Theory of Dimensioning

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Theory of Dimensioning
An Introduction to Parameterizing Geometric Models

• Vijay Srinivasan
• IBM Columbia U.

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How would you dimension a triangle?
and, by the way, how would you parameterize
it?
Euclids Elements Book I Prop. 4 (side-angle-side)
Euclids Elements Book I Prop. 26
(angle-side-angle)
Euclids Elements Book I Prop. 8 (side-side-side)
Aha! Congruence theorems may provide the basis
for a theory of dimensioning
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More on Dimensioning Triangles
 
Are these dimensions valid?
 
Yes
Yes
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Two Types of Congruence
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Chirality
 
 
I call any geometrical figure, or group of
points, chiral, and say it has chirality, if its
image in a plane mirror, ideally realized, cannot
be brought to coincide with itself - Lord
Kelvin (ca. 1904)
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Congruence under rigid motion

• Engineering statement Congruent objects are
  functionally interchangeable.
• This applies only to congruence under rigid
  motion.
• In industrial parlance, congruent objects have
  thesame part number.
• Objects that have the same dimensions must be
  congruent under rigid motion.

• Mathematical statementCongruent objects (under
  rigid motion) belong to an equivalence class,
  because congruence relation is
• reflexive, i.e., A is congruent to A,
• symmetric, i.e., if A is congruent to B,
  then B is congruent to A, and
• transitive, i.e., if A is congruent to B and
  B is congruent to C
  then A is congruent to C.

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Carl Svensens Theory of Dimensioning(Circa
1935)

• Size dimensions
• Location dimensions
• Dimensioning procedure

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Svensens Size Dimensions(Circa 1935)
SPHERE
PRISM
CYLINDER
PYRAMIDS
CONE
POSITIVE
A good, but empirical, classification of size
dimensions.
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Svensens Location Dimensions(Circa 1935)
A good, but empirical, classification of location
dimensions.
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Svensens Procedures in Dimensioning(Circa 1935)

• Divide the object into elementary parts (type
  solids positive and negative).
• Dimension each elementary part (size dimension).
• Determine locating axes and surfaces.
• Locate the parts (location dimensions).
•  

A good two-level hierarchy. In fact, this should
be recursive.
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A Modern Dimensional Taxonomy
Relationaldimensions
Relationaldimensions
Relationaldimensions
Intrinsic dimensions
Intrinsic dimensions
Intrinsic dimensions
Intrinsic dimensions
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Dimensioning Elementary Curves and
Surfaces(Intrinsic dimensions)
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Dimensioning Conics
(Conics Classification Theorem) Any planar curve
of second-degree can be moved by purely rigid
motion in the plane so that its transformed
equation can assume one and only one of the nine
canonical forms given in the following table.
Conic Type Canonical Equation Intrinsic Parameters
1 Ellipse a,b

(Conics Congruence Theorem) Two conics are
congruent if andonly if they have the same
canonical equation.
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Dimensioning Ellipses
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Dimensioning Free-form Curves
(Free-form Curve Invariance Theorem) A free-form
curve is intrinsically invariant under rigid
motion of its control points if and only if its
basis functions partition unity in the interval
of interest.
(Free-form Curve Congruence Theorem) Two
free-form curves, which share the same basis
functions that partition unity, are congruent if
their control polygons are congruent.
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Dimensioning Bézier Curves
Dimensioning a Bézier curve is the sameas
dimensioning its control polygon.
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A General Theorem from Differential Geometry
(Fundamental Existence and Uniqueness Theorem of
Curves) Let ?(s) and ?(s) be arbitrary
continuous functions on a ? s ? b. Then there
exists, except for position in space, one and
only one space curve C for which ?(s) is the
curvature, ?(s) is the torsion and s is a
natural parameter along C.
Therefore, two curves are congruent if and only
if they have the same arc-length parameterization
of their curvature and torsion.
Unfortunately, this theorem is of limited use for
dimensioning curves.
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Dimensioning Elementary Surfaces

• Similar to dimensioning elementary curves
• Dimensioning quadrics
• Quadrics classification theorem ? quadrics
  congruence theorem
• Dimensioning free-form surfaces
• Free-form surfaces are congruent if their
  control nets are congruent.

