Download - Chapter 8 Engineering Geometry
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Chapter 8Engineering Geometry
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Engineering GeometryBasic geometric elements used in design
Why do we care?
What geometry was used in the design of these chairs?
(Break down to primitive 2D shapes to sketch then manipulate to get to final design)
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What defines our designs?– The position of its component elements in space
• Points (Vertices)• Lines (Edges)• Planes (Faces)What makes a circle instead of a rectangle?
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2D Cartesian Coordinate SystemWhat describes where rectangle is?
Points and Lines
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What changes in a 3D coordinate System?
Depth is added
What defines a solid vs. a shape?
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Easiest way to remember
Right hand rule:Point toward
yourself
Thumb = x axisPointer finger = y axisMiddle finger = z axis
Test Question: Rotate around the y axis
clockwise 90ºWhat axis shows depth?What axis shows width?
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World Coordinate SystemOrigin does not move from (0,0,0)
Local Coordinate SystemOrigin can be placed anywhere in space with
and (x,y,z) coordinates
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Points (Node)- Theoretical location (describes exact location in space, but no real geometry is created)
Geometric Relationships
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Line- has length and direction but no thickness (2D only)
3 Categories:StraightCurvedCombo
Regular curved lines- constant radius w/ single center point (Circle, arc)
Irregular- Parabolas, hyperbolas, splines
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Can you identify all of these line conditions?
ParallelTangent
PerpendicularIntersectingNon-Parallel
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Angles- formed by two intersecting lines or planes
Can you indentify all of these?
•Straight•Right angle
•Acute•Obtuse
•Complimentary•Supplementary
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Freeform Curves
Spline- smooth connecting series of control points
Bezier- uses set of control points that only approximate the curveB-spline- approximates curve to
set of control points
Example?
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Where do parabolas, hyperbolas, ellipses come from?
Called Conics- curve formed by intersection of a place with a right circular cone
Parabola- plane is parallel to side
Hyperbola- Plane is parallel to axis
Ellipse- plane to axis is greater than
the axis and sides
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Where are these used in design?
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Elements of a CircleCircle- all points are equal distance from on center point
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2D Shapes
Quadrilaterals- 4 sided where sum of all angles = 360˚
Parallelogram- opposite sides of quadrilateral are parallel to each other
Polygons- multi-sided plane of any # of sides
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Involutes- spiral path of a point on a string unwinding from a line, circle, or polygon
Where can this possible be used?
Gear Teeth
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Helix- curve formed by a point moving at an angular and a linear rate around a
cylinder or cone
Where can this be used?
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Planes- two dimensional surface that wholly contains every straight line
joining any two points lying on that surface
How are planes formed?•3 points
•2 parallel lines•Line and point
•2 intersecting lines (2D only/ Same flat Surface)
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Surface- a finite portion of a plane, or the outer face of an object bounded by a perimeter (2D or 3D)
Any guess who many different types of
surfaces there are?
8
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Computer Modeling TechniquesPolygonal modeling is an approach for
modeling objects by representing or approximating their surfaces using polygons.
The main advantage of polygons is that they are faster than other representations
Polygons are incapable of accurately representing curved surfaces, so a large number of them must be used to approximate curves
http://www.wikipedia.org/High Resolution Model
Low Resolution Model
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Computer Modeling TechniquesNURBS, short for non-uniform, rational B-
spline, is a mathematical model commonly used for generating and representing curves and surfaces. A NURBS curve is defined by its order, a set of weighted control points, and a knot vector.
They are invariant under affine as well as perspective transformations.
They offer one common mathematical form for both standard analytical shapes (e.g., conics) and free-form shapes.
They provide the flexibility to design a large variety of shapes.
They reduce the memory consumption when storing shapes (compared to simpler methods). Source: http://www.wikipedia.org/
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3D Surfaces- restricted or unrestricted by data sets
Coon’s patch
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Homework: Sketch all 4 views of figure 8.141
(Use .5” for depth)(Scale 1:2)