curve, surface and solid

8/14/2019 Curve, Surface and Solid

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Curve Entities

All existing CAD systems provide users

with curve entities, which can be divided

into analytic and synthetic entities.

Analytic entities are points, lines, arcs and

circles, fillets and chambers, and conies

(ellipses, parabolas, and hyperbolas).

Synthetic entities include various types ofspline (cubic spline and B-spline) and

Bezier curves.


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Surface


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Shape design and the representation of complex objectssuch as car, ship, and airplane bodies as well ascastings cannot be achieved utilizing the curves . In suchcases, surfaces must be utilized to describe objectsprecisely and accurately. We create surfaces, and thenwe use them to cut and trim solid features and primitivesto obtain the models of the complex objects. Surfacecreation usually begins with data points or curves.

Surface creation on CAD/CAM systems usually requirescurves as a start. A surface might require two boundarycurves, as in the case of a ruled surface that we cover inthis chapter. All curves can be used to generate

surfaces. In order to visualize surfaces on a computerscreen, a mesh, say m x n in size, is usually displayed.The mesh size is controllable by the user.


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Figure 7.1 Construction of improper and proper surfaces.

Visualization of a surface is aided by the addition of artificial fairing

lines (called mesh), which crisscross the surface and so break it up into

a network of interconnected patches. The default setting of a CAD

system does not display a surface mesh the surface is displayedwith its four boundary curves only. In such a case, the mesh size is 2 x

2, (All surfaces that we create define rectangular patches.) We can

change the default mesh size. CAD systems provide users with a menu

that allows them to specify the mesh size.


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Figure 7.2 shows surfaces of revolutions with mesh sizes of

4 x 4 and 20 x 20. It should be mentioned that a finer mesh

size for a surface does not improve its mathematical

representa-tion; it only improves its visualization. Finally,some CAD/CAM systems do not permit their users to

delete curves used to create surfaces unless the latter are

deleted first.


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Following are descriptions of major surfaces:

1. Plane surface: It is the simplest surface. It

requires three non-coincident points to define an

infinite plane. The plane surface can be used to

generate cross sections by intersecting a solid

with it. Figure 1 shows planar surfaces.2. Ruled (lofted) surface: It is a linear surface. It

interpolates linearly between two boundary

curves that define the surface (rails). Rails can

be any curves. This surface is ideal forrepresenting surfaces that do not have any

twists or kinks. Figure 2 gives some examples.


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Figure 1. Plane surface

Figure 2. Ruled surface


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3. Surface of revolution: It is an axisymmetric surface thatcan model axisymmetric objects. It is generated byrotating a planar curve in space about the axis of

symmetry a certain angle as shown in Figure 3.4. Tabulated cylinder: It is a surface generated by

translating a planar curve a certain distance along aspecified direction (axis of the cylinder or directrix) asshown in Figure 4. The plane of the curve is

perpendicular to the directrix. This surface is not lit-erallya cylinder. It is used to generate extruded surfaces thathave identical cross sections.


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Figure 3. Surface of revolution

Figure 4. Tabulated cylinder


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5. Beziersurface: It is a surface thatapproximates or interpolates given input data. It

is different from the previous surfaces in that it isa synthetic surface. It extends the Bezier curveto surfaces. It is a general surface that permitstwists, and kinks. Bezier surface allows onlyglobal control of the surface. Figure 5 shows a

Bezier surface.

Figure 5. Bezier surface


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6. B-spIine surface: It is a surface that can approximate orinterpolate given input data. Figure 7.8 shows an interpolatingexample. It is a synthetic surface. It is a general surface like aBezier surface but with the advantage of permitting local control ofthe surface.

7. Coons surface: The previously described surfaces are used witheither open boundaries or given data points. A Coons patch is usedto create a surface using curves that form closed boundaries asshown in Figure 7.9.


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8. Fillet surface: It is a B-spline surface that blends twosurfaces together as shown in Figure 7.10. The two originalsurfaces may or may not be trimmed.

9. Offset surface: Existing surfaces can be offset to createnew ones identical in shape but with different dimensions.It is a useful surface to use to speed up surface creation.For example, to create a hollow cylinder, the outer or innercylinder can be created using a cylinder command and theother one can be created by an offset command. The offset

surface command becomes very efficient to use if theoriginal surface is a composite one. Figure 7.11 shows anoffset surface.

