Heute früh wollte ich am Linux-PC eine Reise mit der Bahn planen. Dazu rief ich die Seite im Firefox auf. Leider wurde die Bahn-Seite nicht vollständig geladen. Ein Arbeiten war nicht möglich. Das Gleiche unter Chromium und Opera.
Ein Login war zwar möglich, aber auch die dann folgende Seite erlaubte keinerlei Aktion.
Unter Chromium zeigte sich, dass eine Web-Ressource wegen fehlender Autorisierung nicht ansprechbar war.
Neue Web-Präsenz der Bahn
Anfang letzte Woche funktionierte noch alles. Das aktuelle Layout der nur teilweise anzeigten Seite kam mir auch seltsam vor. Eine kurze Suche im Internet zeigte, dass die Bahn ihre Web-Präsenz vollständig hat neu entwickeln lassen. Die neue Web-Präsenz ging zwischenzeitlich live. Neugierig geworden probierte ich einen Aufruf auf einem Win10-System. Identische Probleme unter Chrome und Opera wie unter Linux. Im dortigen Firefox lief die Bahn-Seite aber unter Windows.
Lösung
Durch das Verhalten von Firefox auf dem Win10-Laptop neugierig geworden experimentierte ich ein wenig herum. Am Schluss war die Lösung einfach:
Man muss Cookies und Website-Daten (für bahn.de) sowie alle entsprechenden Daten im Cache der Browser löschen. Und zwar vollständig. Dies erfordert bei Chrome, Chromium und Opera auch eine Einstellung des Zeitraums für den die Löschung vorgenommen werden soll.
Vermutlich ruft die neue Web-Präsenz alte Cookies/Website-Infos (mit falschen Autorisierungsdaten für bestimmte Web-Ressourcen) auf. Das hätten die Entwickler nach meinem Gefühl bei Tests erkennen und abfangen können. Ein Hinweis an neue Benutzer wäre hilfreich. (Der FF unter Win10 war so eingestellt, dass bei dessen Schließung Cache und Cookies automatisch beseitigt wurden. Deswegen tauchte das Problem dort nicht auf.)
Post III focused on the shearing of a circle, which was centered in the Euclidean coordinate system [ECS] we worked with. The shear operation resulted in an ellipse with an inclination against the coordinate axes of our ECS. This was interesting regarding four points:
A circle, which is centered in a chosen ECS, exhibits a continuous rotational symmetry (isotropy). This obviously allows for a decomposition of a shear operation into a sequence of two affine operations in the chosen ECS: a scaling operation (with different factors along the coordinate axes) followed by a rotation (or the other way round). Equivalently: We could switch to another specific ECS which is already rotated by a proper angle against our originally chosen ECS and just perform a scaling operation there. The rotation angle is determined by the shear parameter λ. This seems to stand in some contrast to the shearing of figures with only discrete rotational symmetries: We saw for rectangles and cubes that an additional rotation was required to replace the shear operation by a sequence of scaling and rotation operations.
Points (x, y) of circles and ellipses are described by quadratic forms in two dimensions (with some real coefficients α, β, γ, δ):
\[
\alpha \,x^2 \, + \, \beta \, x \, y \, + \, \gamma \, y^2 \:=\: \delta
\]
Quadratic forms play a general role in the mathematical description of cone-sections. (Ellipses are the results of specific cone-sections.)
Ellipses also result from projections of multi-dimensional ellipsoids onto two-dimensional coordinate planes. Multi-dimensional ellipsoids are described by quadratic forms in an ECS covering the ℝn.
Hyper-surfaces for constant probability density values of multivariate normal vector distributions form multi-dimensional ellipsoids. Here we have a link to Machine Learning where key properties of certain objects are often ruled by Gaussian distributions.
From the first point we may expect that a shear operation applied to a multi-dimensional sphere will result in a multi-dimensional ellipsoid – and that such an operation could be replaced by scaling the original sphere (with different factors along n coordinate axes of a n-dimensional ECS) followed by a rotation (or vice versa). We will explicitly investigate this for a 3-dimensional sphere in the next post.
If our assumption were true we would get a first glimpse of the fact that a general multivariate standard distribution can be created by applying a sequence of distinct affine (i.e. linear) operations to a spherical probability distribution. This is discussed in detail in another post-series in this blog.
What is a bit confusing at the moment is that a replacement of a shear operation by simpler affine operations in general seems to require at least two rotations, but only one when we work with centered isotropic bodies. We come back to this point when we discuss the decomposition of a shear matrix by the so called SVD-procedure.
