8.1 Position Tolerancing Fundamentals, True Position, & Coordinate Conversions

ASME GDTP Technologist Y14.5-2009

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1 Chapter 1: Scope, Certification Logistics, Fundamental Rules & General Dimensioning

1.1 ASME GDTP Technologist Exam Scope, Prometric Logistics, & Certification Rules 1.2 Fundamental Rules of ASME Y14.5-2009 (Rules a through p) 1.3 Engineering Drawing Standards, Drawing Units, & Basic vs. Toleranced Dimensions 1.4 Dimensioning Features: Diameters, Radii, Chamfers, Countersinks, Counterbores, Spotfaces, Keyseats & Slots 1.5 Application of Dimensions & Location of Features: Lines, Leaders, Reference Dimensions, Rectangular vs. Polar Coordinates

2 Chapter 2: Features, Limits of Size, & Material Condition Principles

2.1 Features, Features of Size, & Limits of Size 2.2 Rule #1 (Taylor Principle / Envelope Rule), Its Exceptions & Overrides 2.3 Material Condition Concepts: MMC, LMC, & RFS Calculations

3 Chapter 3: General Tolerancing & Symbology

3.1 Direct Tolerancing, Limit Dimensioning, & Angular Tolerancing 3.2 The 14 Geometric Characteristic Symbols, Feature Control Frames, & Special Modifiers 3.3 Screw Threads, Gears & Splines, Angular Surfaces, Tapers & Statistical Tolerancing 3.4 Feature Control Frame Placement, Combined Frames, Tolerance Zone Definition & Tabulated Tolerances

4 Chapter 4: Datum Reference Frames & Degrees of Freedom

4.1 Theoretical Datums, Datum Feature Simulators, & Physical Datum Features 4.2 Degrees of Freedom Constraint & The 3-2-1 Datum Precedence System 4.3 Planar, Cylindrical, & Width Datum Feature Establishment

5 Chapter 5: Advanced Datum Applications: Targets, Boundary Modifiers, Customized Frames & Restrained Condition

5.1 Datum Target Points, Lines, Areas, & Movable Datum Simulators 5.2 Material Boundary Modifiers (RMB, MMB, LMB) & Datum Shift Calculations 5.3 Functional Datum Feature Selection, Multiple Datum Reference Frames & Simultaneous vs. Separate Requirements 5.4 Rotational Constraint About a Datum Axis or Point & Customized Datum Reference Frames 5.5 Irregular & Complex Datum Features, Mathematically Defined Surfaces, Restrained Condition & Datum Feature Controls

6 Chapter 6: Form Tolerances

6.1 Flatness & Straightness Controls for Surfaces & Features of Size 6.2 Circularity & Cylindricity Controls, Differences, & Inspection

7 Chapter 7: Orientation Tolerances

7.1 Parallelism Controls for Planar Surfaces & Cylindrical Features of Size 7.2 Perpendicularity & Angularity Tolerances with Basic Angles & Tangent Plane

8 Chapter 8: Location: Position Fundamentals & Calculations

8.1 Position Tolerancing Fundamentals, True Position, & Coordinate Conversions 8.2 Bonus Tolerance & Virtual Condition Calculations for Internal & External Features 8.3 Projected Tolerance Zones (Ⓟ Modifier) & Fixed vs. Floating Fastener Formulas 8.4 Zero Positional and Orientation Tolerancing at MMC and at LMC 8.5 Surface vs. Axis Interpretation of Position & the Boundary Concept for Slots, Tabs & Noncircular Features

9 Chapter 9: Location: Patterns, Composite Controls, Coaxial & Symmetrical Relationships, and Functional Gaging

9.1 Hole & Slot Patterns, Multi-Segment Frames, & Composite Position Tolerancing 9.2 Functional Gaging Principles & Inspection Verification of Position 9.3 Coaxial Holes, Counterbored Holes, Non-Parallel Axes, Spherical Features & Bidirectional Position 9.4 Concentricity & Symmetry Tolerances Under ASME Y14.5-2009 9.5 LMC Applied to Position: Wall Thickness, Breakout Protection & Boundary Calculations

