5 engineering geometry

8/6/2019 5 Engineering Geometry

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Engineering Geometry

EGR 121 Engineering Graphics

R. Twardock


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Introduction EGR121 - Engineering Graphics Slide 2

Coordinate Space

OriginIntersection of 2 or 3 perpendicularaxes

X axes Y-axes Z-axes

Coordinates x, y, and z values locating point


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Right-Hand Rule Right HandRule

Indicates direction of

positive x, y, and z axes

w/ respect to each other.


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Polar and Cylindrical Coordinates

Polar Distance and angle from

origin

Cylindrical Distance from origin,

angle from x axis in x-y plane, and

distance in z-direction


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Spherical Coordinates Spherical locates points on

spherical surface.

Angle in x-y plane

Anglefrom x-y plane

Distance from origin


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Absolute and Relative Coordinates

Absolute Always referenced toorigin (0,0,0)

cartesian coordinate values are 4,7,25

polar coordinates are 4


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Points

Point (node) location w/out width, height, depth

Locus all possible location of point (line,circle,

arc)


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Locus Questions What is the locus of all points that are an equal

distance from a line?

What is the locus of all points that are an equal

distance from a point?

What is the locus of all points that are an equal

distance from a circle or arc?


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Examples of Locus of Points

Constant distance from a

point

Circle w/ point as center


line

Parallel line(s)


circle/arc

Concentric Circle/Arc


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Lines Line geometric primitive that has length and

direction, but no thickness Straightlines - point moving in constant direction

Ray starts at point, infinite in one direction

Construction line infinite in both directions

P

arallel lines constant distance apart PerpendicularLine (normal) - 90 apart

Curvedlines line created by point moving in constantlychanging direction

Single Curved in a plane (e.g. circle, ellipse, parabola, etc)

Double curved no 4 consecutive points in plane (e.g. helix) Regular Curve constant radius arc or circle

Irregular Curve variable radius


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Tangencies Straight line is tangent if it

(or its extension) contacts

circle at only one pointonly one point

Tangent lineperpendicular

to radial line at tangency

2 circles tangent if meet at

only one point


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Line Conditions Parallel

Nonparallel

Perpendicular

(normal)

Intersecting

Tangent


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Circles Circle single curved

surface primitive

All pointsequidistance fromcenter

Concentric vs.

eccentric Circumscribed vs.

inscribed


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Conics Conics Curves formed by intersection

of plane with a right circular cone

Parabola set of points in planeequidistance from focus and directrix

Ellipse set of point in plane for which sum

of distances from foci is constant Major axis

Minor axis

Hyperbola set of points in plane whosedistance from 2 fixed foci have constant

difference


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Freeform Curves Spline

smooth, freeform curve passing through knowncontrol points using interpolation

E.g. cross section of airplane wing BezierSpline

set of points generated mathematically toapproximate curve using approximation

First and last control points are on the curve No local control

B-Spline - allows for local control


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Angles Angles formed by apex of

two intersecting lines or

planes

Review: straight, right, acute,

obtuse, complementary,

supplementary


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Planes Plane 2-D surface containing

every straight line joining any 2

points on surface

Infiniteplane not bounded at

perimeter

Finiteplane extends to perimeter


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Surfaces

Surface

Finite portion of plane, or

Outer face of object

bounded by identifiableperimeter

Represented by path of

moving line generatrix

Diretrix path of

generatrix


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I

ntroduction EGR121 - Engineering Graphics Slide 19

Surfaces (cont) Planar Flat, 2-D, bounded

Single-Curved - e.g. cylinder or cone Straight line generatrix revolved around axis or point directrix

Double-Curved e.g. sphere, ellipsoid, torus

Contains no straight lines

Warped single and double curved transitions

Freeform no set pattern

Ruled- Planar, single curved, or warped; Produced bystraight line generatrix

Developable can be unfolded to plane withoutdistortion

Undevelopable distorts when unfolded onto plane


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Warped Surface Doublecurved ruled 3-

D surface generated by a

straight line moving such

that any two consecutive

positions of the line are

skewed (not in sameplane).


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Quadilaterals 2-D surfaces

4 sides

Interior angles sum to 360


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Polygon 2-D surface

Multisided

Regularsides of

equal length

Interior angles sum

to 180x(n-2); where

n = number sides


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Triangles

3-sided polygon

Interior angles sumto 180

Sides meet at vertex

Equilat

eral,isosceles, scalene,

right, obtuse, acute


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Cylinders Single-curved ruled surface

Right

Oblique

Circular or elliptical base

Truncated or frustrum cone


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Cylinders Single-curved ruled surface Vertical, finite, straight line

generatrix

Generatrix revolved parallel tovertical (or oblique) directrix,tangent to horizontal circular(or elliptical) directrix.

Right Cylinder Oblique Cylinder


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Polyhedra Symmetrical or asymmetrical 3-D surface or

solid with multiple polygonal sides

Sides (faces) are plane ruled surfaces

Lines of intersection are edges


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Regular Polyhedra Regular

polyhedra

sides areregular

polygons


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Polygonal Prisms Polygonal prisms 2

equal faces (bases) w/

lateral faces that areparallelograms

Bases connected by

vertical line (axis) are

called right prisms.


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Surface Modeling Defined by surface

features and edges

Created using variousoperations:

Sweeping

Revolving

Lofting

Others


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Sweeping Defined bymoving generatrix

along directrix

Directrix usually a

curve

Generatrix can be

line, planar curve,

3-D Curve


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Revolving Revolve the

directrix about an

axis (generatrix)


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Lofting Use series of

directrix curves

along ageneratrix path.

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