In the previous post of this series we have used the radius of the circle and the shearing parameter λ to derive analytical expressions for the coordinates of special points with extremal values on our ellipse
Points with maximal and minimal y-coordinate values.
Points with a maximal or minimal distance to the symmetry center of the ellipse. I.e. the end-points of the principal diameters of the ellipse.
From the fact that shearing does not change extremal values along the axis perpendicular to the sharing direction we could easily determine the lengths of the ellipse’s principal axes and the inclination angle of the longer axis with the x-axis of our Euclidean coordinate system [ECS].
What do we have in addition? In another mini-series on ellipses
I have meanwhile described how the geometry of an ellipse is related to its quadratic form and respective coefficients of a symmetric matrix. I call this matrix Aq. It forces the components of position vectors to fulfill an equation based on a quadratic polynomial. Furthermore Aq‘s eigenvalues and eigenvectors define the lengths of the ellipse’s principal axes and their inclination to the axes of our chosen ECS. The matrix coefficients in addition allow us to determine the coordinates of the points with extremal y-values on an ellipse. We will use these results later in the present post.
Objectives of this post: Shearing of a centered, rotated ellipse
In this post I want to show that shearing a given centered, but rotated original ellipse EO results in another ellipse ES with a different inclination angle and different sizes of the principal axes.
In addition we will derive the relations of the shearing parameter λS with the coefficients of the symmetric matrix \(\pmb{\operatorname{A}}_q^S \) that defines ES. I also provide formulas for the dependence of ES‘s geometrical properties on the shear parameter λS.
There are two basic prerequisites:
We must show that the application of a shear transformation to the variables of the quadratic form which describes an ellipse EO results in another proper quadratic form and a related matrix \(\pmb{\operatorname{A}}_q^S \).
The coefficients of the resulting quadratic form and of \(\pmb{\operatorname{A}}_q^S \) must fulfill a mathematical criterion for an ellipse.
We expect point 1 to be valid because a shear operation is just a linear operation.
To get some exercise we approach our goals by first looking at the simple case of shearing an axis-parallel ellipse before extending our considerations to general ellipses with an inclination angle against the coordinate axes of our chosen ECS.
Hat nichts mit Linux zu tun, ist aber für alle Freiheitsliebende relevant. Nach einem Übermaß an Überflutung mit Plakaten der AfD (“Wir vergessen nicht”) und rechten Stammtischparolen in einem Kurzurlaub verweise ich ohne weiteren Kommentar auf folgenden Link:
Ich bin kein FDP-Wähler, aber wo einer der sehr wenigen verbliebenen Aufrechten in dieser Partei leider zu 100 Prozent recht hat, erhält er meine volle inhaltliche Zustimmung.
A reader wrote me a mail and asked what the general direction of this blog is going to be. He wondered about the “flood” of formulas lately. Which, in his opinion, have nothing to do with Linux. In general, his impression was that I seemingly have lost my interest in core Linux topics. He, a German, also complained that I write my posts in English.
My first reaction was: I appreciate that some of my readers care. The criticism is justified. And it deserves an answer and some explanations. The easiest part is the question regarding language.
According to my provider and my own blog-statistics 78% of page requests to this blog come from abroad, i.e. from countries outside Germany. Most requests stem from US-systems. Before Russia’s imperialistic war against Europe there also were connections (and permanent attack trials) from both Russia and China. Their percentage has declined (fortunately). Anyway, the majority of page requests comes from outside Germany. Therefore, I try to write in English. My English certainly is not the best, but it is still easier to read for those who are interested in my posts’ contents. And obviously, these are not German readers. So, I will not go back to German again.
Now to the question regarding the declining number of posts related to Linux. In the time when I worked as a free-lancer (up to 2018), I had some German customers who cared about Linux. It was in my own interest to dig a bit deeper as usual into topics like “virtualized VLANs”, firewalls etc.. The articles on these topics are still the most read ones in this blog.
But then I started to work as an employed consultant for IT-management topics in a Windows-dominated company. I simply had no chance and no time to continue with hard core Linux topics until the end of 2022. The only connection that came up was related to minor Machine Learning topics. Since my retirement I again use my private Linux systems – but what I need there simply works. No need to dig deeper at the moment. I intend to shift all of my HW-platforms and in the wake of such an endeavor typically some Linux topics come up, but all of this requires a period of money saving first. The same holds for a private project concerning central Linux-based audio-station. (Side remark: Due to the systematic destruction of the social system in Germany, ironically and mainly by the politics of social democrats, ca. 10 million of the persons going into retirement the next years will get significantly less than 1500 Euros per month. These are official numbers of the German government. I am on the edge of this wave.)
