Trivial problem 101 : how to avoid transcendental functions.

Engineer and scientist, particularly Physics need to be using linear algebra in their solution and avoid transcendental?functions. There is a way of linearizing many equations that look complicated and often it is easier to opt for a simpler guessing solution based on geometry; for instance rule of thumb. I often see geometry based "brain teasers" that are posted in a form of mathematics Olympiad challenging people to solve a problem that would be considered trivial in linear algebra and Matrices.

I am not a "Genius" but an engineer, I love "geometric puzzles" such as this, lot of my analysis is to do with projected motion in mechanics, flight trajectory and bio mechanics. Most of my software is written around vector projections. This can be done two ways - one is - using rules of geometry and solution is generated using "transcendental?functions" such as sin() and cos() ; perhaps even their derived functions such as tan() and their invers functions the other is linear algebra. .

In ME and EE we use formulas that include sin() and cos() as shortcuts to a solutions. It is there because we intuitively recognize the nature of the problem. If we are writing a software we need to intuitively test the solutions and this approach is given by one of the first people to solve this "challenge".

In Engineering we would see the "challenge" as an equation of a circle, and the attempt would be to solve for the centre of the circle given that we have 3 points on the axis. In my work with mechanics of the rectilinear comb this would be considered a linkage motion. We as engineers, particularly undergraduate engineers see the problems in terms of time and positional derivatives. Take for instance a problem that faced my work several years ago in designing cams that have rocking followers. Relative to the rotation of the cam the follower will rock on the surface - producing temporary retrograde positions.

My approach is: its also a 3 point circle problem, with intersect at [0 0] , give; [ -3 0, 0 6, 8 0] solution for centre is [x y] and R would be a magnitude from that point to any of the given points. I do not like transcendental function so I avoid them, personal preference

The equation of any point on the circle can written in a form of a linear algebra involving no transcendental?functions. Solving this is as trivial as entering it into a mathematical package such as Matlab, Maple, MCAD ...or in Rhino using Python extensions in Grasshopper. Computer do not solve problems engineers do, and Physicist the ones I encountered in CSIRO only know how to ask and the claim credit and ownership over a solution. A solution is a determinate of a matrix numerically calculated cofactors.

Explanation: so why is Det(|A|)=0 The answer is simple and I like to give it in 2D plane, it is simply squashing the circle into a single line represented by a collection of points, so you can imagen a circle being looked like side on. It becomes a linear equation. For instance a location of centre of the circle will be xc which is 1/2 along the length of that vector. If you needed to confirm that this all works Det(A) is in fact a πr2, r which we could obtained by rotating the sub matrix of A to align with the the max value in the plane and subtracting any of the centre Vector. This can also be confirmed by measuring vector difference for all values of (x y) and one will arrive at circumference of a circle 2πr. This also applies to 3D and a Sphere but in that case the Det represents Surface and Volume calculations. How is that applicable - you can do this for most geometric shapes in 2D and 3D but real power lays in higher dimensions.

I am just enjoying this

The solution is very simply if one understand the meaning of submatrix in a determinant and why it was initialy set to zero. So what happens after we get a solution like this we as engineers start wondering if it can be expanded into solving our favourite dimension 3D. Here is the equation of the circle re-written in terms of sphere.

Neat I feel smug - lets see how that works by extending the circle into a slice of the sphere and how we can use that to check our work.I feel inspired do do some graphics for a spherical problem seen how many very good solutions - there is only one for a 2D problem logic as oppose to geometry reasoning. One still need to know how to visualize it as many people contributing to this post have done. Most problems in 3D or any nD can be simply checked by using very reasoning people have presented. Good work to all. Here is problem definition in 3D using linear algebra - this is not a difficult problem but I was having a thoughts thet perhaps I should publish on LinkedIn a generalized linear algebra solution to this.

just extended the problem from plane circle into 3D using the equation for a Sphere, with the previous problem as 2D plane problem now I added z=(0,0,8) but problem can be reduced from 3D back to the original problem. I deliberately set z as x (0,8,0) so zc=xc, and it does. The reason for that was that I could use a simple Pythagorean for a solution check ( previous r, origin in z direction ) result pops out as R= 5√3/√2 previously a solution in 2D plane was r=5√5/2 with zc=5/2.

If you have a look at the sub matrix M(1,5) in det(M) is located as a 45°. this confirms that XY plane has sliced the sphere in the original plane in which circle existed. This can be used to arrive at a Pythagorean solution of { R2 = r2 + r2 } hence the1/√2 in our solution. Yes we are using the skills exhibited by the people that provided a logical solution to the problem. This is an approach that can be extended to many other geometric shapes and trajectories.

This figure shows the prospective of a Sphere from several planes - showing that XY plane the circle required 3 points to be defined and major circle of the sphere is shown as a concentric circle - but this approach would work just as well for an ellipsoid and not necessary in the symmetrical plane ( this includes open and closed geometry, open being such things as Paraboloid ).

