Imagine a spherical ball with radius r that equally coated into two hemisphere P(grey) and S(white) is thrown at the pool. Consider the ball can float (up-down) and rotate. There is an interface separate two different fluid A(air) and B(water). Assume that the interface is flat. α defines the position of the ball and β defines the orientation of the ball. So α larger means ball is more in zone A and vice versa. Range of β and α is $[0,360^0]$ and $[0,180^0]$. The Schematic representation of this is below.   To calculate the surface area of P(grey) coated portion of the ball into A(air) is given by following equation,  $\mathrm{Area}_{P@A}=  r^2\int_{\theta=\frac{\pi}{2}-\beta}^\alpha  \int_{\phi=\arcsin(1/(\tan\theta \tan \beta))} ^{\pi -\arcsin(1/(\tan\theta \tan \beta)} \sin\theta\; d\theta d\phi.$  After solving this equation, we can get the following solution.  $\textrm{Area}_{P@A}= 2 r^2 \left\{ \cos (\alpha) \sin ^{-1}(\cot (\alpha) \cot (\beta)) -\tan ^{-1}\left(\frac{\cos (\beta)}{\sqrt{\sin ^2(\beta)-\cos ^2(\alpha)}}\right)\right\}+\pi  r^2 (1-\cos (\alpha)).$  Note that the solution is defined as a function of α and β. If we change the value of α and β, the area of P coated portion into A(air) will change eventually. Thus we can calculate the area for all valid condition of $\alpha$ and $\beta$.  Now consider a different situation where P and S are not equally coated rather S>P. To show that I have introduced another angle γ also $r1$ the radius of the base of the cap.   Now how can i write $z$ as an increased angle? what will be the limits of integration for a spherical cap? Is there any particular analytical solution exists to calculate the area P(grey) into A(air) as a function of $α,γ,β$?
Given a sphere of radius $r$ with two spherical caps on it defined by the radii ($a_1$ and $a_2$) of the bases of the spherical caps, given a separation of the two spherical caps by angle $\theta$, how do you calculate the surface area of that intersection?   To clarify, the area is that of the curved surface on the sphere defined by the intersection. At the extreme where both $a_1,a_2 = r$, we would be describing a spherical lune.   Alternatively define the spherical caps by the angles  $\Phi_1 = \arcsin(a_1/r)$ and $\Phi_2 = \arcsin(a_2/r)$.
Getting random row through SQLAlchemy How to type special/accented letters in LaTeX? Where can I get a device with 'Ubuntu for phones' pre-installed? Is there a nice solution to get a "presenter mode" for Latex presentations? How to make a Rigid Body fall off a conveyor belt? I've created conveyor belt and and two objects with rigid body physics. But the first object won't fall and second object falls backwards.  How can I fix it? Inconsistency in display of animated avatars It has been discussed previously that animated avatars are disliked because they cause a distraction from the content. However, currently animated avatars are not treated consistently.   On desktop, animated avatars are displayed as a still image, both in posts and on the profile page. On mobile, animated avatars are displayed as a still image in posts, but animated on the profile page.   I'm tagging this  because I'm guessing one of these two is unintended. Either avatars should animate on the profile page on both desktop and mobile, or on neither. (I sincerely hope that we are all agreed there should be no animation of avatars in posts.)  Example animated profile    This appears to animate on mobile, and show a still image on desktop (for myself and for other users who have tried).    I'd also like to hear arguments for and against animating only on the profile page, but the main purpose of this post is to clear up the inconsistency. Missing Microsoft.Data.Services.Client version 5.6 on Azure Websites What is used in the 12.10 full disk encryption? Similar matrices and field extensions How do we prove $n^n \mid m^m \Rightarrow n \mid m$? I'm not sure I've got this right. When proving $a^n \mid b^n \Rightarrow a \mid b$, can we do this indirectly? In short,  "Suppose $a$ does not divide $b$, this implies that $a^n$ does not divide $b^n$. But $a^n \mid b^n$, hence $a$ divides $b$."  How about $n^n \mid m^m \Rightarrow n \mid m$? Can we do this the same way? external variable in awk Suppose $X_n$ is the fortune of a gambler after $n$ th game. Then the game is called fair (Breiman 1968) if   $$E[X_{n+1} \mid X_1, \dots, X_n] = X_n \forall n$$  My question is why a fair game is not defined as the following   $$E[X_{n+1}] = E[X_n] \forall n$$ i.e. $$E[X_{n+1}- X_n]=0$$. This should be the proper definition as a fair game is where avg. gain is zero. Nothing conditioning should be there. Why "hide" Luke from his father somewhere Anakin knows? How can I handle the warning of file_get_contents() function in PHP? Prove that if the identity is written as the product of $r$ transpositions, then $r$ is an even number Theorem. If the identity is written as the product of $r$ transpositions, $id=τ_1τ_2\dots τ_r$, then $r$ is an even number. Proof. We will employ induction on $r$. A transposition cannot be the identity; hence, $r &gt; 1$. If $r = 2$, then we are done. Suppose that $r &gt; 2$. In this case the product of the last two