Cetvel: Precision Measurement Scale Devices

• Cetvel is a graduated measurement scale that converts geometric displacement into precise quantitative readings through dual-slider alignment.
• It operates as both a linear measuring device with fine adjustment and a physical slide-rule that encodes numerical transformations via a monotone function.
• Its hydrostatic application employs ultrasonic timing for continuous leveling, critical for accelerator alignment and structural monitoring.

Searching arXiv for the cited papers and topic context. I’ll proceed using the provided arXiv records and citations, which already identify the relevant papers for this topic. In the cited literature, cetvel appears as a graduated measurement scale, a ruler-like device for resolving lengths below the minimum marked division, and a slide-rule scale that maps numbers to physical positions through a monotone function (Alsmadi et al., 2014, Szalkai et al., 2017). A related metrological use treats the measurement reference as a distributed hydrostatic levelling system for continuous tracking of vertical changes in buildings and ground, where the “scale” is realized through a shared water reference and calibrated sensing chain (Choi et al., 2016). Across these contexts, the common technical theme is the conversion of geometric displacement into a readable quantitative value.

1. Core technical senses of cetvel

The literature associates cetvel with three distinct but related measurement regimes: direct linear measurement, functional scale construction, and distributed height referencing.

Context	Form of cetvel	Defining principle
Linear measurement	Graduated scale with two sliders	Integer part from a millimeter or inch scale; fractional part from measuring points and a measuring line
Slide rules	Physical strip encoding a function	$d = u \cdot f(x)$ for strictly monotone $f$
Accelerator alignment	Hydrostatic reference network	Relative height inferred from ultrasonic timing against a fixed internal reference

This comparison suggests that cetvel is not restricted to a simple ruler. In the linear device, the scale is read directly and then refined by a geometric cursor mechanism (Alsmadi et al., 2014). In the slide-rule setting, the scale is a physical realization of a numerical transformation (Szalkai et al., 2017). In the hydrostatic system, the reference function is distributed over long distances and used for long-term monitoring rather than one-time manual reading (Choi et al., 2016).

2. Linear measuring device: architecture and components

The linear measuring device proposed in “Measuring device suitable for linear distances” comprises three associated longitudinally moving parts, one of which is the scale itself (Alsmadi et al., 2014). The first part is the main body with the graduated scale, traced in millimeters or inches and used for the integer reading. The second is a transparent slider with measuring points, carrying a row of small, equally spaced points arranged along an inclined line. The third is an adjustable transparent slider with a measuring line and two coinciding grids.

The body also includes jaws for gripping the object being measured, together with a fine-adjustment mechanism consisting of a screw, trapezoidal slider, and return spring. This arrangement is intended to position the sliders precisely while preserving the main scale as the primary reference.

The device separates the reading into two parts. The integer part is taken from the standard graduated scale, while the fine measurement, smaller than the minimum scale division, is produced by the interaction of the two sliders. The paper states that the integer part is the graduation point $n$ on the main scale:

$\text{integer part} = n$

Compared with a standard ruler, the device is presented as offering higher precision, improved readability, better exactness in marking the fractional part, and a clearer fine-reading mechanism (Alsmadi et al., 2014). The comparison with a vernier-like cursor system is explicit in spirit but not literal in construction: the mechanism depends on point-line coincidence and controlled relative displacement rather than on classical vernier subdivision.

3. Fine reading, point-line coincidence, and geometric basis

The measurement procedure is specified stepwise (Alsmadi et al., 2014). The lock pin is released, the first measuring point is adjusted so that it “kisses” the measuring line, the object is clamped, the integer part is read from the main scale, the screw is moved until the two coinciding grids align with adjacent graduation lines, and the final reading is obtained from the point intersected by the measuring line.

The measuring points are described as small, equally spaced dots arranged along an inclined line. The paper states that points replace a continuous line because they can give a clearer and more precise indication of intersection. The measuring line is a single line on the second slider, positioned so that it can coincide with one of the measuring points or interact with the graduation lines on the main scale. The decimal part of the reading is related to the fractions of the displacement between the graduated scale and the corresponding measuring line.

The paper formalizes this using a geometric model in which the scale is treated as a one-dimensional line extended into a two-dimensional surface. A point $A_n$ on the scale has a corresponding image point $B_n$ such that

$A_nB_n = AB$

and a sloped line $H$ connects adjacent points with length

$H = \sqrt{4^2 + AB^2}.$

Its projection onto the scale direction is written as $H\cos\theta$. The paper further states that the measuring line can be slid so that the lower end coincides with a graduation point while the other end coincides with the image of the next point, making the fractional reading measurable.

