⚡ electrical engineering

Small Sample Strategies for Measurement of Partial Circles

This paper evaluates and identifies optimal sampling strategies for inspecting quarter, half, and three-quarter partial circles by using Monte Carlo simulations to analyze the uncertainties in radius and center coordinates across various point distributions ranging from three to seven points.

Original authors: Dhanish P B

Published 2026-07-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Dhanish P B

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery, but instead of a crime scene, your clues are tiny dots on a piece of metal. This is the world of coordinate metrology, a branch of science dedicated to measuring the shape and size of objects with extreme precision. Think of it as the ultimate "spot the difference" game, but the stakes are high: if a pump casing or a boiler shell isn't the right shape, it could fail in real life. To figure out if a part is a perfect circle, engineers use machines that poke or scan the surface to grab a few coordinates. But here's the tricky part: you can't measure the entire circle if the object is broken or cut off (like a quarter of a pizza or a half-moon). You only have a slice. The big question is: Where should you place your few measurement dots to get the most accurate picture of the whole shape? If you pick the dots randomly or just space them out evenly like beads on a string, you might end up with a blurry, uncertain guess. This paper dives into that exact puzzle, asking how to arrange a small handful of points (from three to seven) to minimize the "fuzziness" or uncertainty in the final measurement.

The author of this study, Dhanish P B, treats the measurement of these "partial circles" like a game of finding the perfect seating arrangement for a group of friends to get the best view of a stage. The "stage" is the curved edge of the object, and the "view" is the calculated center and radius of that curve. The researcher used a powerful computer simulation technique called Monte Carlo simulation—which is basically running the same measurement experiment a hundred thousand times with tiny, random errors added in—to see which seating chart worked best. The goal was to find the specific angles where the points should sit to make the uncertainty of the radius and the center's location as small as possible.

The results are a bit surprising and definitely break the rules of thumb many people might guess. The most common, "naïve" idea is to space the points out evenly, like slices of a pie. The paper shows that while this uniform strategy is okay, it is not the best way to go. In fact, for many of these partial circles, spreading the points out evenly actually leaves you with more uncertainty than some very specific, non-uniform arrangements.

For example, when measuring a quarter circle (a 90-degree slice) with just four points, the best strategy isn't to space them out at 15-degree intervals. Instead, the simulation suggests you should put two points right at the very ends of the curve and place the other two points at the very center of the arc (at 0 degrees) if your goal is to minimize uncertainty in the radius and the horizontal center. However, if you care most about the vertical center, the best move is actually to place those inner points at 30 degrees. It sounds counterintuitive to cluster points in the middle, but mathematically, it locks down the measurement of the radius and the center's horizontal position much better. Similarly, with five points on a quarter circle, the best move for the radius and horizontal center is to put the two inner points right in the middle (at 0 degrees), but for the vertical center, those inner points should be placed at 45 degrees.

The study also looked at half circles and three-quarter circles. For a half circle with five points, if you want to know the radius and vertical center perfectly, you should put the two inner points at the very ends of the arc. But if you care more about the horizontal position of the center, you should put those inner points right in the middle (at 0 degrees). It's like a game of "choose your own adventure" where the best path depends entirely on which specific detail you are trying to pin down.

Even for a three-quarter circle with just three points, the "best" angle changes based on what you are measuring. If you want the most accurate radius, the outer points should be at 120 degrees. But if you are trying to pin down the horizontal center, the outer points need to be pushed out to 135 degrees (the extreme ends).

The paper explicitly rules out the idea that "more points spread evenly" is always the answer. It also notes that for these tiny samples, they didn't try to calculate the "form error" (how bumpy or wavy the surface actually is) because the sample sizes were too small to do that reliably. Instead, they focused purely on the uncertainty caused by the measurement machine's own tiny errors.

In the end, the author concludes that there is no single "magic bullet" strategy that works for every situation. The best way to measure a partial circle depends on whether you have three, four, five, six, or seven points, and whether you care most about the radius, the horizontal center, or the vertical center. The uniform, equidistant strategy is often just "good enough," but if you need the highest precision, you need to be strategic and sometimes place your points in clusters or at the very edges, rather than spreading them out like a happy, even crowd. The paper provides a detailed map of these optimal arrangements, giving metrology practitioners a guide to plan their inspections so they don't have to guess where to poke their measuring machine.

