Coordinate Graphing Geometry Project: Fill & Download for Free

By sheer mathematics.There is a shop called transmitting station somewhere in the bilge of the battleship where the brain of the ship, the Fire Control Table, is hidden, and the calculations are performed there. See Category:Fire Control on Dreadnought Project webpage.Somewhere high on the superstructure is another shop, Director-Control Tower (DCT), also known as Gunnery Director, where the Fire Control Officer sits. This shop is basically a trainable small gunless turret incorporating the gun-laying sights and often an own rangefinder. From here the gunnery officer could select targets and take the target on his gunsights and fire it by using a central firing key.5–10 miles mean 8 to 16 km. That is point-blank range for a battleship, if you mean statute miles. But since we are at sea, you mean nautical miles. It makes situation more interesting.The eyes of the battleship are the Rangefinders, which are basically horizontal binoculars with extremely long distance between the ‘eyes’ allowing either stereoscopic or coincidental triangulation. By adjusting the prisms and measuring the angle on which the prism is set compared to neutral position, the distance and bearing to the target could be determined. If using metres as unit, it would be a trivial calculation using milliradians or “strokes”.Battleship Fuso of the Imperial Japanese Navy. The rangefinder is located on the top of the pagoda superstructure.The other input data would be provided by ship’s own instruments (speedometer or ‘log’ in Maritimese and gyrocompass) and from weather station (thermometer, anemometer and barometer).The input data would be:Own speed, from logOwn heading, from gyrocompassDistance to enemy, from rangefinderEnemy speed, from rangefinderEnemy heading, from rangefinderRange rate, from rate officerCorrections, from fire control officerAir temperature, from weather stationAir pressure, from weather stationWind speed, from weather stationWind direction, from weather stationThese data are entered in the Transmitting Station on the Fire Control Table, which is basically a mechanical computer. The input data is entered in by setting of various knobs, gauges and devices, and it will calculate two variables as output data: gun barrel azimuth and elevation. Since the propellant charge is pre-measured (and will produce a constant amount of energy), the range is produced by the elevation angle of the cannon barrels. Things like air temperature, air pressure, wind speed and wind direction will effect on the propellant conflagration and must be taken into account.Dreyer Fire Control Table of HMS Hood (WWI vintage)Admiralty Fire Control Table (HMS Belfast, WWII)The heart of the Fire Control Table is a device called Dumaresq. A dumaresq functions by mechanically implementing a flexible means of illustrating the geometry of two independently moving ships to quickly perceive the other ship's motion along and across the line of bearing, these values being called the range rate and the speed-across, respectively. Specifically, it permitted a user to subtract the motion vector of your own ship from that of a target ship to yield the target's relative motion vector which could then be projected onto a Cartesian coordinate graph oriented along the line of bearing, permitting the range rate and speed-across to be read off as a coordinate pair.Dumaresq of HMS Hood. See the illustration of the fire control table; the dumaresq is located at center right.Users commonly set up a dumaresq in this manner:set their own ship's speed and headingset the target ship's speed and heading (by estimation)rotate a pair of coordinate axes along the line of bearing to the targetread the range rate off one axis of the graph and the speed across off the otherHowever, the dumaresq was not a modern computer which used a given set of inputs to a given set of outputs. It was wholly indifferent to which of the variables it represented in its workings were inputs and which were outputs. Altering one variable by rotating a knob or by positioning a slider caused other variables to change to continually satisfy the relationship between two ships moving around and the range rate and speed-across their relative motion would imply. This meant that the sequence of use described above which derives range rate and speed-across from the speed and heading of both vessels and the target bearing could be worked flipped on its head to find the enemy speed and heading from a given range rate and speed-across. This was called a "cross-cut".The dumaresq was supplanted with Argo Clock which kept the variables continuously moving as function of time.Once the firing solution is resolved and all barrels charged, correct azimuth given to turrets and barrels have correct elevation, the Fire Control Officer at Gunnery Director , also kown as director-control tower (DCT), will use the firing key. On Royal Navy, this was a pistol grip mounted on the gunnery director device; in USN, it was a handle-like device connected to the Ford Rangekeeper Computer. The Kriegsmarine firing key was strange: the fire control officer, who operated the rangefinder, blew in a mouthpiece which connected the circuit. Imperial German Navy did not have centralized firing, but instead the guns were fired individually at turrets by using a lanyard.Once the firing key was used, an electric impulse would detonate the primers, which would in turn fire the propellant on cannons and fling the shells to enemy. To encounter the sea swell the firing system was connected with a mercury switch which would enable firing only when the ship was on level position.The first shots were calculated deliberately so that some shells would go long while others would go short. This is