This document discusses 3D modeling systems and solid modeling concepts. It covers key terminology used in 3D modeling like geometry, topology, and faithfulness. Models are analogous representations of real objects that include necessary attributes for design or analysis. Models are stored in databases that contain geometric and topological data, and can be stored explicitly or implicitly. Primary models are used for modeling operations, while secondary models are derived for applications like display or documentation. Associativity connects primary and secondary models so changes propagate. Validity, uniqueness, and expressiveness are important evaluation factors. Solid modeling requires manifolds that represent physically realizable objects with closed interiors and bounded surfaces.

1. 3D modeling systems,
lecture topics:
• terminology used in examining and
comparing systems
• applying geometric modeling (especially
solids) to engineering design
• introduction to solid concepts
3D models create analogous
representations of an object.
• Analogous implies similar, but different.
• Therefore, 3D models are very similar to
the “real world” objects they are
representing but not necessarily identical.
Model is an approximation of
real object.
• Term for the “quality of approximation”
is faithfulness.
• Faithful model incorporates those
attributes necessary to the design or to
the analysis being performed.
• A model need not include all physical
real world features to be “faithful”
Model database
• As previously noted, the heart of a
geometric model is the model database.
• The database can be considered as an
organized form of the information which
describes the model.
• Database information for solids divided
into two general categories, geometric
and topological.
Geometry
• Geometric data relates to dimensional
information, e.g.
– the location of points in space
– the shape and size of geometric features.
Topology
• Topology: refers to the connectivity of
the elements which make up the model,
e.g.
– two faces intersect at an edge
• Inclusion of topological data makes
solid models computationally verifiable.

2. Two objects with the same
topology but different
geometry.
Data format
• model data may be stored:
– explicitly (evaluated data)
• Complete mathematical entity definitions are
stored in the database
– or implicitly (unevaluated data)
• Definitions not stored, but rather computed as
needed
• For example, the curve defined by intersection
of two surfaces or…
• an edge computed for display, then discarded
when not needed (recomputed at next display)
Data / Format
• In practice, database not strictly
evaluated or unevaluated but some
combination
Primary vs. Secondary models
• Systems typically maintain multiple data
representations
• The data used by fundamental modeling
operations is the primary model
• Other representations (which may or
may not be solid) are referred to as
secondary models
Secondary models
• Secondary models derived from primary
model for use is specific applications:
– display, analysis (FEA), manufacturing,
documentation (drawings), data exchange
• Secondary models are maintained to
reduce regeneration times when models
are required.
• Some modelers maintain a log of steps
performed. This log may be considered
a secondary model.
Associativity
• One form of associativity is the direct
connection of primary to secondary
models.
• Associativity can permit alteration of
one model when another is modified
– e.g. “top-down” associativity: alter primary,
secondary documentation file changes

3. Associativity
• An example of this would be the update
of display information when a change to
model geometry is made.
• This concept (associativity) also
appears in the relationship between
parent and child geometry and in the
extraction of mechanical drawings from
a solid model.
Model evaluation factors:
• domain and expressive power
• uniqueness
• validity.
Domain and Expressive
power
• domain refers to the range of model
geometries which may be generated by
a particular modeler.
• some modeling system supply the user
with broader domain
• expressive power: is the primary model
exact or approximate representation?
Uniqueness
• Concept of singularity between model
and database
• Two types of uniqueness may be
considered.
– Interpretation: Can the database represent
more than one object.
– Expression: Can more than one database
represent the same object?
Validity
• whether or not the model represents an
object which can exist in the real world.
• many solid modeling systems include
checks of model validity in their
architecture.
• validity checks often involve topological
checks
Manifold vs.
Non-manifold geometry
• Mathematically manifold geometry
means every point on a surface has
“neighborhood” (infinitesimal sphere)
around it that can be deformed onto a
locally planar surface.
• More simply: manifold geometry rules
out objects which are not physically
realizable such as those with features
joined along a single edge or vertex.

4. Non-manifold geometry Manifolds in solid modeling
• Class of manifolds comprises real
world, manufacturable objects.
• When computationally modeling solids,
is possible to create non-manifolds.
– One case would be a self-intersecting
model.
• Software may perform diagnostic
checks but user should be aware of
what non-manifold geometry is.
Solid Modeling
• “Just as a set of 2D lines and curves
does not need to describe the boundary
of a closed area, a collection of 3D
surfaces and planes does not
necessarily bound a closed volume.”
Solid Modeling
– Many engineering applications of
geometric modeling require the ability to
distinguish between the inside, outside and
surface of an object.
– Several techniques for the computerized
modeling of solid geometries exist and are
in use today.
– The various techniques have different
advantages disadvantages and uses.
Solid Modeling
• What properties are considered important for
of an effective solid modeling system
– Bounded - the boundary must limit and contain the
interior of the solid.
– Finite - Finite in size, model can be defined by a
limited set of information
– Homogeneously 3-dimensional or more simply, no
dangling faces or edges, boundary must always
be in contact with the interior