10 Chapter 10: Profile Tolerances

10.1 Profile of a Line & Profile of a Surface Fundamentals & Basic Geometry 10.2 Bilateral, Unilateral, & Unequally Disposed Profile Boundaries (Ⓤ Modifier) 10.3 Composite Profile Tolerances vs. Multiple Single-Segment Profile Controls 10.4 Profile Verification, Coplanarity Controls, & Non-Rigid Parts 10.5 Profile Applications: Conicity, Extruded and Irregular Shapes, and Profile-Controlled Datum Features 10.6 Modifier Rules for Profile & Combined Controls (Profile with Position, Orientation and Runout)

11 Chapter 11: Runout Tolerances & Comparative Control Analysis

11.1 Circular Runout vs. Total Runout Interpretation & Dial Indicator Verification 11.2 Comparative Analysis: Runout vs. Form, Orientation, & Concentricity Controls

8.1 Position Tolerancing Fundamentals, True Position, & Coordinate Conversions

Key Takeaways Explain This Topic AI

Position tolerancing (ASME Y14.5-2009 Section 7.2) defines a zone within which the center, axis, or center plane of a feature of size is permitted to vary from its theoretically exact location (true position) defined by basic dimensions from a Datum Reference Frame (DRF).

A cylindrical (diametral) position tolerance zone provides 57.08% more usable tolerance area than a square coordinate tolerance zone of the same width, capturing functional parts located in the corner diagonals.

Actual diametral position deviation is calculated from coordinate inspection data using the conversion formula: Actual Deviation = 2 × √(ΔX² + ΔY²), where ΔX and ΔY represent deviations from basic coordinate locations.

Bidirectional position tolerancing governs non-cylindrical features of size (such as rectangular slots or tabs) using two single-segment feature control frames that omit the diameter symbol (⌀), establishing planar tolerance zones bounded by two parallel planes.

Last updated: September 2026

8.1 Position Tolerancing Fundamentals, True Position, & Coordinate Conversions

Quick Answer: In ASME Y14.5-2009 Section 7.2 , position is a geometric tolerance that defines a zone within which the center, axis, or center plane of a feature of size is permitted to vary from its theoretically exact location ( true position ). True position is established by basic dimensions originating from an established Datum Reference Frame (DRF) . Specifying a cylindrical position tolerance zone (preceded by the ⌀ symbol in the tolerance compartment) yields a 57.08% increase in usable tolerance area over traditional square coordinate tolerancing of the same nominal width. For inspection verification, coordinate deviations along orthogonal axes are converted to diametral position deviation via the formula: Actual Diametral Deviation = 2 × Δ X 2 + Δ Y 2 \text{Actual Diametral Deviation} = 2 \times \sqrt{\Delta X^2 + \Delta Y^2} Actual Diametral Deviation = 2 × Δ X 2 + Δ Y 2 . For non-cylindrical features like slots and tabs, bidirectional position tolerancing uses two single-segment feature control frames that omit the ⌀ symbol to define planar zones bounded by two parallel planes.

1. True Position & Theoretical Foundations (ASME Y14.5-2009 Section 7.2)

In modern manufacturing and engineering design, position tolerancing is the most widely applied geometric control. Unlike form tolerances (which never use datums) or orientation tolerances (which control angular relationships without locating), position controls the location, orientation, and relationship of features of size relative to one another or to established datums.