called a ladder shot; its intention was to straddle the enemy (find the corrected range by finding the closest “short” miss and closest “long” miss). By observing the shell sprays hitting the water and the intended range the corrections could now be made. The corrections would now be set to the Fire Control Table. Usually the correct range and thus correct gun azimuth and elevation was found after three or four salvos.Once the output data had been obtained, the Gunnery Director crew was now on charge of firing. A director is the equipment occupying a centralised firing position on a naval ship. It is usually located below the rangefinder. It tells the guns where to point and when to fire, and thus harmonises their destruction in coherent salvoes.There would now be a triangle data link between gunnery director, rangefinder and transmitting station, each providing new data and correction readings on base of observation of the shells hitting.In its fullest implementation, the director firing relies upon a director and its crew to indicate the proper elevation and azimuth for the barrels and turrets of a group of guns, as well as providing the firing impulse for a simultaneous salvo.A director is very similar to a small gun mounting except that it lacks a gun. The output data obtained from Fire Control Table is fed to the director continuously. A typical gunnery director has a calibrated sight that can be set to a desired range and deflection, and hand wheels and telescopes by which crew can elevate and train it such that the crosshairs are on the object to be fired upon. The elevation angle and training angle resulting from these motions is continually signaled by electro-mechanical data circuits to the guns so the gun crews can mimic the proper motions. When the crosshairs are correctly positioned, a press on a pistol-type trigger sends an electrical impulse to cause the guns to fire.Directors varied in their particulars according to nation and type. Amongst the most complete of the World War I era was the Royal Navy's Vickers Tripod-Mounted Director, which housed 4 crew: a Fire Control Officer acting as the director layer, a director trainer, a sightsetter, and a talker.The talker was equipped with a telephone headset and often a flexible speaking tube snaking in from the spotting top above. He would act as the communication interface for any miscellaneous commands and criticisms that were not fully conveyable by the data receivers and gongs nearby.The sightsetter had a follow-the-pointer sight to adjust to the range and deflection (gun elevation and azimuth) being continually indicated upon it from the transmitting station. As on any pedestal-mounted gun with a settable sight, this action rotated a pair of telescopes in a backwards manner so the elevation and deflection angles would be applied in the opposite direction.The director trainer would use his handwheels to rotate the entire director assembly (a small bathtub-like cupola on the lip of which all 4 crew sat as if attending a tea party) and the top of the external housing until he saw the target aligned in his telescope.The fire control officer would use his handwheel to elevate his crosshairs (whose elevation had been altered as the sightsetter dialed in the desired gun range). When the target was aligned and a small gun ready display board showed a sufficient number of barrels were ready to fire, he would pull a trigger on a device-mounted pistol grip with his free hand to signal the firing impulse. This in turn would send an electric impulse with the velocity of light to all turrets, which would detonate the primers and fire the guns simultaneously. In the Imperial German Navy, the Fire Control Officer would send a gong signal to the turrets, where the guns were fired individually by pulling a lanyard.Here is a scheme of the Gunnery Director (above) and Transmitting Station (below) and how they work together. Note this is a WWII five-man gunnery director instead of WWI four-man.Gun Range is the range to which the guns should fire to hit the target. This is distinct from clock range, which is the distance from the gun to the target, as in most long range actions, the range to the target will change during the time-of-flight (the time it takes the shell to fly from the barrel to the enemy). The time of flight could be measured with a shell clock.The difference between clock range and gun range was regarded as so fundamental that the Royal Navy saw fit to ensure that their Dreyer Fire Control Tables were adapted to plot both values clearly by use of separate pencils on their range plots. The difference between the two values in the Royal Navy was expressed as a straddle correction and spotting corrections entered into the Spotting Corrector of a Dreyer table.Once radar was introduced in WWII, it revolutionized the gunnery. Now precise range, precise speed of target and precise heading could be obtained, and gunnery became science instead of art.This is reflected by all battleship actions in WWII; the USN battleships simply butchered their Japanese opponents, who did not have decent radar, in very short order.EDIT: The fire control officers and transmitting station ratings were a selected and well trained lot. I visited HMS Belfast in London some years ago, and was amazed of the transmitting station settings.A piece of useless trivia: Finnish architect Viljo Revell was conscripted as the fire control officer of coastal defence ship Ilmarinen, and he survived her sinking. He later went on to design this:Toronto City Hall, designed by Lieutenant Viljo Revell of FNS IlmarinenSo to become a Fire Control Officer, you had to have excellent eyesight, very high intelligence and a good sense of three-dimensional perception. Architects indeed are good on that.