A second point which obviously has an impact on this blog is that I have an education in physics and an inborn interest in math. One of the best aspects of retirement is that you gain a lot of freedom regarding your real interests. No employer longer forces you to focus on things you only work with to earn a living. In my case the physicist woke up in spring 2023. I started to read a lot of books on theoretical physics and cosmology. To find out that I needed to revive some university level mathematics. At the same time I got interested in some admittedly special aspects of my own ML-experiments and network simulations in general. And suddenly you find yourself applying some basic linear algebra and calculus again. An easy but not very thoughtful way to start collecting some simple, but useful results was using this blog. I admit: It has turned the blog’s focus away from Linux.
The solution is clear: This blog has to be split up. I will do this as soon as I find some motivation for the boring task to set up a new blog, database, etc. For the time being I have changed the subtitle of this blog to indicate that other topics have come up.
What I cannot promise is that Linux topics will dominate my interests in the future. As said: What a retired person needs on PCs and laptops normally works perfectly under the control of Linux. Thanks to all the fantastic people of the Open Source community.
This post requires Javascript to display formulas!
A centered, rotated ellipse can be defined by matrices which operate on position-vectors for points on the ellipse. The topic of this post series is the relation of the coefficients of such matrices to some basic geometrical properties of an ellipse. In the previous posts
we have found that we can use (at least) two matrix based approaches:
One reflects a combination of two affine operations applied to a unit circle. This approach led us to a non-symmetric matrix, which we called AE. Its coefficients ((a, b), (c, d)) depend on the lengths of the ellipses’ principal axes and trigonometric functions of its rotation angle.
The second approach is based on coefficients of a quadratic form which describes an ellipse as a special type of a conic section. We got a symmetric matrix, which we called Aq.
We have shown how the coefficients α, β, γ of Aq can be expressed in terms of the coefficients of AE. Another major result was that the eigenvalues and eigenvectors of Aq completely control the ellipse’s properties.
Furthermore, we have derived equations for the lengths σ1, σ2 of the ellipse’s principal axes and the rotation angle by which the major axis is rotated against the x-axis of the Cartesian coordinate system [CCS] we work with.
We have also found equations for the components of the position vectors to those points of the ellipse with maximum y-values.
In this post we determine the components of the vectors to the end-points of the ellipse’s principal axes in terms of the coefficients of Aq. Afterward we shall test our formulas by a Python program and plots for a specific example.
Reduced matrix equation for an ellipse
Our centered, but rotated ellipse is defined by a quadratic form, i.e. by a polynomial equation with quadratic terms in the components xe and ye of position vectors to points on the ellipse:
The quadratic polynomial can be formulated as a matrix operation applied to position vectors vE = (xE, yE)T. With the the quadratic and symmetric matrix Aq
Method 1 to determine the vectors to the principal axes’ end points
My readers have certainly noticed that we have already gathered all required information to solve our task. In the first post of this series we have performed an eigendecomposition of our symmetric matrix Aq. We found that the two eigenvectors of Aq for respective eigenvalues λ1 and λ2 point along the principal axes of our rotated ellipse:
This is trivial regarding the algebraic operations, but results in lengthy (and boring) expressions in terms of the matrix coefficients. So, I skip to write down all the terms. (We do not need it for setting up ordered numerical programs.)
Remember that you could in addition replace (α, β, γ) by coefficients (a, b, c, d) of matrix AE. See the first post of this series for the formulas. This would, however, produce even longer equation terms.
We pick the yE with the positive term in the following steps. (The way for the solution with the negative term in yE is analogous.) The square of yE is:
A detailed analysis also for the other yE-expression (see above) leads to further solutions for the coordinates (=vector component values) of points with extremal values for the radii. These are the end-points of the principal axes of the ellipse:
I leave it to the reader to expand the convenience variables into terms containing the original coefficients α, β, γ.
Plots
It is easy to write a Python program, which calculates and plots the data of an ellipse and the special points with extremal values of the radii and extremal values of ye. The general steps which I followed were:
Step 0: Create 100 points a unit circle. Save the coordinates in Python lists (or Numpy arrays). Use Matplotlib’s plot(x,y)-function to plot the vectors.
