Theory of Dimensioning Vijay Srinivasan IBM & 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   30   7 Are these dimensions valid? Yes  7 No, because … Yes

Two Types of Congruence Congruent under rigid motion in the plane Congruent under isometry; congruent under rigid 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 are not 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 is 1.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 dimensions  Location dimensions  Dimensioning 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 Relational dimensions Intrinsic dimensions Relational dimensions

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. Conic TypeCanonical Equation Intrinsic Parameters 1Ellipse a,b … (Conics Congruence Theorem) Two conics are congruent if and only 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. (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 B é zier Curves Dimensioning a B é zier curve is the same as 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 have the same arc-length parameterization of their curvature and torsion. 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 Relational dimensions Intrinsic dimensions Relational dimensions

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) (S 1,S 2, …,S n ) = (P 1,P 2, …,P n ) if and only if S i =P i for all i.  (Tuple Rigid Motion) r(S 1,S 2, …,S n ) = (rS 1,rS 2, …,rS n ).  Informally, tuple represents a collection of objects rigidly welded together by an invisible welding material.

Some Elementary Cases  Let p 1, p 2, p' 1 and p' 2 be points, in a plane or in space. Then (p 1,p 2 ) is congruent to (p' 1,p' 2 ) if and only if d(p 1,p 2 ) = d(p' 1,p' 2 ).  …  Let l 1, l 2 be two skew lines in space, and l' 1, l' 2 be two other skew lines in space. Then (l 1, l 2 ) is congruent to (l' 1, l' 2 ) if and only if they have the same chirality, d(l 1, l 2 ) = d(l' 1, l' 2 ) and  (l 1,l 2 ) =  (l' 1,l' 2 ).

Pair of Skew Lines is Chiral! l2l2 l1l1 l3l3

Tuple Congruence Question  Has the relative positioning of two geometric objects changed when each of them is subjected to arbitrarily different rigid motions?  Is (S 1, S 2 ) congruent to (r 1 S 1, r 2 S 2 )?  (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 TypeSimple Replacement 1SphericalPoint (center) 2CylindricalLine (axis) 3PlanarPlane 4HelicalHelix 5RevoluteLine (axis) & point-on-line 6PrismaticPlane & line-on-plane 7GeneralPlane, 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 Constraints Are these dimensions valid? A B C D E A B C D E F Simultaneous constraints are resolved by inducing a hierarchy

Dimensioning Solids P2P2 P1P1 C Constraints : P 2 // P 1 Axis of C  P 1 Parameters : Distance h between P 1 and P 2 (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. Introduction 2. Congruence 3. Dimensioning Elementary Curves 4. Dimensioning Elementary Surfaces 5. Dimensioning Relative Positions of Elementary Objects 6. Symmetry 7. General Theory of Dimensioning Relative Positions 8. Dimensional Constraints 9. Dimensioning Solids Intrinsic dimensions Relational dimensions Book to be published by Marcel Dekker Inc in October, 2003

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 AD).  Supplements ASME Y (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.