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It is more than simple geometric transformations or projections. A brief overview of the history of nomography is given in a later paper by Evesham; the thesis, of course, goes into much greater mathematical detail.
I have been meaning to write an essay on this calculator for some time. Lalanne is the one who termed this process anamorphosis.
The Saint-Robert criterion says this is possible if. I created a nomogram last summer for this equation for an upcoming blog essay on nomograms to model 2-, 3- and 4-factor linear equations in Design of Experiments DOE methodology; this equation is the general form for a weighted, linear 2-factor equation.
Can we draw it as three parallel scales in x, y and z? Saint-Robert also provides a way of finding the functions X x , Y y and Z z to plot on the scales:.
Click on this figure and all figures in this essay for full-resolution versions. Only then did I think to algebraically derive it.
For example, an equation of the form. In fact, the variables x, y and z can be replaced here by any function f 1 x , f 2 y and f 3 z and this formulation holds.
Warmus uses exhausting algebraic techniques based on classifications of equations to either find the determinant elements or show that it is not possible.
This is followed by a detailed look at J. In fact, I learned that Clark instead authored in a series of articles on nomography in a French mechanical engineering journal, which to my surprise I subsequently found digitized at Google Books see pages and of here and pages and of here.
As Soreau points out, this can actually be drawn as a basic parallel scale addition nomogram. Then you just plot the tangent of each value on the other side of the scales, and voila!
Clark looked at what nomographic forms are possible with given orders of equations. More on that later!
Evesham also mentions that a homographic transformation can always transform this parabola into another conic such as a circle.
A circle is far easier to draw by hand, and I think it provides a very striking nomogram. Evesham does not explain how this is done.
The thing that Clark did that is most astonishing to me is to find a way to generalize his equation form and provide a means of getting all three scales to lie along a single curve!
Clark deduced that the extraneous factor f 1 - f 2 in the determinant for the conical determinant is responsible for aligning two of the three scales f 1 and f 2 here but we can use any two of the variables on a common curve, so he searched for a factor that would align all three scales.
For an equation in the general form. You can see from the x- and y-elements of each row that the three scales lie on exactly the same curve, although if the functions are not identical the scale divisions will be different.
Now we can add the first column to the third, divide by the second column, and shift the first column to the third column to get this into standard nomographic form:.
In fact, the general equation form breaks into three distinct nomographic curve types that cannot be transformed into each other, represented by the following forms:.
We have just seen the general acnodal form in our tangent addition nomogram. You may have seen the crunodal form below in my other essays or nomography calendar.
This curve is also known as the folium of Descartes. The cuspidal form is used for the harmonic relation, which can represent, for example, the focal length of two lenses in series or the equivalent resistance of two resistors in parallel: With work you can find this determinant equation and its corresponding nomogram: Clark called all of these cubic forms of nomograms because the common curve in each case is given by a third-degree equation.
Evesham concludes his thesis with a discussion of Russian advances in nomography from the s, including the use of oriented transparencies of scales laid across other printed scales as treated by Khovanskii.
This type of nomogram can incorporate more variables in a single alignment. If the presentation above intrigues you, then you will find real treasures in this book.
The bibliography in the book is valuable in itself for searching out more details. I enjoyed reading the book immensely.
Evesham does a great service in making these advancements available to those of us today with an interest in the field of graphical calculation.
Again, the book can be found here or here. This entry was posted on Saturday, February 12th, at You can follow any responses to this entry through the RSS 2.
Both comments and pings are currently closed. Thank you for this simply wonderful post. I visited your website thanks for providing the Translate option and I read some of your material.
You obviously do not have the problem I have with reading old mathematics texts in other languages! It seems that people who are interested in graphical calculation live far apart.
Geographically speaking, the only difference between one place and another might be access to libraries with this type of material, but I can certainly share research materials like that.
Oct 20 - Nov 18, Oct 14 - Nov 10, History can also mean the period of time after writing was invented. Scholars who write about history are called historians.
It is a field of research which uses a narrative to examine and analyse the sequence of events, and it sometimes attempts to investigate objectively the patterns of cause and effect that determine events.
Historians debate the nature of history and its usefulness. This includes discussing the study of the discipline as an end in itself and as a way of providing "perspective" on the problems of the present.
The stories common to a particular culture, but not supported by external sources such as the legends surrounding King Arthur are usually classified as cultural heritage rather than the "disinterested investigation" needed by the discipline of history.
Events of the past prior to written record are considered prehistory. Amongst scholars, the fifth century BC Greek historian Herodotus is considered to be the "father of history", and, along with his contemporary Thucydides, forms the foundations for the modern study of history.
Their influence, along with other historical traditions in other parts of their world, have spawned many different interpretations of the nature of history which has evolved over the centuries and are continuing to change.
The modern study of history has many different fields including those that focus on certain regions and those which focus on certain topical or thematical elements of historical investigation.
Often history is taught as part of primary and secondary education, and the academic study of history is a major discipline in University studies.
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