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A Modern Dimensional Taxonomy
Relationaldimensions
Relationaldimensions
Relationaldimensions
Intrinsic dimensions
Intrinsic dimensions
Intrinsic dimensions
Intrinsic dimensions
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Dimensioning Relative Positions(Relational
dimensions)

• Special theory of relative positioning
• Involving only points, lines, planes, and
  helices.
• General theory of relative positioning

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Tuples

• A tuple is an ordered collection whose members
  are symbolically enclosed by parentheses.
• (Tuple Equality) (S1,S2,,Sn) (P1,P2,,Pn) if
  and only if SiPi for all i.
• (Tuple Rigid Motion) r(S1,S2,,Sn)
  (rS1,rS2,,rSn).
• Informally, tuple represents a collection of
  objects rigidly welded together by an invisible
  welding material.

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Some Elementary Cases

• Let p1, p2, p'1 and p'2 be points, in a plane or
  in space. Then (p1,p2) is congruent to (p'1,p'2)
  if and only if d(p1,p2) d(p'1,p'2).
• Let l1, l2 be two skew lines in space, and l'1,
  l'2 be two other skew lines in space. Then (l1,
  l2) is congruent to (l'1, l'2) if and only if
  they have the same chirality, d(l1, l2) d(l'1,
  l'2) and ?(l1,l2) ?(l'1,l'2).

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Pair of Skew Lines is Chiral!
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Tuple Congruence Question

• Has the relative positioning of two geometric
  objects changed when each of them is subjected to
  arbitrarily different rigid motions?
• Is (S1, S2) congruent to (r1S1 , r2S2)?
• (Tuple Replacement Theorem) The answer to the
  tuple congruence question remains unaltered if
  we replace the point-sets by those in the same
  symmetry class.

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Seven Classes of (Continuous) Symmetry
Type Simple Replacement
1 Spherical Point (center)
2 Cylindrical Line (axis)
3 Planar Plane
4 Helical Helix
5 Revolute Line (axis) point-on-line
6 Prismatic Plane line-on-plane
7 General Plane, line point.
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Hierarchy of Basic Constraints

• Projective transformation
• Preserves incidence, cross-ratio

• Affine transformation
• Preserves parallelism, ratio

• Isometric transformation
• Preserves angles (e.g., perpendicularity),
  distance

• Rigid motion transformation
• Preserves chirality

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Dimensional ConstraintsAre these dimensions
valid?
Simultaneous constraints are resolved by inducing
a hierarchy
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Dimensioning Solids
Constraints P2 // P1 Axis
of C ? P1 Parameters Distance h between
P1 and P2 (relational
dimension) Diameter d of C
(intrinsic dimension)
Dimensions and constraints should be imposed on a
solid representation.
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TOC of Columbia Lecture Notes on Theory of
Dimensioning

• Introduction
• Congruence
• Dimensioning Elementary Curves
• Dimensioning Elementary Surfaces
• Dimensioning Relative Positions of Elementary
  Objects
• Symmetry
• General Theory of Dimensioning Relative
  Positions
• Dimensional Constraints
• Dimensioning Solids

Book to be published by Marcel Dekker Inc
in October, 2003
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Summary

• The modern theory of dimensioning is a synthesis
  of several ideas.
• They range from results in classical Euclidean
  geometry (ca. 300 BC) to Lie group
  classification (ca. 1996 AD).
• Supplements ASME Y14.5.1 (Mathematical
  Definition of Dimensioning and Tolerancing
  Principles).
• Supplements ISO/TC 213 standards (Geometric
  Product Specifications and Verification).
• Theory of dimensioning is also a theory of
  parameterizing geometric models.
• Supplements ISO STEP standards.