Valley Average of Lines (VAL) and of Directions (VAD) in 3D

doi:10.1016/j.cad.2020.102957

Computer-Aided Design

Volume 131, February 2021, 102957

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Abstract

Extending (Sati et al., 2016) to 3D, we propose definitions and closed-form expressions for what we call the Valley Average of Directions (VAD), $\vec{T}$ , of a set ${{\vec{T}}_{i}}$ of directions (unit vectors) and for what we call the Valley Average of Lines (VAL), $L$ , of a set ${L_{i}}$ of lines. $L$ is independent of the input order. When the ${L_{i}}$ are parallel, $L$ is parallel to them and passes through the centroid of their intersections with any normal plane. When all ${L_{i}}$ contain point $P$ , $L$ passes through $P$ .

Introduction

The concept of averaging is pervasive in Geometric Dimensioning and Tolerancing (GD&T) [1], [2] and is used to fit planes and lines to measured points.

In this note, we consider the problem of averaging a set of lines in 3D. The importance of our contributions may be illustrated by the following anticipated application: When defining GD&T for manufacturing quality control and for testing conformance for assembly [3], one may wish to compute, for a selection of machined parts, the “average” of the axes of a cylindrical feature estimated from acquired sensor data (such as a Coordinate Measuring Machine). The variability of these axes with respect to this average axis and the deviation of this average from the desired datum axis make it possible to distinguish between noise and systemic error. This distinction it not possible when only statistics on the deviation between the individual axes and the desired datum are collected.

Section snippets

Prior art

It is reasonable to expect an average of a set of lines to possess the following list of properties:

•
Order invariance: The average should be invariant under permutations of the inputs.
•
Parameterization invariance: The average should not depend on the parameterization or a particular choice of representation for the input lines.
•
Independence from sampling: The average should not depend on a particular sampling of input lines.

It is not trivial to construct an average that satisfies these desired

Valley average of directions (VAD)

We first present a particular definition of the average $\vec{T}$ of a set ${{\vec{T}}_{i}}$ of directions (unit vectors) in 3D. We call it the Valley Average of Directions (VAD).

Other options for defining $\vec{T}$ , such as a linear average or the geodesic squared-distance minimizer on the Gaussian sphere [8], may be suitable for some applications, but, none of these options lead to a formulation of an average of lines that satisfies the desired properties listed above. We show in Section 4 that VAD does.

Let ${{\vec{T}}_{i}}$ be a

Valley average of lines (VAL)

We use the VAD to define our average of lines. Let ${L_{i}}$ be a set of lines in 3D, where line $L_{i} = L i n e (P_{i}, {\vec{T}}_{i})$ is defined by a point, $P_{i}$ , that it contains and by its direction, ${\vec{T}}_{i}$ .

We define the average of ${L_{i}}$ as $L = L i n e (P^{*}, \vec{T})$ , where $\vec{T}$ is the VAD of ${{\vec{T}}_{i}}$ and where $P^{*} = a r g m i n Q (P)$ , with, $Q (P) = \sum_{i}^{n} {‖ \vec{P P_{i}} ‖}^{2} - \sum_{i} {(\vec{P P_{i}} \cdot {\vec{T}}_{i})}^{2} = P^{T} K P + B^{T} P + d$ being the sum of the squared distances from $P$ to the lines ${L_{i}}$ . In the above matrix–vector expression for $Q (P)$ , the quadratic terms have been encoded and expressed with the

Multi-line, order independent correspondence

Given lines $L_{i}$ in 3D, one may need a correspondence (common parameterization) between them that is independent of the order in which the lines are given. The VAL proposed here suggests two different solutions: CPontoVAL and CPfromVAL. For both solutions, we consider a parameterization of the average line, where $C (s)$ is a point on the VAL $L$ and, for each $i$ , use it to define the corresponding point $C_{i} (s)$ on $L_{i}$ .

Let $P r o j (P, K)$ return the point on line $K$ that is closest to point $P$ .

1.
CPontoVAL (Common

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Technical NoteValley Average of Lines (VAL) and of Directions (VAD) in 3D

Abstract

Introduction

Section snippets

Prior art

Valley average of directions (VAD)

Valley average of lines (VAL)

Multi-line, order independent correspondence

Declaration of Competing Interest

Comput Aided Des

Comput Aided Des

Comput Aided Des

Comput Aided Des

Technical Note
Valley Average of Lines (VAL) and of Directions (VAD) in 3D