True Position Defined

True position is defined as the theoretically exact location of a feature of size (a hole, pin, slot, tab, or spherical feature) established by basic dimensions . True position has zero tolerance itself; it serves as the absolute target coordinate in 3D space about which a geometric tolerance zone is centered. TRUE POSITION & BASIC DIMENSIONS

Datum Plane B
| ◄─────────── [50.00] BASIC ───────────►
| │
| ┌──────────────────────────────┼────────────────┐
| │ │ │
| │ ▼ │
| │ (True Position Axis) │
| │ │ │
| │ │ │
| │ │ [30.00] BASIC │
| │ │ │
| │ │ │
| │ ▼ │
└─────────┴───────────────────────────────────────────────┴────
Datum Plane C
The Role of Basic Dimensions & Datum Reference Frames

Basic Dimensions : A basic dimension represents a theoretically exact value used to describe the size, profile, orientation, or location of a feature. On drawings, basic dimensions are enclosed in a rectangular box (e.g., [50] ). Because basic dimensions carry no intrinsic tolerance, title-block plus-and-minus tolerances never apply to basic dimensions.

Datum Reference Frame (DRF) : Position tolerancing requires datums. A position feature control frame must reference one, two, or three datums to constrain the required degrees of freedom (up to three translations and three rotations) and locate the tolerance zone unambiguously in space.

Rule #2 Application : Under ASME Y14.5-2009, Regardless of Feature Size (RFS) is the default condition for position tolerances and datum references if no material condition modifier (Ⓜ or Ⓛ) is indicated.

2. Cylindrical (Diametral) Tolerance Zones vs. Square Coordinate Tolerancing

Before geometric dimensioning and tolerancing, engineering drawings relied on coordinate (plus-and-minus) tolerances to locate hole centers (e.g., X = 50.00 ± 0.10 X = 50.00 \pm 0.10 X = 50.00 ± 0.10 , Y = 30.00 ± 0.10 Y = 30.00 \pm 0.10 Y = 30.00 ± 0.10 ). This legacy approach suffers from three major functional flaws:

Square / Rectangular Tolerance Zones : Coordinate tolerancing generates a square or rectangular boundary for the feature center.

Tolerance Accumulation : Tolerances stack up across sequential coordinate chains.

Artificial Rejection of Functional Parts : Parts with equal clearance in all radial directions are rejected at the orthogonal extremes.

The 57.08% Usable Area Advantage

Consider a hole located by coordinate tolerances of ± 0.10 mm \pm 0.10\text{ mm} ± 0.10 mm along the X-axis and ± 0.10 mm \pm 0.10\text{ mm} ± 0.10 mm along the Y-axis:

The coordinate tolerance zone is a square with side length S = 2 × 0.10 = 0.20 mm S = 2 \times 0.10 = 0.20\text{ mm} S = 2 × 0.10 = 0.20 mm .

The total area of this coordinate square is: Area square = S 2 = ( 0.20 mm ) 2 = 0.0400 mm 2 \text{Area}_{\text{square}} = S^2 = (0.20\text{ mm})^2 = 0.0400\text{ mm}^2 Area square = S 2 = ( 0.20 mm ) 2 = 0.0400 mm 2

The maximum permissible displacement in this square occurs at the four 45° corner diagonals: r corner = ( 0.10 mm ) 2 + ( 0.10 mm ) 2 = 0.0200 ≈ 0.1414 mm r_{\text{corner}} = \sqrt{(0.10\text{ mm})^2 + (0.10\text{ mm})^2} = \sqrt{0.0200} \approx 0.1414\text{ mm} r corner = ( 0.10 mm ) 2 + ( 0.10 mm ) 2 = 0.0200 ≈ 0.1414 mm

If the mating bolt or clearance assembly can function successfully when the hole center is displaced by 0.1414 mm 0.1414\text{ mm} 0.1414 mm at a 45° angle, then the hole should functionally be permitted to displace by that same radial distance ( 0.1414 mm 0.1414\text{ mm} 0.1414 mm ) in any radial direction! A round hole mating with a round fastener has radial symmetry; it does not care whether a displacement occurs horizontally, vertically, or diagonally. COORDINATE SQUARE VS. DIAMETRAL CYLINDRICAL ZONE

