Advanced Computer Graphics: Constructive Modelling

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Advanced Computer GraphicsConstructive Modelling

• James Gain
• Department of Computer ScienceUniversity of Cape
  Town
• jgain_at_cs.uct.ac.za

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Objectives

• To introduce a range of modelling techniques
• Curves and Surfaces
• Surfaces of Revolution
• Generalized Cylinders
• Spatial Deformation
• Volumetric Techniques
• Constructive Solid Geometry
• To consider Constructive Solid Geometry in detail
• To discuss graphics in the real world

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Modelling a Sword
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Modelling Techniques

• Building polygon mesh models by hand is
  infeasible.
• Need modelling techniques which are
• Versatile (great variety of achievable shapes)
• Usable (easy to understand controls and specify
  shapes)
• Suitable for particular texturing, rendering and
  animation purposes
• Compact (low space consumption)
• Efficient (interrogation algorithms execute
  rapidly)
• How are shapes modelled in practice?

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Parametric Surfaces

• A bi-parametric surface is a
  rectangular function of two variables and
  , which produces a point on the surface.
• Shape is governed by control points, which either
  interpolate or approximate the surface.
• Objects are created by joining the edges of
  surfaces. Like stitching together rubber sheets.
• Very widespread technique but has problems with
  joins and topology.

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Surfaces of Revolution

• Circular symmetric objects.
• 3D surface created by revolving a 2D profile
  curve around an axis of rotation in space.
• Closed profile curves generate closed surfaces.
• Examples
• Circular cylinder - The profile
  is a line segment parallel but
  not coincident with the axis of
  rotation. The closed
  version requires
  a rectangle.
• Truncated cone - The line
  segment profile is slanted with
  respect to the axis.

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Surfaces of Revolution

• Further examples
• Torus - The profile is a
  circle inset in a plane aligned
  with the axis.
• Complex circularly symmetric shapes -
  employ a Bézier or
  B-spline profile.

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Generalized Cylinders

• Extrusion sweep a 2D shape along a
  (non-circular) path.
• Some objects can be generated by extrusion or
  revolution. e.g. a cylinder (an extruded circle
  or a revolved line).
• Generalized cylinders extend the concept of
  extrusions and surfaces of revolution to the
  extreme.
• Total control over all sweep parameters.
• But can produce degeneracies, e.g.
  self-intersection.

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Generalized Cylinder Parameters

• Can vary
• Cross Section 2D shape does not have to be a
  circle and can change shape as it is swept.
• Sweep Path path does not have to be a straight
  line or revolution, can be any space curve.
• Twist the cross section can be rotated as it
  moves along the path
• Scale the size of the cross section can change
  along the path
• Normal vector direction usually the vector
  normal to the cross section points along the
  path, but even this can be varied
• And any other parameters.

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Spatial Deformation

• Principle
• Indirectly deform an object by warping
  the surrounding space.
• Jelly metaphor
• A shape is set within a block of jelly.
• Flexing the jelly results in a corres-
  ponding distortion of the shape.
• Mechanism
• Object vertices are embedded in a
  parametric hyperpatch, which is a
  3D generalization of curves.

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Volumetric Techniques

• Spatial Occupancy Enumeration Objects are
  constructed Lego-like from adjoining simple
  solids.
• Often restricted to cubes. These Voxels (Volume
  Elements) are the 3D analog of square Pixels
  (Picture Elements)
• Objects for Biomedical Applications
• Medical Scans (CT, MRI) produce image slices with
  a regular grid of sample values (for skin, bone,
  muscle, etc).
• The isosurface at a particular isovalue is very
  difficult to visualize from separate 2D scans.
• Can be converted to a Polygon-Mesh using the
  famous Marching Cubes algorithm.

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Implicit (Blobby) Objects

• Blobby (or Soft) objects are used to create and
  animate smooth shapes.
• Use implicit rather then
  parametric surfaces.
• A skeleton is defined by a set of key points,
  which radiate energy to the surrounding space.
• The object boundary is an isosurface at a
  particular energy level.
• The position and orientation of keys can be
  animated.

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Constructive Solid Geometry

• Constructive Solid Geometry (CSG) consists of
  regularized boolean set operations on closed 3D
  objects.

