# Geometric Elements by peteronline125

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```									3D Spatial Principles and
Modeling Techniques
Geometric Elements

3D Entities
Cones
– Right cone?
Axis perpendicular to base
– Oblique cone?
Axis not perpendicular to base
– Truncated cone or frustum of a
cone?
Tip of cone cut off
3D: Cones
Geometric Elements
3D Entities
Defining Cylinders
– A single-curved ruled surface formed by a
vertical, finite, straight-line element
(generatrix) revolved parallel to a vertical or
oblique axis directrix and tangent to a
horizontal circular or elliptical directrix.
– Right cylinder
Axis perpendicular to base
– Oblique cylinder
Axis not perpendicular to base
– If base is circle, cylinder is circular. If base is
elliptical, cylinder is elliptical
3D: Cylinders
Geometric Elements
3D Entities
Defining Polyhedra
– Polyhedron: a symmetrical or
asymmetrical 3-D geometric
surface or solid object with
multiple polygonal sides. The
sides are plane ruled surfaces.
– Regular polyhedra
Have regular polygons for faces
tetrahedron, hexahedron,
octahedron, dodecahedron,
icosahedron
3D: Polyhedra
Geometric Elements
tetrahedron
n., pl. -drons or -dra (-drə).
A polyhedron with four faces.
[Late Greek tetraedron, from Greek, neuter of tetraedros, four-faced :
tetra-, tetra- + hedrā, face of a geometric solid.]
Etymology: New Latin, from Late Greek tetraedron, neuter of tetraedros having
four faces, from Greek tetra- + hedra seat, face
Translations for: Tetrahedron
Nederlands (Dutch) viervlak
Français (French) tétraèdre
Deutsch (German) Tetraeder (Vierflächner, dreiseitige Pyramide)
Ελληνική (Greek) τετράεδρο
Italiano (Italian) tetraedro
Português (Portuguese) tetraedro (m)
Русский (Russian) тетраэдр, четырехгранник, (воен.) пирамидальный
противотанковый надолбник
Español (Spanish) tetraedro
Svenska (Swedish) tetraeder (geom.)

Geometric Elements
3D Entities
Defining Prisms
– Polygonal prism
Polyhedron with two equal parallel faces,
called bases, and lateral faces that are
parallelograms
Right or oblique
– Pyramid
Polyhedron that has a polygon for a base
and lateral faces that have a common
intersection point called a vertex.
Right, oblique, or truncated(frustum)
3D: Prisms
3D: Pyramids
3D: Math!
3D: Math!
Cool Stuff

http://mathworld.wolfram.com/Polyhedron.html
Why 3D?

3D is Reality
3D is Easier to Understand
(Visualization)
3D is More Fun
3D Models

Three Types of 3D
Models:
–Wireframe
–Surface
–Solid
Wireframe Models
Vertices (Points)
Edges (Lines, Arcs,
etc.)
No Surface Information
No Volume Information
Wireframe Models

–Easy to Construct
–Fast!
–Visualization Difficulties
–No Data
Surface Modeling
Defines Surface Features as
Well as Edges
Development Driven by
Shipping, Automotive,
Aerospace Industries
Highly Mathematical;
‘Control Point’ Concept
Solid Modeling
Defines Volumetric Data as
Well as Surfaces, Edges
‘Real World’ Design - Used
by All Industries
Concurrent Engineering
–Smart Models
Solid Primitives
– Cube, Cylinder, Wedge, etc.
– Can Be Combined to Create Complex
Geometry
– Simple and Fast for Computer to
Process and Display
– Limited Flexibility in Design (e.g.:
impossible to make a hole)
Solid Modeling

CSG - Constructive Solid
Geometry
–Boolean Operations
Union
Difference
Intersection
–Makes Primitives More Useful
Solid Modeling
B-rep : Boundary Representation
– Object Surfaces Used to Define the
Solid Model
– Surfaces Have ‘Inside’ & ‘Outside’
– Display of Curved Surfaces can be a
Problem – slow system
Mathematically Intense
Solution: Faceted Representation
Solid Modeling
Hybrid Modelers- combine CSG & BREP
– CSG type commands – easier to model
with
– BREP output – easier to display to screen
– Most modelers store data in both
structures
– Support both Boolean operations and
complex surfaces as modeling tools to
make solids
– Exact precision may be lost if both
structures are utilized, as data is moved
from one to the other
Solid Modeling

Constraint-based Modeling
– Relates geometry together/modifiable
features
– Changes update throughout model
– Mating parts can be associated
Change Often – Allows for
Modification
Goal: Capture Designer’s Intent –
Critical!
Solid Modeling

Building the Features
–Basic Sweep (protrusion)
–Revolved Sweep