How is this applicable?

In my work at CSIRO and my PhD I had to solve many problems in dimensions greater the 3. Wavelets and neo-Wavelets as used in my PhD was a problem of solving a serious of linear equations as vectors in Fourier and Wavelets are actually rotating constructs instead of sinusoidal waves we represent then on the graphs requiring time to be displayed rather the "frequency" . NOTE: In Wavelets all concept of "frequency" is very vague, so vague that I initially call it "Shape based Transfer" of geometry defined objects rotating at non stationary or variable meaner. It would have been a concept that would be very scary for like of my CSIRO group leader and his "mate". Being anointed to steal my work and not having any background in linkage cam simulation, so I basically provided the Physicist with the solution using transcendental?functions and vector manipulation. I could tell that Ken and Budgie were ill equipped to follow the actual analysis so I gave them a dumb down version of the solution - while in 1989 I was writing this with this in C++ , Lisp and Pascal - those days had no software or hardware that we have post 2010. eventually all of my Patent and IP was lost to Schlumberger, because CSIRO division was run by Physicist with no appreciation of skill level an actual well trained engineer may posses. Here is a story how this is applicable directly to a problem I had in redesigning the NSC Schlumberger comb. I have unloaded on CSIRO but my problem was also the hubris of Schlumberger underestimating my skill level and commitment to solve that worsted combing problem. But biggest Hubris is located in management of science csiro.

This was a part of my PhD rig in investigating Wavelets application to "Sokolov Cam" [1] system overtaking the "German Cam" system as most efficient required motion delivery system in mechanics. It all happened for trying to write a software in 1989 to solve trivial problems in motion and dynamics and exclude use of Vector solutions involving sin(), cos() , tan() and non-right angle triangle solutions. Why was that a problem: the answer is simple - during each cycle there are always two solutions in the positive and two solutions in the negative, one needed to write extra code to trach motion between 0 and 2π noting in which quadrant the solution was - this got exceptionally messy but it was doable. However there are always an easier ways and that is in the form of linear algebra.

One can see a complexity of the mechanical motion in two cam rocking followers and a rocking arm delivering the motion. One needed to uses a system of mathematics that could cope with complexity of the Vector motion. Added to this was a need to dynamically make the follower arm spring inert - no spring deflection so at high speed there would be no resonance effects from the spring thus loosing the precision of the delivered motion. That was the definition of the "Sokolov Cam" system.

Notes:

This is an intro into interpretation of Linear Algebra for Engineers and Scientist. It is also a valuable tool in Wavelet analyses as interpretation of a signal as a vector becomes more appropriate. I like "sticking it up", csiro management as I know how little or have forgotten undergraduate university math an average Physicist knows as the compose majority of management of science in some division relegating engineers to secondary positions. It is bit unfair of me to say that as there are exceptions to that rule. Are you a #csiro management genius or not ? - I am certainly not that, but I do possessing IP and csiro patents which only qualified my position as non-management for redundancy. This is not my normal publication but I was inspired by a problem published by Nabil Elwan and then solution provided by Krunoslav Popovic. Well done guys - Keep up the good work.

As a member of an engineering and scientific fraternity we all like stealing from Mathematicians. Most notable theft from a Mathematicians by Physicist was Heisenberg uncertainty principle, way I say theft - well its been known for almost 100y that if one knows the frequency in transform one loses the positional information a problem approximated by Wavelets.

Most important part of mathematics transformation into application is interpreting what they do and relating it back to a physical meaning - this publication is just that, bring it back to a familiar ground of calculus and avoiding messy integration of let say elliptical problems using integrals. If you explore the field of linear algebra and matrices you will find that "integral under a curve" is just a high school and undergraduate introduction or simplification. Most people think of math for engineers as being 2D and 3D problem but its a nD or multidimensional problem. It sound all complicated and long winded or simply hidden in fancy notation BUT its not, it is there to take our software and analysis or signal processing to that next level. Publication presents a simplified case of a circle Sphere and perhaps an evolution into 4th dimension and not just (x y z ) 3D we are so familiar with - ps let not the time as variable be an obstacle it is just another dimension in number crunching. Time like Area, Volume and other properties at higher dim can be negative - just do not think that it is a way to travel into the past.Just try not to mind my need to kick CSIRO management at every opportunity as they are the hindrance to progress in science and technology. Problem for then since 1990's was the hubris of "management of science" and ephemeral values rather then simply just do the "work". I hope this publication has inspired you to read more Linear Algebra.

Ref: read the full story:?https://lnkd.in/gHWXCwga?or or start from here;?https://lnkd.in/g_U4usuR

References

1. "xcsiro ToD21thCentury 106-110: Australian Worsted Comb Story"?Sokolov N.N.,Nov25, 2021

MATLAB symbolic code: Not the most elegant but a start to get the centre follow the suggestions in previous Figures in this publication. Note this is a Symbolic solution approach.