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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
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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Engineering Geometry
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.
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2-D Surfaces
A triangle is apolygon with
three sides. Thesum of theinterior anglesequals 180
degrees.
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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.
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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.
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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).