+Y ▲
│ /‾‾‾‾‾‾‾‾‾‾‾‾‾‾\ ◄── Cylindrical Zone
│ / ┌────────┐ \ (Diameter = 0.2828)
+0.10 ─┼─┼────│────────│────┼─
│ │ │ • │ │
0.00 ─┼─┼────┼──(TP)──┼────┼─► +X
│ │ │ │ │
0.10 ─┼─┼────│────────│────┼─ Square Zone
│ \ └────────┘ / ◄── (0.20 x 0.20)
│ \______________/
│
─┴──────┴────────┴──────
0.10 +0.10

By converting this maximum functional radial displacement ( r = 0.1414 mm r = 0.1414\text{ mm} r = 0.1414 mm ) into a cylindrical tolerance zone:

The diameter of the circumscribing cylindrical tolerance zone is: D = 2 × r corner = 2 × 0.1414 mm = 0.2828 mm D = 2 \times r_{\text{corner}} = 2 \times 0.1414\text{ mm} = 0.2828\text{ mm} D = 2 × r corner = 2 × 0.1414 mm = 0.2828 mm

The area of this circular zone is: Area circle = π 4 D 2 = π 4 ( 0.2828 mm ) 2 ≈ 0.06283 mm 2 \text{Area}_{\text{circle}} = \frac{\pi}{4} D^2 = \frac{\pi}{4} (0.2828\text{ mm})^2 \approx 0.06283\text{ mm}^2 Area circle = 4 π D 2 = 4 π ( 0.2828 mm ) 2 ≈ 0.06283 mm 2

Calculating the percentage increase in usable tolerance area: Ratio = Area circle Area square = π 4 ( 2 2 t ) 2 ( 2 t ) 2 = π 4 ( 8 t 2 ) 4 t 2 = 2 π 4 = π 2 ≈ 1.5708 \text{Ratio} = \frac{\text{Area}_{\text{circle}}}{\text{Area}_{\text{square}}} = \frac{\frac{\pi}{4} (2\sqrt{2} t)^2}{(2t)^2} = \frac{\frac{\pi}{4} (8t^2)}{4t^2} = \frac{2\pi}{4} = \frac{\pi}{2} \approx 1.5708 Ratio = Area square Area circle = ( 2 t ) 2 4 π ( 2 2 t ) 2 = 4 t 2 4 π ( 8 t 2 ) = 4 2 π = 2 π ≈ 1.5708 Percentage Increase = ( 1.5708 − 1 ) × 100 % = 57.08 % \text{Percentage Increase} = (1.5708 - 1) \times 100\% = 57.08\% Percentage Increase = ( 1.5708 − 1 ) × 100% = 57.08%

By replacing a ± t \pm t ± t coordinate tolerance with a diametral position tolerance of diameter 2 2 t 2\sqrt{2}t 2 2 t , manufacturing gains 57% additional usable tolerance area without reducing assembly clearance or functional reliability. Conversely, if a cylindrical zone of diameter 2 t 2t 2 t (matching the width of the coordinate box) is used, the inspection zone is invariant in all directions, eliminating the directional bias of coordinate tolerancing.

3. Coordinate-to-Position Conversion Formulas

When inspecting features of size on Coordinate Measuring Machines (CMMs) or open-setup surface plates, measurements are recorded in Cartesian coordinates ( X , Y X, Y X , Y ). The quality engineer or inspector must convert these linear coordinate readings into the equivalent diametral position deviation to verify compliance with the feature control frame.