For a chosen calibration, the numerical example gives

• $f$0,
• $f$1,
• $f$2,

and the device equation reduces to

$f$3

The general reading principle remains

$f$4

with the fractional part determined by which measuring point aligns with the measuring line or the corresponding scale graduation. A plausible implication is that the device operationalizes decimal subdivision without requiring finer engraved base graduations.

4. Cetvel as a function on slide rules

In “Constructing and Understanding New and Old Scales on Slide Rules,” a scale is defined by placing each value $f$5 at a distance

$f$6

from the left end $f$7, where $f$8 is strictly monotone and $f$9 is a scale unit determined by the available length and intended numeric range (Szalkai et al., 2017). In this formulation, a cetvel is a geometric encoding of a monotone function.

The paper applies this framework to traditional and new scales. For the reciprocal scale $n$0, the law is

$n$1

so the order of markings reverses. In the example, $n$2 are placed at distances $n$3, and no finite number appears at the very start because $n$4 is impossible; the left endpoint is conceptually associated with $n$5. For the traditional $n$6 and $n$7 scales, the distance is logarithmic:

$n$8

which explains why they begin at $n$9, since $\text{integer part} = n$0.

A central clarification in the paper is that the symbolic meaning of a scale and its actual distance law are not identical. In common slide-rule language, $\text{integer part} = n$1 and $\text{integer part} = n$2 are described as “$\text{integer part} = n$3-scales,” $\text{integer part} = n$4 as an “$\text{integer part} = n$5-scale,” $\text{integer part} = n$6 as a “$\text{integer part} = n$7-scale,” and $\text{integer part} = n$8 as an “$\text{integer part} = n$9-scale.” Mathematically, however, the distance functions behind them differ: $A_n$0 and $A_n$1 use $A_n$2, $A_n$3 uses $A_n$4, $A_n$5 is equidistant with $A_n$6, and the $A_n$7 scales involve nested logarithmic structure (Szalkai et al., 2017). This distinction addresses a common misconception: the printed algebraic role of a scale is not necessarily its literal placement rule.

5. Range, zoom, readability, and non-redundant scale design

A slide rule has only a short physical length, so a single scale can display only a bounded interval. The paper stresses that one cannot show very small and very large numbers in the same way, and that the local slope of the function directly controls readability (Szalkai et al., 2017). The geodetic or horizon-related scale

$A_n$8

is cited as an example: it is steep for small $A_n$9 and nearly horizontal for large $B_n$0, which spreads marks on one end and compresses them on the other.

To manage this, the paper formalizes zooming. If the base formula is $B_n$1, then the physical distance may be written as

$B_n$2

with any positive factor $B_n$3. The starting point $B_n$4 still corresponds to the same mathematical value determined by $B_n$5, but the maximum value can be changed by selecting the zoom. This is used to explain why quadratic scales can appear in different versions such as $B_n$6 and $B_n$7, and why traditional scales $B_n$8, $B_n$9, and $A_nB_n = AB$0 are related by constant factors:

$A_nB_n = AB$1

The paper then introduces homogeneity:

$A_nB_n = AB$2

which includes power laws $A_nB_n = AB$3. For comparable scales of the same functional type and equal physical length,

$A_nB_n = AB$4

and the range ratio is

$A_nB_n = AB$5

Points align across the two scales exactly when

$A_nB_n = AB$6

The practical rule is explicit: one should avoid choosing $A_nB_n = AB$7 to be a power of $A_nB_n = AB$8 or a simple rational number, because then one scale becomes effectively useless relative to the other. The paper recommends more irrational-looking ratios such as $A_nB_n = AB$9, $H$0, $H$1, or $H$2.

Readability is treated as a quantitative constraint. To distinguish $H$3 from $H$4, the physical separation must satisfy

$H$5

For homogeneous $H$6, this becomes

$H$7

Because spacing is not uniform, a scale must be designed so that the least favorable end still meets the threshold. The paper notes that on a quadratic scale spacing increases with $H$8, while on a reciprocal scale it decreases.

The quadratic scale $H$9 is then examined for the computation

$H = \sqrt{4^2 + AB^2}.$0

With

$H = \sqrt{4^2 + AB^2}.$1

the corresponding angle bounds are

$H = \sqrt{4^2 + AB^2}.$2

and the resulting range of $H = \sqrt{4^2 + AB^2}.$3 is

$H = \sqrt{4^2 + AB^2}.$4

The worked example uses $H = \sqrt{4^2 + AB^2}.$5 mm, $H = \sqrt{4^2 + AB^2}.$6 mm, and $H = \sqrt{4^2 + AB^2}.$7, giving

$H = \sqrt{4^2 + AB^2}.$8

with

$H = \sqrt{4^2 + AB^2}.$9

For $H\cos\theta$0, the paper derives

$H\cos\theta$1

and

$H\cos\theta$2

Hence the scale can solve triangles with

$H\cos\theta$3

and opposite angle between $H\cos\theta$4 and $H\cos\theta$5. Here cetvel is not merely a display; it is a physical computational domain whose usable interval is fixed by length, zoom, and readability.

