated mass under their surfaces.

• Solid models provides mass property information: volume, center of gravity, mass moment of inertia.

• The example of solid model is shown below.

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It appears very similar to a surface model

But, if we sliced in in half, as demonstrated in the center model, it show truly solid.

Basic rule for solid modeling – all surfaces must touch another surface

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Example of Solid Modeling

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Solid Modeling

• 3 different types of solid modeling:

– Primitive modeling

– Constructive solid geometry (CSG)

– Feature-based modeling (FBM)

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1. Primitive Modeling

• Objects described using basic geometrical forms.

• Common geometric primitives.

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Example:

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2.Constructive Solid Geometry (CSG)

• More flexible and powerful than primitive.

• Allow Boolean Operations:

- union, difference & intersection

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Boolean operation

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Example using CSG:

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Example using CSG:

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3. Feature-based Modelling (FBM)

• 3D model is built using series of features, such as hole, slot, square block, etc.

• Each feature can be independent or linked to other feature.

• The geometry of each feature is controlled by modifiable constraints and dimensions.PTT105: Engineering

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FBM: 3D operations

Basic concept– 2D cross-section or profile is

produced– Depth is given to the profile

• Generally 4 types**– Extrude– Revolve– Sweep– Blend

**different terms might be used in different software/books

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1. 3D Ops: Extrude

A linear sweep, where the profile is given a depth in straight line, perpendicular to the profile plane

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2. 3D Ops: Revolve

The profile is rotated around a defined axis, 0 – 360 degree

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3. 3D Ops: Sweep

The new command and is similar to the EXTRUDE command, but it concentrates on using paths to define the direction of the extrusion.

This command SWEEP a 2D object along a path.

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3D Ops: Sweep & Blend

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Steps in building 3D object

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Examples of FBM + Boolean

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