Figure 7.10 Fillet surface. Figure 7.10 offset surface.


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Example


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Solid models are known to be complete, valid,

and unambiguous representations of objects.

Simply stated, a complete solid is one whichenables a point in space to be classified relative

to the object, if it is inside, outside, or on the

object. This classification is sometimes call

spatial addressability. A valid solid is one that

does not have dangling edges or faces. An

unambiguous solid has one and only one

interpretation. Solid modeling achievescompleteness, validity, and unambiguity of

geometric models.


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CAD systems offer two approaches to creatingsolid models: primitives and features. The

former approach allows designers to usepredefined shapes (primitives) as building blocksto create complex solids. Designers must useBoolean operations to combine the primitives.This approach is limited by the restricted shapesof the primitives. The features are more flexibleas they allow the construction of more complexand elaborate solids than what the primitivesoffer. Some CAD systems offer bothapproaches, while others offer only the featuresapproach.


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Consider the object shown in Figure 9.1 to illustrate thetwo approaches. We can create a block and subtract sixcylinders from it using the primitives approach. Or, we

can create a rectangle with six circles inside it in the Topsketch plane and extrude it using the features approach.The resulting solid is the feature in this case.

Figure 9.1 A typical solid model.


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Geometry and Topology

A solid model of an object consists of both thetopological and geometrical data of the object. Thecompleteness and unambiguity of a solid model areattributed to the fact that its database stores both its

geometry and its topology. The difference betweengeometry and topology is illustrated in Figure 9.2.Geometry (sometimes called metric information) is theactual dimensions that define the entities of the object.The geometry that defines the object shown in Figure 9.2

is the lengths of lines L1, L2, and L3, the angles betweenthe lines, and the radius Rand the centerP1 of the halfcircle.


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Topology (sometimes called combinatorialstructure) is the connectivity and associativity of

the object entities. It has to do with the notion ofneighborhood; that is, it determines therelational information between object entities.The topology of the object shown in Figure 9.2bcan be stated as follows: L1 shares a vertex(point) with L2and C1, L2shares a vertex withL1, and L3, L3 shares a vertex with L2and C1,L1 and L3 do not overlap, and P1 lies outsidethe object. Based on these definitions, neither

geometry nor topology alone can completelydefine objects.


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While solid models are complete and unambiguous, they are not unique. Anobject may he constructed in various ways. Consider the object shown inFigure 9.3. Using the primitive approach, one can construct the solid modelof the object by dividing it into two blocks and a cylinder. We can add thetwo blocks first and then subtract the cylinder (Figure 9.36), or we cansubtract the cylinder from a block and add the other block to the resulting

subsolid (Figure 9.3c). Figure 9.4 shows two alternatives (create differentcross sections and extrude them) if we use the features approach.Regardless of the order and method of construction, the resulting solidmodel of the object is always complete and unambiguous. However, therewill always be one way that is more efficient than others to construct solidmodels, as is the case with curves and surfaces.


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General types of solid


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More explanation on solid


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Primitives are considered building blocks.

Primitives are simple, basic shapes which

can be combined by a mathematical set of

Boolean operations to create the solid.

Primitives themselves are considered valid

off-the-shelf solids. The user usuallypositions primitives as required before

applying Boolean operations to construct

the final solid.


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All these primitives can be created using the

features approach. They are all 21/2 D objects.

The block, cylinder, and wedge are uniformthickness. The cone, sphere, and torus are

axisymmetric. This explains why some CAD

systems such as Pro/E,SolidWorks and CATIA

do not offer them the user can generate themvia sketching. This simplifies software

development as there is no need to write

separate primitives' functions.


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Figure 1. Most common primitives


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Two or more primitives can be combined to form a solid.To ensure the validity of the resulting solid, the allowedcombinatorial relationships between primitives are

achieved via Boolean (or set) operations. The availableBoolean operators are union (U or +), intersection (n orI), and difference ( - ). The union operator is used tocombine or add together two objects or primitives.Intersecting two primitives gives a shape equal to their

common value. The difference operator is used tosubtract one object from the other and results in a shapeequal to the difference in their volumes.


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End of lecture


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curve, surface and solid

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