Intersection
Sphere (A) and Parallelepiped (B)
Union
Difference
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Validity

• Ordinary Boolean set operations may not yield a
  solid. For instance, the intersection of two
  cubes can produce a solid, a plane, a line, a
  point or null.
• For robustness we require set operations which
  take valid closed objects as input and produce a
  valid closed solid as output.
• Regularization ( notation) ensures this

B
B
B
B
A
A
A
A
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Regularization

• Object can be partitioned into interior and
  boundary points.
• Boundary occurs where distance from the object
  and its complement is zero
• Exterior Space Interior Boundary.
• Problems
• Boundary points need not be part of an object.
  Closure prevents this by forming the union of a
  set with its boundary.
• Dangling boundaries are part of an object but not
  adjacent to any interior. Regularization prevents
  this by forming the closure of the interior.

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Regularization Example

• KEY
• Boundary
• (within object)
• Boundary
• (outside object)
• Interior

Nasty Object
Closure
Interior
Regularization
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Set Op Components
B
A
i interior b boundary
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Regularized Boolean Set Ops
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Constructive Solid Geometry

• Constructive Solid Geometry (CSG) is an effective
  means of storing and interrogating complex
  objects
• Objects stored as a tree
• Leaf nodes are usually closed primitives.
  Sometimes half-spaces are used, e.g. cube is the
  intersection of six half-spaces. Although useful
  these may cause validity problems.
• Internal nodes are Boolean operations, affine
  transformations or deformations.
• Ordering of edges is important because set ops
  are not all commutative
• Evaluated using a depth first walk

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CSG Example
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CSG Ray Intersections

• Ray tracing requires the calculation of points of
  intersection between an object and ray.
• Solution intersect arguments and ray, track
  inside/outside status along ray, points are
  retained according to lookup table.

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Exercise CSG-Ray Intersection

• Question create the CSG intersection lookup
  table for the set operation
• Answer

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CSG Summary

• A compact representation
• Easy to manipulate
• Regularized Boolean Set Operations depend on the
  complexity of the arguments and can be expensive.
• Intuitive but can sometimes be confusing since
  there are many different ways of achieving the
  same shape.
• Sometimes supported by a particular rendering
  architectures.

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Exercise Constructive Modelling

• Specify the steps required for construction of
  the jug pictured below. Note some stages can be
  achieved in several different ways.

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Constructive Modelling Solution

• Jug - Surface of Revolution
• Handle - Extruded Ellipse
• Spout - Spatial Deformation
• Combination - Union of Deformed Jug and Handle

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Versatility

• Representing Complex Shapes
• Creases is it possible to introduce creases and
  corners?
• Smoothness can continuity be guaranteed where
  required?
• Topology are complex topologies (with a variety
  of holes) possible?

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Usability

• Some desirable ease-of-use properties in
  modelling (c.f. HCI)
• Closeness of mapping (between problem and
  solution)
• Simplicity (make things as simple as possible
  but no simpler).
• Consistency (similar operations are expressed in
  a similar way).
• Flexibility (technique does not require
  experience but does rewards it)
• Interactive Feedback (shape updates immediately)
• Fluidity (easy to make modelling changes)
• Reversibility (undo changes)
• Order Independence (can design operations be done
  in any order)
• Direct manipulation (there is no indirection in
  control of the shape).
• Some are properties of the modelling system
  rather than the underlying technique.
• CSG, blobby objects and generalized cylinders are
  relatively intuitive (hide mathematical
  underpinnings).

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Suitability for Renderingand Animation

• Rendering
• Polygon scan conversion requires that all
  representations be converted to a polygon mesh.
• Ray tracing requires ray-object intersection
  tests.
• Animation
• Requires controls which allow the shape to vary
  over time.
• Best Match
• Ray Tracing CSG.
• Polygon Scan Conversion parametric surfaces and
  generalized cylinders.
• Animation blobby objects, spatial deformation.
• Space Economy
• Most representations are compact (before
  conversion to Polygon Mesh) except volumetric
  objects.

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Beyond the Ivory Tower

• CAD/CAM
• Parametric surfaces and Constructive Solid
  Geometry.
• Visualization
• Volumetric techniques for Biomedical
  applications.
• Simulation, 3D Games
• Any technique amenable to pre-processed
  conversion to polygon-mesh and which produces low
  polygon counts.
• Often use decimation techniques to reduce number
  of polygons.
• Film
• Almost all techniques (surfaces - Geris Game,
  Toy Story 2, blobby objects - Flubber).
• But, often model objects in clay and then laser
  scan directly to polygon mesh (Star Wars,
  Godzilla).