Theory of Dimensioning

Vijay SrinivasanIBM & Columbia U.

An Introduction to Parameterizing Geometric Models

…and, by the way, how would you parameterize it?

Euclid’s Elements Book I Prop. 4 (side-angle-side)

Euclid’s Elements Book I Prop. 26 (angle-side-angle)

Euclid’s Elements Book I Prop. 8 (side-side-side)

Aha! “Congruence theorems may provide the basis for a theory of dimensioning”

How would you dimension a triangle?

More on Dimensioning Triangles 

 

10

100 3

10

11030

10

30

7

Are these dimensions valid?

Yes

10

30

7

No, because… Yes

Two Types of Congruence

 

Congruent under rigid motion in the plane

Congruent underisometry; congruent underrigid motion only if

allowed to move outside the plane

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)

  

A

B

C

D

A

B

C

D

Chiral objects are congruent under isometry; but, they arenot congruent under rigid motion.

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 the

same “part number”.– Objects that have the same dimensions must be

congruent under rigid motion.• [Mathematical statement]

Congruent objects (under rigid motion) belong to an “equivalence class”, because congruence relation is1. reflexive, i.e., A is congruent to A,2. symmetric, i.e., if A is congruent to B,

then B is congruent to A, and 3. transitive, i.e., if A is congruent to B and

B is congruent to C then A is congruent to C.

Carl Svensen’s Theory of Dimensioning(Circa 1935)

Size dimensionsLocation dimensionsDimensioning procedure

Svensen’s Size Dimensions(Circa 1935)

PRISM CYLINDER CONE PYRAMIDS SPHERE

POSITIVE

A good, but empirical, classification of size dimensions.

Svensen’s Location Dimensions(Circa 1935)

CENTER TO CENTER SURFACE TO CENTER SURFACE TO SURFACE

A good, but empirical, classification of location dimensions.

Svensen’s Procedures in Dimensioning(Circa 1935)

1. Divide the object into elementary parts (type solids positive and negative).

2. Dimension each elementary part (size dimension). 3. Determine locating axes and surfaces. 4. Locate the parts (location dimensions). 

A good two-level hierarchy. In fact, this should be recursive.

A Modern Dimensional Taxonomy

Intrinsic dimensions

Intrinsic dimensions

Relationaldimensions

Intrinsic dimensions

Intrinsic dimensions

Relationaldimensions

Relationaldimensions

Dimensioning Elementary Curves and Surfaces

(Intrinsic dimensions)

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.

bab

y

a

x ,1

2

2

2

2

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.

Dimensioning Ellipses

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.

n

iii tptp

0

)()(

(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.

Dimensioning Bézier Curves

Dimensioning a Bézier curve is the sameas dimensioning its control polygon.

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 havethe same arc-length parameterization of their curvature andtorsion.

Unfortunately, this theorem is of limited use for dimensioning curves.

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.

A Modern Dimensional Taxonomy

Intrinsic dimensions

Intrinsic dimensions

Relationaldimensions

Intrinsic dimensions

Intrinsic dimensions

Relationaldimensions

Relationaldimensions

Dimensioning Relative Positions

(Relational dimensions)Special theory of relative positioning

Involving only points, lines, planes, and helices.

General theory of relative positioning

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 Si=Pi 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.

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).

Pair of Skew Lines is Chiral!

l2

l1

l3

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.

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.

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

Dimensional ConstraintsAre these dimensions valid?

A

B C

D

E

A

B

C

D

E

F

Simultaneous constraints are resolved by inducing a hierarchy

Dimensioning Solids

P2

P1

C

Constraints: P2 // P1 Axis of C P1

Parameters: Distance h between P1 and P2

(relational dimension)Diameter d of C (intrinsic dimension)

h

d

Which solid is it?

Dimensions and constraints should be imposed on a solid representation.

TOC of Columbia Lecture Notes on “Theory of Dimensioning” 1. Introduction2. Congruence3. Dimensioning Elementary Curves4. Dimensioning Elementary Surfaces5. Dimensioning Relative Positions of

Elementary Objects6. Symmetry7. General Theory of Dimensioning

Relative Positions8. Dimensional Constraints9. Dimensioning Solids

Intrinsicdimensions

Relationaldimensions

Book to be published by Marcel Dekker Inc in

October, 2003

SummaryThe 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.