6. Hydrostatic levelling as a distributed reference scale

At PAL-XFEL, the relevant measurement problem is not a single linear dimension but the continuous monitoring of vertical changes in buildings and ground that affect accelerator alignment (Choi et al., 2016). The facility is intended to maintain bunch-to-bunch beam parameters of 60 Hz, 10 GeV, 200 pC, 60 fs, and Emittance X/Y 0.481 / 0.256 mm·rad, while alignment tolerances are ±100 μm for the linear accelerator and ±50 μm for the undulator. The paper states that if the ground or building vertically changes after alignment, then magnets, RF structures, and other components move out of position, producing electron beam trajectory errors and changes to beam parameters.

The BINP ultrasonic-type Hydrostatic Levelling System (HLS) is introduced as a precision, long-term height-monitoring system for the Linac, undulator / insertion-device region, and beam line / beam transfer line. It monitors ground sinking, ground uplifting, building floor deformation, and foundation settlement, and the measured data support accelerator alignment and re-alignment decisions.

The operating principle is hydrostatic leveling in a connected water system, but the BINP design uses an ultrasonic transducer rather than a capacitive sensor. The transducer sends pulses toward an absolute reference reflector called an “absolute ruler” and toward the water surface. The fixed reference distance is

$H\cos\theta$6

and the unknown distance is obtained from the echo timing ratio:

$H\cos\theta$7

Because the same electronic chain affects both timing measurements, the ratio method removes timing jitter and time-delay errors and is therefore the basis of the system’s self-calibration.

The paper gives the distance resolution as

$H\cos\theta$8

The electronics chain includes an ultrasonic transducer, comparator, Time-to-Digital Converter (TDC), microcontroller, temperature sensor, ADC, power-over-Ethernet (PoE) / DC-DC power conversion, and an 8 MHz system clock oscillator. The TDC resolution is 125 ps. Since sound velocity in water is approximately 1500 m/s at 25°C, the paper also emphasizes temperature measurement and temperature-aware correction of the measured height.

7. Installation, calibration, and interpretation of long-term reference data

The PAL-XFEL installation concept distributes the HLS throughout the Linac tunnel, Undulator hall, and Beam transfer line / beam line sections (Choi et al., 2016). The paper gives a water pipe total length of 291 m and an accelerator water pipe total length of 709 m. Section lengths shown in the installation figures include 82 m for the Beamline Experiment Hall, 245 m for the Hard X-ray Undulator Hall, 57 m for the Beam Transfer Line, and 718 m for the Linac tunnel in the foundation figure.

The pipe material is anti-corrosive stainless steel SUS304 with sanitary surface treatment. Hydraulic behavior is treated as part of the measurement design: the paper states that the pipe must allow good fluidity, the diameter must be selected according to pipe length, thermal deformation of the support should be minimized, and support length should be shortened to reduce temperature-induced deformation. The selected pipes are Diameter 60.5 mm with water depth 30.25 mm and Diameter 76.3 mm with water depth 38.15 mm. The system is half-filled, because the paper argues that a half-filled pipe gives accurate measurement behavior and acceptable stabilization time.

Calibration is divided into two modes. Absolute calibration is performed using a laser tracker and should be done once or twice a year, because it takes time. The relation is written as

$H\cos\theta$9

After this, relative routine measurement uses one HLS station as a reference. In the given example, 0 mm means the same level as the reference, +1 mm means uplift, and −1 mm means sinking. The reference HLS itself should be rechecked with a laser tracker once or twice a month.

A crucial interpretive limitation is that the HLS signal does not represent only true ground motion. The paper explicitly lists temperature, atmospheric pressure, gravity/tidal effects, vibration, humidity/evaporation, water-surface disturbances, and drift and irregular noise as contributors to the observed data. To separate these effects, the paper mentions BAYTAP-G, a Bayesian tidal-analysis program that decomposes the record into tidal part, temperature part, irregular part, and drift part. This addresses a common misconception in long-baseline metrology: a precise reference network is not automatically a direct measure of structural displacement; environmental decomposition is part of the measurement problem itself.

The foundation context reinforces the need for such monitoring. The facility was built on terrain with an elevation around 62 m, with removal of weak or weathered-zone soil, replacement with concrete in some regions, and bedrock-based construction rather than pile foundation. Even under those conditions, the paper emphasizes that the building floor can still undergo deformation due to subsidence or uplift of the foundation. In this sense, the hydrostatic system functions as a large-scale cetvel for alignment maintenance: a distributed reference against which extremely small vertical changes can be detected continuously over time.