1. Chapter 8 Engineering Geometry
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2. Engineering Geometry Basic 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)

4. 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?

5. 2D Cartesian Coordinate System
What describes where rectangle is?
Points and Lines

6. What changes in a 3D coordinate System?
Depth is added
What defines a solid vs. a shape?

7. Right hand rule: Point toward yourself Easiest way to remember
Thumb = x axis
Pointer finger = y axis
Middle finger = z axis
Test Question:
Rotate around the y axis clockwise 90º
What axis shows depth?
What axis shows width?

8. World Coordinate System Local Coordinate System
Origin does not move from (0,0,0)
Local Coordinate System
Origin can be placed anywhere in space with and (x,y,z) coordinates

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

10. Line- has length and direction but no thickness (2D only)
3 Categories:
Straight
Curved
Combo
Regular curved lines- constant radius w/ single center point (Circle, arc)
Irregular- Parabolas, hyperbolas, splines

11. Can you identify all of these line conditions?
Parallel
Tangent
Perpendicular
Intersecting
Non-Parallel

12. Angles- formed by two intersecting lines or planes
Can you indentify all of these?
Straight
Right angle
Acute
Obtuse
Complimentary
Supplementary

13. Freeform Curves Spline- smooth connecting series of control points
Bezier- uses set of control points that only approximate the curve
B-spline- approximates curve to set of control points
Example?

14. Where do parabolas, hyperbolas, ellipses come from?
Called Conics- curve formed by intersection of a place with a right circular cone
Hyperbola-
Plane is parallel to axis
Parabola-
plane is parallel to side
Ellipse-
plane to axis is greater than the axis and sides

15. Where are these used in design?

16. Circle- all points are equal distance from on center point
Elements of a Circle
Circle- all points are equal distance from on center point

17. 2D Shapes Polygons- multi-sided plane of any # of sides
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

18. Where can this possible be used?
Involutes- spiral path of a point on a string unwinding from a line, circle, or polygon
Where can this possible be used?
Gear Teeth

19. Helix- curve formed by a point moving at an angular and a linear rate around a cylinder or cone
Where can this be used?

20. 2 intersecting lines (2D only/ Same flat Surface)
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)

21. Any guess who many different types of surfaces there are?
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

22. Computer Modeling Techniques
Polygonal 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
Low Resolution Model
Over the course of time many computer modeling techniques were developed. Without going into technical details let’s have an overview of the most important ones. Polygonal modeling is the most widely used technique in CAD programs and computer games. The main advantage of this approach is that polygon models are faster to rebuild then other methods. The downside of this technology is that for curved shapes a large number of polygons have to be generated, which requires significant amount of memory (RAM).
High Resolution Model

23. Computer Modeling Techniques
NURBS, 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).
The NURBS technology solves the problem of curved shapes as it is using mathematical models to describe them. On the other the implementation of NURBS in a program requires extensive programming. For this reason NURBS are primarily used in freeform modelers.
Source:

24. 3D Surfaces- restricted or unrestricted by data sets
Coon’s patch

25. Sketch all 4 views of figure 8.141
Homework:
Sketch all 4 views of figure 8.141
(Use .5” for depth)
(Scale 1:2)