Step 1: Create an axis-parallel ellipse with values for the axes ha = 2.0 and hb = 1.0 along the x- and the y-axis of the Cartesian coordinate system [CCS]. Do this by applying a diagonal scaling matrix Dσ1, σ2 (see the first post of this series).
Step 2: Rotate the ellipse bei π/3 (60 °). Do this by applying a rotation matrix Rπ/3 to the position vectors of your ellipse (with the help of Numpy). Alternatively, you can first create the matrices, perform a matrix multiplication and then apply the resulting matrix to the position vectors of your unit circle.
(The limiting lines have been calculated by the formulas given above.)
Step 3: Determine the coefficients of combined matrix AE = Rπ/3 ○ Dσ1, σ2
I got for the coefficients ( (a, b), (c, d) ) of AE :
A_ell =
[[ 1. -0.8660254 ]
[ 1.73205081 0.5 ]]
Step 3: Determine the coefficients of the matrix Aq by the formulas given in the first post of this series. I got
A_q =
[[ 3.25 -1.29903811]
[-1.29903811 1.75 ]]
For δ I got:
delta = 4.0
which is consistent with the length-values of the principal axes.
Step 4: Determine values for the eigenvalues λ1 and λ2 from the Aq-coefficients by the formulas given in the first post. Also calculate them by using Numpy’s
eigenvalues, eigenvectors = numpy.linalg.eig(A_q). Theory tells us that these values should be exactly λ1 = 4 and λ1 = 1. I got
Eigenvalues from A_q: lambda_1 = 4. :: lambda_2 = 1.
Step 5: Determine the components of the normalized eigenvectors with the help of numpy.linalg.eig(A_q). I got:
Components of normalized eigenvectors by theoretical formulas from A_q coefficients:
ev_1_n : -0.8660254037844386 : 0.5000000000000002
ev_2_n : 0.5000000000000001 : 0.8660254037844385
Eigenvectors from A_q via numpyy.linalg.eig():
ev_1_num : 0.8660254037844387 : -0.5000000000000001
ev_2_num : 0.5000000000000001 : 0.8660254037844387
The deviation between ev_1_n and ev_1_num is just due to a difference by -1. This is correct as the eigenvectors are unique only up to a minus-sign in all components.
Step 6: Calculate the sinus of the rotation angle of our ellipse from Aq– and Aq-coefficients. The theoretical value is sin(2 π/3) = sin(2 pi/3) = 0.8660254037844387. I got:
sin(2. * rotation angle) of major axis of the ellipse against the CCS x-axis from A_E coefficients:
sin_2phi-A_E = 0.8660254037844388
sin(2. * rotation angle) of major axis of the ellipse against the CCS x-axis from from eigenvectors of A_q:
sin_2phi-ev_A_q = 0.8660254037844387
sin(2. * rotation angle) of major axis of the ellipse against the CCS x-axis from A_q-coefficients:
sin_2phi-coeff-A_q = 0.8660254037844388
Perfect!
Step 7: Plot the end-points of the normalized eigenvectors of Aq:
Note that in our example case the end-point of the eigenvector along the minor axis must be located exactly on the elliptic curve as the ellipses minor axes has a length of b=1!
Step 8: Calculate the components of the vectors to data-points of the ellipse with maximal absolute ye-values from the Aq-coefficients given in the previous post. Plot these data-points (here in green color).
Step 9: Calculate the components of the vectors to data-points of the ellipse with maximal values of the radii with the help of the complex formulas presented in this post and plot these points in addition.
Conclusion
In this mini-series of posts we have performed some small mathematical exercises with respect to centered and rotated ellipses. We have calculated basic geometrical properties of such ellipses from the coefficients of matrices which define ellipses in algebraic form. Linear Algebra helped us to understand that the eigenvectors and eigenvalues of a symmetric matrix, whose coefficients stem from a quadratic equation (for a conic section), control both the orientation and the lengths of the ellipse’s axes completely.
This knowledge is useful in some Machine Learning [ML] context where elliptic data appear as projections of multivariate normal distributions. Multivariate Gaussian probability functions control properties of a lot of natural objects. Experience shows that certain types of neural networks may transform such data into multivariate normal distributions in latent spaces. An evaluation of the numerical data coming from such ML-experiments often delivers the coefficients of defining matrices for ellipses.
In my blog I now return to the study of with shearing operations applied to circles, spheres, ellipses and 3-dimensional ellipsoids. Later I will continue with the study of multivariate normal distributions in latent spaces of Autoencoders. For both of these topics the knowledge we have gathered regarding the matrices behind ellipses will help us a lot.