Broadly speaking, machine learning algorithms are “happiest” when presented training data in a vector space. The reasons are not surprising: in a vector space (including the infinite-dimensional Hilbert space generalization), one can compute angles between vectors, which allows doing “projections”, one of the fundamental operations underlying most machine learning algorithms.But, the major challenge in machine learning is that data come from all over: text documents, images, graphs, sensor streams, and so on. Most of these entities do not “live” in a vector space. For example, a graph is not a vector. You cannot “multiply” a graph by a scalar, or add two graphs.Similarly, words in English do not “live” in a vector space. I can’t multiply 3.5 times “happiness” to create more “happiness” (sometimes I wish I could!). Similarly, I cannot add “more” and “money” to increase my bank account.So, whenever you see the word “embedding” in machine learning, what that means is that the authors are exploring some technique to take non-vectorized data and “embed” it into a vector space.Let’s take a couple of examples to get the hang of the concept. For example, let’s say I want to “embed” a graph in a vector space. How would I go about doing that? A simple way to think about this concept is to imagine drawing the graph on paper. That exercise forces you to take every vertex and plonk it down somewhere on the paper. So, in effect, you have constructed a mapping from the set of vertices V of a graph G to pairs of real numbers (the (x,y) coordinate of the point in space with respect to some arbitrary coordinate system). In other words, you have constructed an embedding of the graphSimilarly, for word embeddings, suppose I want to know what the word “Democrat” means. Well, one way to ascribe meaning to this word is to find some way to map “Democrat” into an N-dimensional vector (say, N = 100), so that every instance of “Democrat” in a text document is replaced by that 100-dimensional vector.In both these examples of graph embeddings and word embeddings, the trick of course is to construct an embedding that is “meaningful”. For example, if you map “Democrat” to some N-dimensional vector, and do so for every word in your vocabulary, then the embedding is useful if when I compute the nearest neighbors of “Democrat”, I get words like “Obama” or “Nancy Pelosi” (two famous Democrats!), and I should certainly not get “Trump” ( a famous “Republican”!).How about for graphs? Well, suppose you are trying to infer some function defined on a graph (e.g., some property of users on a social network, which defines the graph). You are given the graph, but only a small number of vertices are labeled (say the social network labels some vertices “Democrat” and some vertices “Republican”, but most of the vertices are unlabeled, meaning the political affiliation of the individuals corresponding to these vertices is unknown).So, how should we design an algorithm to “fill in” the labels of the unknown vertices, while exploiting the structure of the graph? A random method would be to plonk down the social network on paper, look at the (x,y) coordinate of every vertex with labels, create a training set of the labeled vertices, and then voila’, you have a simple 2D classification problem. This works rather poorly for the simple reason that it completely ignores the spatial connectivity of the graph.So, you’d like to construct an embedding that respects the “geometry” or the “topology” of the graph, meaning that vertices that are “close” to each other in terms of the graph distance should receive “similar” labels. This captures the property that in a social network, people with similar tastes tend to hang around each other. So, “Democrats” tend to hang around with other “Democrats” and similarly, “Republicans” tend to hang out with other “Republicans”. For a real-world example of why this labeling is smooth on the graph, let's look at this image from the polling website FiveThirtyEight, which did a “what-if” scenario prediction of the midterm results from 2018, assuming that only women voted (that’s why it’s a “counterfactual”, meaning a “what-if’ type of analysis): as you can see, the red “Republican” labels are very smooth across spatial neighborhoods, as are the “Blue” democratic labels.OK, how does one construct an embedding of a graph such that smoothness of the label function across the graph topology is preserved. This leads us to one of the most beautiful results in machine learning of the last 25 years: the graph Laplacian. The Laplacian has been termed “the most beautiful object in all of mathematics and physics” (Nelson, Tensor Analysis). The reasons are not hard to see: every major equation in physics has the Laplacian operator (sometimes called the div-grad operator).In a discrete graph, the graph Laplacian is simply L = D - A, where A is the adjacency matrix of an undirected graph, D is a diagonal matrix of the valencies of the graph (meaning the degree of each vertex). It is about as simple a matrix as one can define, but my, what beauties does it reveal! One never wants to analyze a graph by looking at the adjacency matrix! That tells you very little. Now, the graph Laplacian, ah, that's an entirely different kind of operator. It’s as revealing as a high precision microscope into everything that you'd like to know about a graph!It would take too long to get into the beauties of this topic, but for those interested, I highly recommend Fan Chung’s brilliant monograph on “Spectral Graph Theory”. This brings the full bore power of differential geometry to the discrete setting, and Fan Chung’s book goes into much depth on this topic. I also recommend Spielman’s lecture notes at Yale, which have some nice examples of graph