Prisma

A tool to accurately assess the geometry of a prismatic rule

Prisma is a tool intended to assess the geometry of a triangular prismatic rule. Such rules are used in machining for example to mark and then scrap machine dovetails.

As the sides of the rules are slanted, often at 45°, 55° or 60°, it is not straightforward to check accurately their angles.

This tool allows to measure the horizontal distance of two sides at various heights and to deduce the overall geometry, i.e. the included angles at the three vertices and the global scale.

Measurement principle

The cross-section of a prismatic rule is a triangle. If the rules are properly scraped, their sides should be really close to theoretical straight line segments. Their vertices, on the other hand, cannot be assumed to be perfect points joining the segments, they present either tiny chamfers or radius, and these features are not consistent along the length of the rule. The vertices therefore cannot reliably be used as the basis for measuring the triangular cross-section.

The classical way to measure slanted surfaces is to use a cylindrical gauge pin of known diameter $d_k$ that will touch the slanted surface at an intermediate point and to use a caliper or micrometer to measure up to the external side of the pin. As the contact point between the cylindrical pin and the slanted surface is an intermediate point and not one of the vertices, it is accurately located on the slanted surface. As the pin is convex, the measurement point is easy to access. As the pin has known diameter, taking into account its effect on the measurement is possible, hence allowing to recover the position of the contact point on the slanted surface.

The following sketch shows how this principle can be used to check the triangular cross-section of a prismatic rule. The cross-section is represented by the green triangle with vertices $A_1$, $A_2$ and $A_3$. The unknowns are the three included angles $\alpha_1$, $\alpha_2$ and $\alpha_3$ and the radius $R$ of the triangle circumscribed circle.

This sketch was created using Geogebra. The source file is included at the top of this repository, it has sliders to play with the height $h_k$, the diameter $d_k$, the radius $R$ and the two angles $\alpha_1$ and $\alpha_2$ (choosing these two angles enforces the value of the third angle $\alpha_3$).

In order to be checked, the prismatic rule must be laid on a reference flat (typically a surface plate). Two cylindrical gauge pins of known diameter $d_k$ (displayed in light orange) are laid on two spacer blocks of height $h_k$ (hatched in black) in such a way they touch the sides of the prismatic rule with tangential contact points $T_{A_2,A_3}$ and $T_{A_1,A_3}$. The operators use a caliper or a micrometer to perform measurement $m_k$ as the horizontal distance between the external points $C_1$ and $C_2$ of the cylindrical pins. It is of course possible to not use any spacer blocks at all, which corresponds to have $h_k=0$ in all equations below.

It is important to use the same diameter $d_k$ for both cylindrical pins and to use the same height $h_k$ for both spacer blocks. This ensures that the points $C_1$ and $C_2$ are at the same height (which is $\frac{2h_k+d_k}{2}$) and hence that the measurement is accurate as the caliper or micrometer anvils touch the cylindrical pins properly. This property remains fulfilled even despite the contact points $T_{A_2,A_3}$ and $T_{A_1,A_3}$ between the cylindrical pins and the prismatic rule are not at the same height with respect to the surface plate. These contact points would be at the same height only if the two included angles $\alpha_1$ and $\alpha_2$ were equal. This measurement principle therefore works regardless of the shape of the prismatic rule; it works even for irregular prismatic rules.

The expression of the measurement $m_k$ is: $$m_k=2R\sin(\alpha_1+\alpha_2)+\frac{d_k(1+\sin\alpha_1)-(d_k+2h_k)\cos\alpha_1}{2\sin\alpha_1}+\frac{d_k(1+\sin\alpha_2)-(d_k+2h_k)\cos\alpha_2}{2\sin\alpha_2}$$

This measurement depends only on the radius $R$, on the two angles $\alpha_1$ and $\alpha_2$, on the cylindrical pins diameter $d_k$ and on the spacer blocks height $h_k$. Using several different diameters $d_k$ and several different heights $h_k$ therefore leads to several different measurements $m_k$. The goal of the prisma program is to almost invert the equation above, i.e. to extract $\alpha_1$, $\alpha_2$ and $R$ given a set of measurements $m_k$. The third angle $\alpha_3$ can be deduced from the two other ones because the sum of the three angles of a triangle is always 180° (or $\pi$ radians).