transpositions, $τ_{r−1}τ_r$, must be one of the following cases: $(ab)(ab) = id\\ (bc)(ab) = (ac)(bc)\\ (cd)(ab) = (ab)(cd)\\ (ac)(ab) = (ab)(bc),$ where $a$, $b$, $c$, and $d$ are distinct. The first equation simply says that a transposition is its own inverse. If this case occurs, delete $τ_{r−1} τ_r$ from the product to obtain $id=τ_1τ_2···τ_{r−3}τ_{r−2}$. By induction $r − 2$ is even; hence, $r$ must be even. In each of the other three cases, we can replace $τ_{r−1} τ_r$ with the right-hand side of the corresponding equation to obtain a new product of $r$ transpositions for the identity. In this new product the last occurrence of $a$ will be in the next-to-the-last transposition. We can continue this process with $τ_{r−2} τ_{r−1}$ to obtain either a product of $r − 2$ transpositions or a new product of $r$ transpositions where the last occurrence of $a$ is in $τ_{r−2}$. If the identity is the product of $r − 2$ transpositions, then again we are done, by our induction hypothesis; otherwise, we will repeat the procedure with $τ_{r−3}τ_{r−2}$. At some point either we will have two adjacent, identical transpositions canceling each other out or $a$ will be shuffled so that it will appear only in the first transposition. However, the latter case cannot occur, because the identity would not fix $a$ in this instance. Therefore, the identity permutation must be the product of $r − 2$ transpositions and, again by our induction hypothesis, we are done. Question: That is the proof written in the book, but unfortunately after more than $10$ times reading I don't understand the last two paragraph (i.e., from &quot;In each of the other three cases&quot;). Is there another simpler proof or could someone please clarify what is going on? Thank you. I want to shade the area of overlap for adjacent normal curves, shown in the lower graph. I can get shading between the first two curves but not for subsequent curves. I need shading at overlapping areas indicated by the arrows. I applied the "logic" of the first fill for the subsequent curves but that does not work.   The arrows and letters in the example image are not part of the MWE.   \documentclass{article}  \usepackage{pgfplots} \pgfplotsset{ticks=none} \pgfplotsset{compat=1.7}  \usepgfplotslibrary{fillbetween}  \pgfmathdeclarefunction{gauss}{2}{%normal distribution where #1 = mean and #2 = sd}     \pgfmathparse{exp(-((x-#1)^2)/(2*#2^2))}% }  \pgfplotsset{baseplot/.style={%     no markers,      domain=1:4.5,      samples=100,     smooth,     axis lines*=left,     height=5cm, width=12cm,     enlargelimits=upper, clip=false, axis on top,     xlabel = near ticks,     xlabel={Resource} }}  \begin{document}  \begin{tikzpicture}   \begin{axis}[baseplot]   \addplot+[name path=A]{gauss(2.7,0.4)};   \addplot+[name path=B]{gauss(3.2,0.4)};   \fill[gray!20, intersection segments ={of= B and A}]; \end{axis} \end{tikzpicture}  \vspace*{3\baselineskip}  \begin{tikzpicture}   \begin{axis}[baseplot]   \addplot+[name path=A]{gauss(1.7,0.15)};   \addplot+[name path=B]{gauss(2.2, 0.15)};   \addplot+[name path=C]{gauss(2.7, 0.15)};   \addplot+[name path=D]{gauss(3.2, 0.15)};   \addplot+[name path=E]{gauss(3.7, 0.15)};   \fill[gray!20, intersection segments ={of= B and A}];   \fill[gray!20, intersection segments ={of= C and B}];   \fill[gray!20, intersection segments ={of= D and C}];   \fill[gray!20, intersection segments ={of= E and D}]; \end{axis} \end{tikzpicture}  \end{document} Set limit to one author when using "et al." in biblatex Why doesn't C3PO recognize Leia in R2's recorded message? Can you nest html forms? I have reviewed this:   but it does not answer my question. The suggested fixes in this question is to set up FB, IG, and other social media pages. We have had these for months. We have 576 Instagram followers, 293 followers on Facebook, 17,425 views on Vimeo, Youtube, Pinterest and LinkedIn presence. Here is the response in a YouTube search:  Let's be clear. If you search for my company name and product name YT changes the spelling to something that doesn't exist and suggests a video with no matching result terms (using either spelling) disparaging OTHER luxury handbags, while insinuating it is referring to my handbags. When a user searches for our business, , Google and Youtube changes the spelling to Aslan Vaughn and displays these results:  For the first few months it just seemed like it was taking some time to percolate, but over 7 months we have had 17,500 unique visitors and nearly 90 backlinks.  The only results for Aslan Vaugn is the birth announcement of the second child of a musician in New Jersey. It is a Facebook post and has 14 likes. Google is favoring a spelling that has only two results with 14+ views over dozens of links and tens of thousands of views. How is this possible? An FB post with 14 likes is that much more popular than a business with 17,500 unique visitors (despite the Google obfuscation)? Is my company on some list? On other search engines my company covers the first few pages in results. Through Google, you cannot find our company no matter how many pages in you go. Unless you search for the term in quotes, . In which case we cover the first few pages here as well: