Mathematical Definition of Dimensioning and Tolerancing Principles
|Publication Date:||1 January 1994|
SCOPE AND DEFINITIONS
This Standard presents a mathematical definition of geometrical dimensioning and tolerancing consistent with the principles and practices of ASME Y14.5M-1994, enabling determination of actual values. While the general format of this Standard parallels that of ASME Y14.5M-1994, the latter document should be consulted for practices relating to dimensioning and tolerancing for use on engineering drawings and in related documentation.
Textual references are included throughout this Standard which are direct quotations from ASME Y14.5M-1994. All such quotations are identified by italicized type. Any direct references to other documents are identified by an immediate citation.
The definitions established in this Standard apply to product specifications in any representation, including drawings, electronic exchange formats, or data bases. When reference is made in this Standard to a part drawing, it applies to any form of product specification.
Units. The International System of Units (SI) is featured in the Standard because SI units are expected to supersede United States (U.S.) customary units specified on engineering drawings.
Figures. The figures in this Standard are intended only as illustrations to aid the user in understanding the principles and methods described in the text. In some instances figures show added detail for emphasis; in other instances figures are incomplete by intent. Any numerical values of dimensions and tolerances are illustrative only.
Notes. Notes shown in capital letters are intended to appear on finished drawings. Notes in lower case letters are explanatory only and are not intended to appear on drawings.
Reference to Gaging. This Standard is not intended as a gaging standard. Any reference to gaging is included for explanatory purposes only.
This Subsection describes the mathematical notation used throughout this Standard, including symbology (typographic conventions) and algebraic notation.
Symbology. All mathematical equations in this Standard are relationships between real numbers, three-dimensional vectors, coordinate systems associated with datum reference frames, and sets of these quantities. The symbol conventions shown in Table 1.3 are used for these quantities.
These symbols may be subscripted to distinguish between distinct quantities. Such subscripts do not change the nature of the designated quantity.
Technically, there is a difference between a vector and a vector with position. Generally in this Standard, vectors do not have location. In particular, direction vectors, which are often defined for specific points on curves or surfaces, are functions of position on the geometry, but are not located at those points. (Another conventional view is that all vectors are located at the origin.) Throughout this Standard, position vectors are used to denote points in space. While there is a technical difference between a vector and a point in space, the equivalence used in this Standard should not cause confusion.
Algebraic Notation. A vector can be expanded into scalar components (with the components distinguished by subscripts, if necessary). Let i, j, and k be the unit vectors along the x, y, and z axes, qspectively, of a coordinate system. Then a vector V can be uniquely expanded as:
The vector can be written V = (a,b,c). The magnitude (length) of vector V is denoted by V and can be evaluated by:
A unit vector V is any vector with magnitude equal to one. The scalar product (dot product; inner product) of two vectors V = (a1, b1, c1) and V2 = (a2, b2, c2) is denoted by V1 • V2. The scalar product is a real number given by:
and is equal in value to the product of the lengths of the two vectors times the cosine of the angle between them. The vector product (cross product; outer product) of two vectors V1 and V2 is denoted by V1 × V2. The cross product is a vector V3 = (a3, b3, c3) with components given by:
The magnitude of the cross product is equal in value to the product of the lengths of the two vectors times the sine of the angle between them.
For a given feature, the notation r (P, Γ) will denote the distance from a point P to true position (see Subsection 1.4) in datum reference frame Γ. When the datum reference frame is understood from the context, the notation r (P) will be used. Figure 1-1 shows a case of a true position axis. If the axis is represented by a point P on the axis and a direction N (a unit vector), then r (P) can be evaluated by either of the following formulas:
The first equation is a version of the Pythagorean Theorem. The second equation is based on the properties of the cross product.