embeddings.So, how does one construct graph embeddings using the Laplacian L = D-A. Simple. Just compute the eigenvectors of L, where each eigenvector is a vector of dimensionality equal to the number of vertices. The first eigenvector is all 1’s, and corresponds to the constant function. The associated eigenvalue is 0. The second eigenvector is a thing of beauty, the so-called “Fiedler” eigenvector after the mathematician who first studied it. There are many beautiful theorems about the second eigenvalue associated with the second eigenvector (it tells you how fast a random walk on the graph “mixes”, which is useful for so many things, like routing, or how information spreads in a social network, or how diseases spread across populations).With this example out of the way, let's get to constructing word embeddings. How do we embed “Democrat” into a 100-dimensional space? Well, now we are dealing with sequential datasets, namely text documents. Suppose we read through all of Wikipedia (many gigabytes!), but a fast PC can scan all of Wikipedia in several hours now. So, every time the word “Democrat” appears in Wikipedia, we keep track of the surrounding “context” words (say, the 5 words preceding it and the 5 words succeeding it).So, we build a gigantic matrix M, whose rows correspond to words like “Democrat” and whose columns correspond to all possible contexts C. This is a very large matrix, since there are hundreds of thousands of words in Wikipedia (including proper nouns), and millions of contexts. No matter, we have lots of memory and powerful machines. We take this giant matrix M, and find its singular value decomposition M = U S V’, where S is the diagonal matrix of singular values. The matrix U gives us an embedding of words.Of course, we’d like an incremental way of computing word embeddings, which is far more efficient. Can’t we use a “neural network” to do this, and approximate what our linear algebraic approach does? This was exactly the approach taken by Mikolov in his beautiful work on “word2vec”, and you can download his rather elegant C code that computes embeddings of all the words in Wikipedia in one afternoon on a reasonably fast PC. This is an amazing piece of work, and has brought new life into NLP and ML. Many extensions have been explored, but Mikolov’s original paper is still worth reading.https://arxiv.org/pdf/1301.3781.pdfHere’s a nice picture of the concept of word embeddings and how to reason with them (spatial vector displacements allows one to compute analogical relationships, like King is to Queen as Man is to X, where X is of course “woman”).The Tensorflow web page explains the basic ideas nicely.Vector Representations of Words | TensorFlowNow, are word embeddings Euclidean? That would be amazing, if true. A few years ago, while on sabbatical at IBM Research, when I was working with the Watson Deep Learning group, I explored how to reason about word embeddings using a non-Euclidean “manifold” approach. This produces far better results than Mikolov’s original approach. My paper on Arxiv on this topic can be downloaded here (warning: the math is a bit more heavy going than Mikolov’s work!).Reasoning about Linguistic Regularities in Word Embeddings using Matrix ManifoldsHope this has been helpful! Good luck exploring the wonderful world of embeddings. What does the brain do? How are meanings stored in our brains? Philosophers have been pondering this question for centuries. This is what ML research on embeddings is trying to also explain.

Descartes said his original invention of coordinates was because he didn’t like the cleverness required to do synthetic proofs a la Euclid. He had a vision like the OP’s: turn the geometry into algebra, turn the crank and solve the algebra mindlessly, proof done.Cartesian coordinates have been an incredibly successful technology, permeating mathematics, science, even culture and news where large data sets are often conveyed using Cartesian graphs. To a large extent Descartes’ vision has come true.Still, in practice it’s not as automatic as Descartes had hoped. Sometimes analytic geometry leads to algebra that’s hard. Solving more than a quadratic equation is often difficult. Sometimes a nice synthetic Euclidean proof is the easiest path.There are three main approaches, and lots of variations within approaches. A usual Euclidean straightedge and compass proof doesn’t generally involve any measurement. Then there are formulas, geometric and trigonometric, which let us write equations about the various lengths in the problem which can then be solved. That’s a kind of analytic geometry that doesn’t use a grid — the various measurements refer to aspects of the problem itself. Then there’s analytic geometry on the grid, true Cartesian geometry. It takes some experience to choose the best method, or to combine methods effectively.This is really only referring to Euclidean Geometry. There are many kinds of geometry, even restricting ourselves to planar geometry. Projective geometry is the geometry of lines, with no parallelism or perpendicularity. Affine geometry adds parallelism, but there’s no generalized notion of distance, no meaningful rotations, just displacements parallel to the axes. We can make carpenters’ geometry, using a T square to define perpendicularity. Beyond that we can add a metric, a notion of distance. Depending on how we do it we can get Euclidean geometry or various 2D non-Euclidean geometries. I’m sure there are plenty more geometries I don’t know about.Anyway, there are many different ways of approaching geometric proofs. I’m an algebra guy myself — I tend to prefer the analytic methods. But it’s best to be familiar with as many ways as possible, so you can choose the best approach for any given problem.