Step-by-Step Mathematical Conversion

Determine Coordinate Deviations from Basic Location : Δ X = X measured − X basic \Delta X = X_{\text{measured}} - X_{\text{basic}} Δ X = X measured − X basic Δ Y = Y measured − Y basic \Delta Y = Y_{\text{measured}} - Y_{\text{basic}} Δ Y = Y measured − Y basic

Calculate True Radial Displacement ( r r r ) : r = Δ X 2 + Δ Y 2 r = \sqrt{\Delta X^2 + \Delta Y^2} r = Δ X 2 + Δ Y 2

Calculate Actual Diametral Position Deviation ( D actual D_{\text{actual}} D actual ) : D actual = 2 × r = 2 × Δ X 2 + Δ Y 2 D_{\text{actual}} = 2 \times r = 2 \times \sqrt{\Delta X^2 + \Delta Y^2} D actual = 2 × r = 2 × Δ X 2 + Δ Y 2

Evaluate Compliance :

Compare D actual D_{\text{actual}} D actual to the total allowable position tolerance ( T allowable T_{\text{allowable}} T allowable ): Pass Condition: D actual ≤ T allowable \text{Pass Condition: } D_{\text{actual}} \le T_{\text{allowable}} Pass Condition: D actual ≤ T allowable

Worked Example: CMM Inspection Verification

An engineering drawing specifies a hole location with basic dimensions X = 45.00 mm X = 45.00\text{ mm} X = 45.00 mm and Y = 60.00 mm Y = 60.00\text{ mm} Y = 60.00 mm . The feature control frame specifies: [ Position | Ø 0.25 | A | B | C ] (applied Regardless of Feature Size).

During CMM inspection, the actual center of the hole is probed at:

X measured = 45.09 mm X_{\text{measured}} = 45.09\text{ mm} X measured = 45.09 mm
Y measured = 59.93 mm Y_{\text{measured}} = 59.93\text{ mm} Y measured = 59.93 mm

Calculation :

Δ X = 45.09 − 45.00 = + 0.09 mm \Delta X = 45.09 - 45.00 = +0.09\text{ mm} Δ X = 45.09 − 45.00 = + 0.09 mm
Δ Y = 59.93 − 60.00 = − 0.07 mm \Delta Y = 59.93 - 60.00 = -0.07\text{ mm} Δ Y = 59.93 − 60.00 = − 0.07 mm

Radial deviation: r = ( 0.09 ) 2 + ( − 0.07 ) 2 = 0.0081 + 0.0049 = 0.0130 ≈ 0.1140 mm r = \sqrt{(0.09)^2 + (-0.07)^2} = \sqrt{0.0081 + 0.0049} = \sqrt{0.0130} \approx 0.1140\text{ mm} r = ( 0.09 ) 2 + ( − 0.07 ) 2 = 0.0081 + 0.0049 = 0.0130 ≈ 0.1140 mm

Actual diametral position deviation: D actual = 2 × 0.1140 mm = 0.228 mm D_{\text{actual}} = 2 \times 0.1140\text{ mm} = 0.228\text{ mm} D actual = 2 × 0.1140 mm = 0.228 mm

Evaluation: Since 0.228 mm ≤ 0.25 mm 0.228\text{ mm} \le 0.25\text{ mm} 0.228 mm ≤ 0.25 mm , the hole location is in conformance .

Critical Note : Under traditional coordinate tolerancing of ± 0.08 mm \pm 0.08\text{ mm} ± 0.08 mm , this hole would have been erroneously scrapped because Δ X = 0.09 mm \Delta X = 0.09\text{ mm} Δ X = 0.09 mm exceeds 0.08 mm 0.08\text{ mm} 0.08 mm , even though its true radial position error is fully acceptable.

4. Bidirectional Position Tolerancing (ASME Y14.5-2009 Section 7.2)

While cylindrical tolerance zones (preceded by ⌀) are standard for cylindrical holes and pins, non-cylindrical features of size—such as rectangular slots, keyways, and tabs —frequently have different functional alignment requirements in their length and width directions. BIDIRECTIONAL POSITION ON A RECTANGULAR SLOT

Datum B
│
◄─── [40.0] BASIC ────►│
┌──────────────────────┼────────────────────────┐
│ │ │
│ ┌────────────┼────────────┐ │
│ │ │ (Center) │ │ ▲
│ │ ───────*─────── │ │ │ [25.0] BASIC
│ │ │ │ │ │
│ └────────────┼────────────┘ │ ▼
└──────────────────────┴────────────────────────┴───────── Datum C

Width Direction (X): `[ Position | 0.15 | A | B | C ]` (No ⌀)
Length Direction (Y): `[ Position | 0.60 | A | B | C ]` (No ⌀)
Rules for Bidirectional Position Callouts

Absence of Diameter Symbol (⌀) : When controlling slots, tabs, or planar surfaces, the tolerance zone is not a cylinder. Therefore, the diameter symbol (⌀) must be omitted from the tolerance compartment.