In order to be able to perform the measurements $m_k$, a few conditions must be fulfilled. Given some spacer block height $h_k$, the diameter $d_k$ of the cylindrical pin must not be too small, otherwise either the spacer block would come too close to the base of the triangle and eventually collide with it or the pin would fall between the slanted surface and the spacer block and would not be laid on the top of the spacer block. On the other hand having a too large diameter $d_k$ would imply the contact point between the cylindrical pin and the triangle goes upwards and finally exceeds the height of the triangle, so there would be no tangential contacts anymore. For a given spacer block height $h_k$, there are therefore both a lower limit and an upper limit on the diameter $d_k$ of the cylindrical pins that can be used. This can be seen clearly by playing with the sliders in the Geogebra application.

The conditions to be fulfilled for the cylindrical pin on the right hand side are: $$h_k\frac{2\cos\alpha_2}{1-\cos\alpha_2} < d_k < \frac{4R\sin\alpha_1\sin\alpha_2-2h_k}{1-\cos\alpha_2}$$

Similar conditions apply to the cylindrical pin on the left hand side, swapping $\alpha_1$ and $\alpha_2$ in the previous equation.

If for example the prismatic rule is an isosceles triangle (all 3 angles are 60° angles or $\pi/3$ radians), then the limits for the pins diameter are: $2h_k < d_k < 6R - 4h_k$, which is a non-empty interval only if $h_k < R$.

How it works

The equation above that gives $m_k$ cannot be inverted directly (i.e. we cannot find the unknowns $R$, $\alpha_1$ and $\alpha_2$ directly from several measurements $m_k$). The prisma program works by using several measurements $m_k$ (as many as the operator is willing to perform), and finding the set $R$, $\alpha_1$, $\alpha_2$ and $\alpha_3$ that best match the measurements, using the Levenberg-Marquardt method to solve the least squares problem. At least three measurements are needed, but the more, the better.

Using several diameters $d_k$ and several heights $h_k$ allows to perform multiple measurements with contact points at varying heights along the sides of the triangle. If a sufficient number of measurements is taken, then it is theoretically possible to retrieve the full geometry (i.e. $R$, $\alpha_1$, $\alpha_2$ and $\alpha_3$) without moving the rule at all. It is however recommended to perform three series of measurements, turning the prismatic rule around between each series. A first series should be performed with the $(A_1,A_2)$ side on the surface plate as shown, a second series should be performed with the $(A_2,A_3)$ side on the surface plate and a third series should be performed with the $(A_3,A_1)$ side on the surface plate. Using multiple independent series leverages measurements errors.

The measurements $m_k$ are provided to the prisma program as a simple text file with each line giving the name of the top vertex ($A_1$, $A_2$ or $A_3$), the diameter $d_k$ of the pins, the height $h_k$ of the spacer blocks, and the value $m_k$ that was measured with these settings. The program solves the least squares problem and provides both the geometrical characteristics $R$, $\alpha_1$, $\alpha_2$, and $\alpha_3$ with an estimate of the accuracy of these values as well as the global RMS (Root Mean Square). It also optionally provides the residuals $m_k -\tilde{m}_k$ where $\tilde{m}_k$ are the theoretical measurements that should have been obtained with the estimated values for $R$, $\alpha_1$, $\alpha_2$, and $\alpha_3$ if the measurements were perfect. This gives an indication of how good or bad the raw measurements were. The least squares problem solved is really finding $R$, $\alpha_1$, $\alpha_2$, and $\alpha_3$ that minimize $\sum_k \left(m_k -\tilde{m}_k\right)^2$.

License

Prisma is licensed by Luc Maisonobe under the Apache License, version 2.0. A copy of this license is provided in the LICENSE.txt file.