Technical Summary: Small Sample Strategies for Measurement of Partial Circles

Problem Statement
In coordinate metrology, the inspection of partial circle features—common in engineering components such as pump casings and boiler shells—presents a specific sampling strategy problem. When only a portion of a circle is available for measurement, the metrologist must determine the optimal number of sampling points (ranging from 3 to 7 in this study) and their angular distribution to minimize measurement uncertainty. While roundness testers may capture thousands of points, manual coordinate measuring machines (CMMs), profile projectors, and measuring microscopes often rely on limited samples. A smaller sample size reduces inspection time but increases the uncertainty in the derived parameters (radius and center coordinates). The challenge lies in identifying a sampling strategy that yields acceptable uncertainty levels without the computational or temporal cost of dense sampling.

Methodology
The study systematically evaluates sampling strategies for three types of partial circles: quarter circles, half circles, and three-quarter circles. The research assumes a standard radius of 100 mm and models the uncertainty of x and y coordinate measurements as independent normal distributions with a standard uncertainty of 1 µm. Form error is not explicitly modeled, as the authors posit that for small samples, its effect would uniformly inflate uncertainties regardless of the strategy, and angular variations due to form error would be negligible.

The core methodology employs Monte Carlo simulation (100,000 trials) in accordance with GUM Supplement 1 to propagate uncertainties. For each partial circle type and sample size (3 to 7 points), various angular distributions were tested. The "best fit" was determined using the Least Squares Circle criterion. The study calculates the standard uncertainties for the radius ( $u(r_0)$ ) and the x and y coordinates of the center ( $u(x_0)$ and $u(y_0)$ ) for each configuration. These results are compared against a "naïve" uniform equidistant sampling strategy to identify superior arrangements.

Key Contributions and Results
The paper identifies that the "best" sampling strategy is not universal; it depends heavily on the specific parameter (radius, x-center, or y-center) being optimized and the geometry of the partial circle. Contrary to intuitive expectations, uniform spacing does not generally yield the minimum uncertainty.

Quarter Circles:
- 3 Points: The optimal strategy places outer points at the extreme ends of the arc and the middle point symmetrically in the center ( $\alpha=0^\circ, \delta=45^\circ$ ), which coincides with uniform distribution.
- 4 to 7 Points: The results become counter-intuitive. For minimizing uncertainty in the radius ( $r_0$ ) and x-coordinate ( $x_0$ ), the optimal strategy often involves clustering inner points at the center of the arc ( $\alpha=0^\circ$ ) while keeping outer points at the extremes. Conversely, minimizing uncertainty in the y-coordinate ( $y_0$ ) often requires shifting inner points away from the center (e.g., to $30^\circ$ or $45^\circ$ ). Uniform sampling consistently yields higher uncertainties than these optimized strategies.
Half Circles:
- 3 Points: Similar to quarter circles, the optimal strategy places outer points at the extremes and the middle point at the center ( $\alpha=0^\circ, \delta=90^\circ$ ).
- 4 to 7 Points: For $r_0$ and $x_0$ , the best strategy involves placing outer and intermediate points at the extreme ends of the arc, with innermost points at the center. For $y_0$ , the innermost points should be positioned at specific angles (e.g., $75^\circ$ for 6 points) rather than the center. Uniform sampling again performs sub-optimally.
Three-Quarter Circles:
- As the arc approaches a full circle, the optimal strategies shift. For 3 points, the best radius uncertainty is achieved at $\delta=120^\circ$ , not the extreme ends.
- For 4 to 7 points, the optimal strategies for $r_0$ and $x_0$ often involve placing points along the axis of symmetry or at specific angles (e.g., $120^\circ$ or $135^\circ$ ) rather than strictly at the arc's physical extremes. Uniform sampling provides a "generally satisfactory" but not optimal result.

Significance and Claims
The paper claims to provide metrology practitioners with specific, pre-calculated data to plan inspection strategies for partial circles when sample sizes are limited (3–7 points). The primary significance lies in demonstrating that the intuitive approach of uniform equidistant sampling is generally inferior to optimized, non-uniform distributions for minimizing specific parameter uncertainties.

The authors emphasize that the "best" strategy is not obvious and varies significantly based on the target parameter. Consequently, they argue that sampling strategies must be planned in advance based on the required uncertainty for the specific feature being measured. The provided data serves as a reference to help practitioners meet specific uncertainty requirements without resorting to excessive sampling, particularly for the complex case of quarter circles where the deviation from uniform strategies is most pronounced. The study refrains from addressing form error computation, noting that small samples are generally inadequate for that purpose.

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