Planar Tolerance Zone Geometry : The tolerance zone consists of two parallel planes separated by the specified tolerance value, symmetrically centered on the true position basic center plane.

Two Single-Segment Callouts : To control location in both orthogonal directions with different tolerances, two separate single-segment feature control frames are used:

One FCF is associated with the width size dimension , controlling center plane displacement in the width direction.

A second FCF is associated with the length size dimension , controlling center plane displacement in the length direction.

Datum Precedence in Bidirectional Control : Both feature control frames typically reference the same primary, secondary, and tertiary datums, but the directional orientation of the parallel planes is governed by which feature dimension the frame is attached to.

5. Master Comparison Table: Coordinate vs. Position Tolerancing
| Attribute / Parameter | Coordinate Tolerancing (Plus/Minus) | Position Tolerancing (True Position) |
| :--- | :--- | :--- | |

| Tolerance Zone Geometry | Square or rectangular box | Cylindrical volume (for cylinders) or parallel planes (for slots) |

| Zone Orientation | Locked to arbitrary coordinate grid lines | Locked to functional Datum Reference Frame (DRF) |

| Usable Tolerance Area | Fixed square boundary ( S 2 S^2 S 2 ) | 57.08% larger usable area ( ( π / 2 ) × S 2 (\pi/2) \times S^2 ( π /2 ) × S 2 ) for radial zones |

| Material Condition Modifiers | Prohibited (strictly size-based) | Permitted (RFS default, Ⓜ for bonus, Ⓛ for wall thickness) |

| Functional Gaging | Impossible (requires point-by-point CMM) | Directly inspectable via fixed-pin functional receiver gages |

| Tolerance Stacking | High risk of cumulative stack-up | Eliminated via basic dimensions from common datums |

6. Common Exam Traps: Position Fundamentals

Trap 1: Forgetting the Factor of 2 in Diametral Deviation : In CMM conversion problems, candidates frequently compute r = Δ X 2 + Δ Y 2 r = \sqrt{\Delta X^2 + \Delta Y^2} r = Δ X 2 + Δ Y 2 and compare r r r directly to the specified diameter tolerance. The radial displacement r r r is only the zone radius; it must be multiplied by 2 to obtain the actual diametral deviation ( D = 2 r D = 2r D = 2 r ).

Trap 2: Adding a Diameter Symbol (⌀) to Bidirectional Slot Controls : A drawing showing [ Position | Ø 0.2 | A | B ] applied to a slot width is a syntax error. Slots are bounded by two parallel planes; adding ⌀ indicates an impossible cylindrical zone.

Trap 3: Applying Title Block Tolerances to Basic Dimensions : Basic dimensions are untoleranced references. Applying a title-block tolerance (such as ± 0.25 \pm 0.25 ± 0.25 ) to a boxed basic dimension is an outright exam error.

Trap 4: Assuming Square Zones Allow More Variation : Candidates often mistakenly think square zones allow more variation because they have corners. In reality, the corners allow displacement along the 45° angle, but the cylindrical zone permits that same maximum displacement in all 360° directions , capturing 57% more acceptable parts.

Trap 5: Confusing Single-Segment Bidirectional with Composite FCFs : Bidirectional tolerancing of a slot uses two distinct single-segment frames (one for width, one for length). It is not a composite frame; composite frames control pattern location in the upper segment and internal feature-to-feature orientation in the lower segment.