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Chapter 8Engineering Geometry

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)

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?

2D Cartesian Coordinate SystemWhat describes where rectangle is?

Points and Lines

What changes in a 3D coordinate System?

Depth is added

What defines a solid vs. a shape?

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?

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

Points (Node)- Theoretical location (describes exact location in space, but no real geometry is created)

Geometric Relationships

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

Can you identify all of these line conditions?

ParallelTangent

PerpendicularIntersectingNon-Parallel

Angles- formed by two intersecting lines or planes

Can you indentify all of these?

•Straight•Right angle

•Acute•Obtuse

•Complimentary•Supplementary

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?

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

Where are these used in design?

Elements of a CircleCircle- all points are equal distance from on center point

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

Involutes- spiral path of a point on a string unwinding from a line, circle, or polygon

Where can this possible be used?

Gear Teeth

Helix- curve formed by a point moving at an angular and a linear rate around a

cylinder or cone

Where can this be used?

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)

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

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

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/

3D Surfaces- restricted or unrestricted by data sets

Coon’s patch

Homework: Sketch all 4 views of figure 8.141

(Use .5” for depth)(Scale 1:2)