Digital Measuring Tapes. Our Answer to Dude would you hold my tape
We have an idea about digital measuring tapes motivated from repeatedly overhearing the question “Dude would you hold my tape?” on work sites. We think it’s absurd question, and two people are not needed to take an accurate measurement. Our goal is a “point and click” tool to find distances between arbitrary points in an open space volume.
If we are measuring distances inside a volume \(V\) with boundary \(\partial V\), this means evaluating distances for arbitrary pairs $ (x,y) V V$. All the laser pointers currently on the markey, e.g. Amazon, strictly find distances between boundary points \((x,y) \in \partial V \times \partial V\). This distinction between distances in the volume and distance on the boundary is key.
Formally the problem is this: let \(C\subset {\bf{R}}^3\) be a set of centres in space. Suppose we are given two arbitrary points \(x, y\in {\bf{R}}^3\) and we are given the pairwise distances from \(x,y\) to the centres, i.e. we are given the distances $ { dist(x,c)}$ and \(\{dist(y,c)\}\) for every \(c\in C\). The problem is how to decide \(dist(x,y)\) from these measurements. In the literature this problem is called “trilation” as opposed to “triangulation” where distances are inferred from distances instead of angles. There is direct interpretation of the trilation problem via sphere intersections. We also explore a Choquet max entropy method below.
Remark. We must admit there is wonderful simplicity to conventional measuring tapes. The contractor spans the tape across the distance, hooking the edge of the tape at a given edge, and then estimates the distance with his eye looking at the “end” of the tape in their hand. This basic application requires the tape have a well layed position and be human readable. For simple situations this works fine. But the contractor has many instances throughout a day where awkward measurements need be taken. Here we think the point and click distance finder could work. No physical spans are required.
Sphere Intersections
The measuring tape problem is related to the fast solving of sphere intersections. If \(S_1 = S(c_1, r_1)\) and \(S_2 = S(c_2, r_2)\) are spheres in \({\bf{R}}^3\), then their intersection \(S_1 \cap S_2\) is a sphere orthogonal to the vector \(c_{12}:=c_1-c_2\). This is immediately seen in figure below [insert image]. Thus the immediate problem is to decide when spheres have empty or nonempty intersection. Generically the intersection is either empty or a codimension one sphere orthogonal to \(c_{12}\). Thus we have a bisection type method to decide sphere intersections, namely $S_1 S_2 S_3 $ is equal to \(S_{12} \cap S_{34} \cap \cdots\) which is equal to \(S_{1234} \cap \cdots\) etc.
Remark. If the centres \(c\in C\) are infinitely far away from the points \(x,y\). Then the spheres centred at \(c\) will intersect \(x,y\) at large radius and appear as affine subspaces. Therefore when the centres \(C\) are at infinity, the problem reduces to linear algebra. Given normal vectors \(\{n_i\}\), decide the distance \(dist(x,y)\) from the values of the linear functionals \(\{\langle n_i, x \rangle\}_i\), \(\{\langle n_i, y \rangle\}_i\) for given \(x,y\). This means solving an inhomogeneous linear system of equations.
Euclidean Distance Formula from Choquet Representation.
Now our idea is to use Choquet representation theorem. Given the distance to centres \(dist(x,c)\), \(c\in C\), we construct the maximal entropy measure \(\lambda_x\) supported on \(C\) which represents \(x\). The measure \(\lambda_x\) represents \(x\) in the sense of Choquet Representation theorem [ref] if \[\ell(x)=\int_C \ell(\bar{x}) d\lambda_x(\bar{x})\] for every linear functional \(\ell\) on \({\bf{R}}^3\). We emphasize the linearity of \(\ell\).
If we study the Euclidean distance, then we deduce the following representation formula:
\[dist(x,y)^2:=|x-y|^2~=~\iint_{C \times C} \langle c_i, c_j \rangle ~~ d(\lambda_x - \lambda_y) \otimes d(\lambda_x - \lambda_y).\] Therefore we find the squared euclidean distance is a \(\lambda_x\) weighted sum of the signed dot products \(\langle c_i, c_j \rangle\) of centres .
We define \(\lambda=\lambda_x\) as the unique probability measure on \(C\) which maximizes the entropy \(H(\lambda)\) subject to the linear constraints \(\sum_i \lambda_i = 1\) and \(\sum_i \lambda_i c_i=x\). For [eq1] to be an efficient formula, we need a fast algorithm to represent \(\lambda_x\) given the distance to centres \(dist(x,c)\).
