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8/3/2019 Engineering Bertoline

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Engineering Geometry


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Objectives

Describe the importance of engineeringgeometry in design process.

Describe coordinate geometry andcoordinate systems and apply them toCAD.

Review the right-hand rule.List major categories of geometricentities.


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Objectives

Explain geometric conditions thatoccurs between lines.

Explain tangent conditions betweenlines and curves.

List and describe surface geometric

formDescribe engineering applications ofgeometry.


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Engineering geometry is thebasic geometric elements and

forms used in engineering design.Engineering and technical graphicsare concerned with the

descriptions of shape, size, andoperation of engineered products.


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Shape Description

Shape description of an object relatesthe positions of its componentgeometric elements (e.g., vertices,edges, faces) in space.

Coordinate Space In order to locate points, lines, planes,

or other geometric forms, theirpositions must first be referenced tosome known position, called areference point or origin ofmeasurement


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Coordinate Space

The Cartesiancoordinate system,commonly used inmathematics and

graphics, locates thepositions of geometricforms in 2-D and 3-Dspace. A 2-D coordinate

system establishes anorigin at theintersection of twomutuallyperpendicular axes,labeled X (horizontal)and Y (vertical).


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Coordinate Space

In a 3-D coordinatesystem, the originis established at

the point wherethree mutuallyperpendicular axes(X, Y, and Z) meet.The origin isassigned thecoordinate valuesof 0,0,0.


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Coordinate Space

The right-hand rule isused to determinethe positive direction

of the axes. Theright-hand ruledefines the X, Y, andZ axes, as well as

the positive andnegative directions ofrotation on eachaxes.


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Coordinate Space

Polarcoordinates are

used to locatepoints in the X-Yplane. Polarcoordinates

specify a distanceand an anglefrom the origin(0,0).


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Coordinate Space

Cylindricalcoordinates locatea point on the

surface of a cylinderby specifying adistance and anangle in the X-Y

plane, and thedistance in the Zdirection.


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Coordinate Space

Sphericalcoordinateslocate a point onthe surface of asphere byspecifying anangle in one

plane, an angle inanother plane,and one height.


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Coordinate Space

Absolute coordinates are alwaysreferenced to the origin (0,0,0).

Relative coordinates are alwaysreferenced to a previously definedlocation and are sometimes

referred to as delta coordinates,meaning changed coordinates.


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Coordinate Space

The world coordinate system uses aset of three numbers (x,y,z) located on

three mutually perpendicular axes andmeasured from the origin (0,0,0).

The local coordinate system is amoving system that can be positioned

anywhere in 3-D space by the user, toassist in the construction of geometry.


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Coordinate Space


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Geometric Elements

A point is a theoretical location thathas neither width, height, nor depth.Points describe an exact location inspace. Normally, a point is representedin technical drawings as a small crossmade of dashes that are approximately1/8" long. In computer graphics, it is

common to use the word node to meana point. A locus represents all possiblepositions of a point.


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Points


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Lines

A line is ageometricprimitive that haslength anddirection, but notthickness. A linemay be straight,

curved, or acombination ofthese.


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Lines Relations


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Curve lines

A curved line is the path generated by apoint moving in a constantly changingdirection, or is the line of intersection between

a 3-D curved surface and a plane. single-curved (circle, ellipse, parabola)

double-curved (cylindrical helix, conical helix)


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Tangent Conditions

A tangentcondition exists

when a straightline is in contactwith a curve at asingle point.


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Tangent Conditions

In 3-D geometry,a tangentcondition existswhen a planetouches but doesnot intersectanother surface

at one or moreconsecutivepoints


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Tangent Conditions


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Tangent Conditions


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Circles

A circle is a single-curved-surfaceprimitive, all points of which areequidistant from one point, the center.

A circle is also created when a planepasses through a right circular cone orcylinder and is perpendicular to the axisof the cone

The elements of a circle: diameter,radius, chord, circumference, secant,arc, tangent, concentric.


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Circle


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Conic Curves

A parabola is the curvecreated when a planeintersects a right circularcone parallel to the side

of the cone. A parabola isa single-curved-surfaceprimitive.Mathematically, aparabola is defined asthe set of points in aplane that areequidistant from a givenfixed point, called afocus, and a fixed line,called a directrix.


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Conic Curves

A hyperbola is thecurve of intersectioncreated when a

plane intersects aright circular conethat makes a smallerangle with the axis

than do theelements.


