Constructive Solid Geometry

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Constructive Solid Geometry

• Mel Slater

http//graphics.lcs.mit.edu/legakis/legakis_ucdav
is/Gallery/CSG1.jpg
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Introduction

• Quadric Surfaces
• Ray Intersections with Quadrics
• Set Operations
• Ray Classification

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Quadric Surfaces

• Consider a sphere
• And also a cylinder
• These are special cases of a general class of
  object called quadrics.

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Quadric Surfaces

• Let Q be a 44 matrix
• Let p (x,y,z,1)
• s(x,y,z) pTQp 0.
• The quadric is the boundary and the interior of
  the surface defined by the equation, and hence
  consists of pTQp ? 0.

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Quadric Surfaces

• Note that the highest power of any coordinate is
  2 (hence quadric).
• Note that the a,,j are given constants
• Special cases a,,j of define different types of
  surface.
• Multiplying out we get

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Special Cases

• Plane
• Sphere
• Cylinder

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Special Cases

• Cone
• Paraboloid

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More Properties

• Given a quadric the normal at point p is given by
  2pTQ.
• If all the points p on a quadric are transformed,
  eg, q pM then the result is also a quadric
• where M is a transformation matrix.

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Ray Intersection

• The beauty of the quadric representation is that
  the unified representation requires only one
  ray-intersection algorithm.
• Let p(t) p tv, t ?0
• Be the parametric equation of the ray that starts
  at p in direction p.
• Substitute p(t) into pTQp 0 and solve the
  quadratic equation for t.
• This will give
• Complex results if no intersection
• Two equal values of t if tangent
• Two real values the entry and exit points
  otherwise.
• (What happens in the case of a plane?)

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Set Operations

• The quadrics represent solid objects
• Unlike boundary representations for mesh and
  B-Spline approaches
• We can use set operations such as union,
  intersection and difference to combine such
  solids together.

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Regularising Operation

• Given a point p we define the open ball.
• A boundary point in any point-set S is a point p
  such that for any ?gt0 B(p, ?) will contain points
  in S and also in not-S.
• S is an open set if it does not contain its
  boundary.
• Closure(S) S ?boundary(S)

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Regularising Operation

• Interior(S) S boundary(S)
• Regularisation(S) closure(interior(S))
• For any set operation it is the regularisation of
  the result that is required.
• A op B regularisation(A op B)
• where op is any operator such as union,
  difference, intersection

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Example
Intersection results in illegal set

• Is the set, which is closed in (b) with interior
  in (c)
• and regularisation in (d)

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CSG Operations

• We can define set operations such as ?, ?, ?(not)
  and difference -.
• We can make formulae expressing more complex
  objects as combinations of simpler objects, eg
• A ?B C
• ((A ?B) ?C)-D
• Each can be expressed as a binary operator tree.

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((A ?B) ?C)-D
-
?
Each expression canbe represented as a binary
operator tree.
D
?
C
A
B
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CSG Data Structure

• typedef struct _csgtree
• Operator op
• Quadric primitive
• struct _csgtree right
• struct _csgtree left
• CSGTreePtr
• Each leaf node represents a primitive (usually a
  quadric primitive).
• op can also be leaf rather than a combination
  operator.

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Intersecting a Ray

• Pseudo code for ray-tree intersection

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Intersecting a Ray

• The result will be a sequence of intersection
  points along the ray, represented by the
  parametric values
• t0, t1, t2, ,tn
• where there are n intersections
• Each individual intersection with a solid will
  result (typically) in two values.

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Combine Function

• Shows how the result of the left and right solids
  combine together.
• Note that the set operations therefore occur at a
  2D level.

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Examples
ccvweb.csres.utexas.edu/ccv/projects/VisualEyes/v
isualization/domainpara/algebraic2/parameterizatio
n/gallery/Reconstruction/Csg/
www-2.cs.cmu.edu/afs/cs/misc/rayshade/all_mach/om
ega/doc/Examples/jpg/csg.jpg
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Summary

• Unified representation for quadric surfaces
  provides a simplicity and elegance for a wide
  class of shapes.
• CSG provides a simple methodology for
  combinations via set operations of solids
• CSG does not require quadrics, it can work other
  types of solid representation.