The method of Lagrange multipliers is based on the observation that \(H\) is optimized given the constraints when \(\nabla_\lambda H\) is linearly dependant with the constraint gradients. The Lagrangian for this optimization problem is \[L(\lambda, \alpha, \beta):=H(\lambda)+\alpha(\sum_i \lambda_i -1) + \langle \beta, ~~\sum_i \lambda_i c_i -x \rangle\] where \(\lambda \in {{\bf{R}}^I}\), \(\alpha \in \bf{R}\), \(\beta \in {\bf{R}}^3\).
If \(\lambda\) is maximizer, then we have vanishing partial derivatives \[\frac{\partial L}{\partial \lambda_i}=\log \lambda_i +1 + \alpha + \langle \beta, ~ c_i \rangle=0, ~~\frac{\partial L}{\partial \alpha}=0, ~~ \nabla_\beta L=0.\]
Now we need solve for the variables \(\alpha, \beta, \lambda\) using the above equations. We reproduce the calculation below.
First we have \(\lambda_i = e^{-(1+\alpha)} ~~ e^{-\langle \beta, c_i \rangle}.\)
The condition \(\sum \lambda_i=1\) implies \(e^{1+\alpha} =\sum_i e^{-\langle \beta, c_i \rangle}\). Consequently we find
\[\lambda_i = \frac{1}{\sum_i e^{-\langle \beta, c_i \rangle}}~~ e^{-\langle \beta, c_i \rangle}.\]
Next the condition \(\sum \lambda_i c_i = x\) implies \[\sum_i e^{-\langle \beta, c_i \rangle} ~ (c_i - x)=0.\] Thus we have reduced everything to solving for \(\beta \in {\bf{R}}^3\) in this equation. In this case the auxiliary mapping \(F:{\bf{R}}^3 \to {\bf{R}}^3\) defined by \(F(\beta):= \sum_i e^{-\langle \beta, c_i \rangle} ~ (c_i - x)\) is nonlinear. The max entropy measure \(\lambda\) is defined by the equation [lambda] where \(\beta\) is the solution to \(F(\beta)=0\). We observe that \(F(0)=\sum_i c_i-x\). The basic idea is that \(F\) is monotone in \(\beta \in {\bf{R}}^3\), i.e. the partial derivatives \(\frac{\partial F}{\partial \beta_1}\), \(\frac{\partial F}{\partial \beta_2}\), \(\frac{\partial F}{\partial \beta_3}\) are nonvanishing and have constant sign.
Action at a Distance: How to Physically Measure Distances
We have commented on the simplicity of the physical measuring tape. But are there other physical principles to determine distances? Evidently the measuring tape requires a physical tape connecting two distinct points in space. But can we measure distance at a distance? How did the Mars rovers determine their positions on the Mars surface. Was position and distance found indirectly via satellite imagery? For example, how does one really decide the Earth-Moon distance?
We are inspired to reconsider action at a distance forces.
Can we use forces to measure distances?
Sir Isaac Newton’s 1640s gravitational force law introduced instantaneous action at a distance into physics. Newton unified the forces of the tides, the moon, orbits of the planets, falling bodies. Before Newton it was believed that the orbits of the planets were circular, and that uniform circular motion was perfect and requiring no external causes to persist. However for our purposes gravity is too weak a force to use for interaction measurements.
Electrostatic forces between a proton-electron pair has a magnitude \(39 x\) greater than gravitational attraction. For example, one might naively consider using Coloumb’s law of electrostatic repulsion to determine distance. This would involve a unit charge at position \(x_1\), and a unit charge at \(x_2\), and measuring the net force. However the net resultant force \(F\) at \(x_1\) is not necessarily equal to the Coulomb force \(F_{21}\). This is because the system is not necessarily electrically isolated! Instead we consider \[F_{\text{net}, 1}=F_{21} + F_{\text{env},1}\] where \(F_{\text{env},1}\) is the resultant environmental force on \(x_1\). In this formula assume that the introduction of the electrical charge at \(x_1\) and \(x_2\) does not affect the environmental forces \(F_{\text{env},1}\), \(F_{\text{env},2}\). This is an idealized situation. The Coulomb force is deduced only after subtracting the net background force \(F_{\text{env},1}\) from the resultant force. And this is the main idea: we have to make two measurements for \(F_{\text{env}}\) and \(F_{\text{net}}\) and deduce the interaction as a difference.
We emphasize the deduction because this is a logical construct. As we have commented there is possibility that the introduction of the electric charges at \(x_1\), \(x_2\) has an effect on the environmental forces.
The basic interaction \(F_{21}\) does not necessarily have to be Coulomb’s interaction. We could replace the point charges at \(x_1, x_2\) with, say, ring currents or dipoles.
[to be continued … JHM]