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Conic Curves

An ellipse is asingle-curved-surface primitive and

is created when aplane passesthrough a rightcircular cone obliqueto the axis, at an

angle to the axisgreater than theangle between theaxis and the sides.


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Conic Curves

A spiral is a single-curved surface thatbegins at a point called a pole and

becomes larger as it travels around theorigin in a plane.


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Conic Curves

A cycloid is the curve generated by themotion of a point on the circumference of acircle as the circle is rolled along a straight line

in a plane.


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Conic Curves

An involute is a spiral path of a point on astring unwinding from a line, circle, orpolygon.


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Doubled-curved lines

A double-curved line is a curve generated by a pointuniformly moving at both an angular and a linear ratearound a cylinder or cone.

cylindrical helix

Spiral staircases, worm gear, drill bits, spring

conical helix


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Freeform Curves

If the curves arecreated by smoothlyconnecting the

control points, theprocess is calledinterpolation.

If the curves arecreated by drawing a

smooth curve thatgoes through most,but not all thecontrol points, theprocess is called

approximation.


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Freeform Curves

A spline curve is asmooth, freeformcurve that connects

a series of controlpoints. Changing anysingle control pointwill result in a

change in the curve,so that the curve canpass through thenew point


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Freeform Curves

The Bezier curve,which uses a set ofcontrol points that

only approximate thecurve.

The B-spline curve,which approximatesa curve to a set of

control points anddoes provide forlocal control.


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Angles

Angles areformed by theapex of two

intersecting linesor planes Straight

Right

Acute

Obtuse

Complementary


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Planes

A plane is aninfinite,unbounded, two-

dimensionalsurface thatwholly containsevery straight line

joining any twopoints lying onthe surface.


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Surfaces

A surface is afinite portion of a

plane, or theouter face of anobject boundedby an identifiable

perimeter.


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2-D Surfaces

Quadrilateralsare four-sided

plane figures ofany shape. Thesum of the anglesinside a

quadrilateral willalways equal 360degrees.


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2-D Surfaces

A polygon is a multisided plane of anynumber of sides.
http://highered.mcgraw-hill.com/olc/dl/25965/fig3-0052_PNG.htmlhttp://highered.mcgraw-hill.com/olc/dl/25965/fig3-0052_PNG.html


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2-D Surfaces

A triangle is apolygon with

three sides. Thesum of theinterior anglesequals 180

degrees.
http://highered.mcgraw-hill.com/olc/dl/25965/fig3-0053_PNG.html


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Ruled Surfaces

Single-curved surfaces aregenerated by moving a straight

line along a curved path such thatany two consecutive positions ofthe generatrix are: either parallel (cylinder),

intersecting (cone), tangent to a double-curved line

(convolute).


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Ruled Surfaces

A cone is a single-curved-surfaceprimitive formed by

a line (generatrix)moving along acurved path suchthat the line always

passes through afixed point, calledthe vertex.
http://highered.mcgraw-hill.com/olc/dl/25965/fig3-0055_PNG.html


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Ruled Surfaces

A cylinder is a single-curved ruled surfaceformed by a vertical,finite, straight-line

element (generatrix)revolved parallel to avertical or oblique axisdirectrix and tangent to ahorizontal circular orelliptical directrix. The

line that connects thecenter of the base andthe top of a cylinder iscalled the axis.
http://highered.mcgraw-hill.com/olc/dl/25965/fig3-0056_PNG.htmlhttp://highered.mcgraw-hill.com/olc/dl/25965/fig3-0056_PNG.html


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Ruled Surfaces

A convolute is asingle-curvedsurface generated by

a straight linemoving such that itis always tangent toa double-curved line.


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Ruled Surfaces

polyhedron is asymmetrical orasymmetrical 3-D

geometric surface orsolid object withmultiple polygonalsides. The sides areplane ruled surfaces,

and are called faces,and the lines ofintersection of thefaces are called theedges.


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Ruled Surfaces

polygonal prism is apolyhedron that has twoequal parallel faces,called its bases, andlateral faces that areparallelograms. Theparallel bases may be ofany shape and areconnected by

parallelogram sides. Aline connecting thecenters of the two basesis called the axis.


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Ruled Surfaces

pyramid is a polyhedron that has a polygonfor a base and lateral faces that have acommon intersection point called a vertex. The

axis of a pyramid is the straight lineconnecting the center of the base to thevertex.


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Ruled Surfaces

warped surface isa double-curvedruled 3-D surface

generated by astraight line movingsuch that any twoconsecutive positions

of the line areskewed (not in thesame plane).

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