Unity: Animate Sprite Shape Sprite? Is it possible to animate a sprite shape?  I don't see the Sprite Shape Profile in the animator properties, so I was just wondering if there's another way to do that. To clarify: Our game is 2D animated by hand, and one level has conveyor belts that span the level.  We can animate it normally, but that will end up taking more video memory than if we were able to animate the conveyors as tiled sprite shapes.  It may not end up being worth the effort to save the memory, but I figured it was worth a shot.  Because the sprites span the level, they end up taking up lots of space in a sprite atlas.
How to efficiently animate 2D scene spanning sprites in Unity? Our game is a 2D point-and-click adventure game.  One of the levels is set in a factory with conveyor belts that span the scene.  The game is hand animated in 2D and so animating these conveyor belts could potentially take up lots of memory since the sprites are as long as the screen (it's an HD game - 1920 x 1080). I'm not sure if tiled sprite shapes can be animated, so I was wondering if there's a prescribed way to handle larger sprite animation like this - efficiently.
How can I find $\gcd(a^m+1,a^n+1)$ with $a,m,n$ positive integers? How can I find $\gcd(a^m+1,a^n+1)$ with $a,m,n$ positive integers?    I have this idea:  Let $d=\gcd(m,n)$. Then there exist positive integers $x,y$ such that $mx-ny=d$ (WLOG). We shall find $G=\gcd(a^m+1,a^n+1)$.  If $m,n$ are odd, then $d$ is odd, therefore one and only one of $x,y$ is even. We have:  $$a^{ny}(a^d+1)=a^{mx}+a^{ny}=(a^{mx}-1)+(a^{ny}+1)=(a^{mx}+1)+(a^{ny}-1).$$  If $x$ is even and $y$ is odd, then $a^{m}+1\mid a^{mx}-1$ and $a^{n}+1\mid a^{ny}+1$, therefore $G\mid a^{ny}(a^d+1)$. If $x$ is odd and $y$ is even, then $a^{m}+1\mid a^{mx}+1$ and $a^{n}+1\mid a^{ny}-1$, thus $G\mid a^{ny}(a^d+1)$. However, since $\gcd(a^m+1,a^{ny})=\gcd(a^n+1,a^{ny})=1$, so $\gcd(G,a^{ny})=1$, hence $G\mid a^d+1$. We also have $a^d+1\mid a^{m}+1$ and $a^d+1\mid a^{n}+1$, so $a^d+1\mid G$. Thus $G=a^d+1$.  If $v_2(m)=v_2(n)=v_2(d)=k&gt;1$, then there exist some odd numbers $m_1,n_1,d_1$ such that $m=2^km_1,n=2^kn_1,d=2^kd_1$. We shall have $m_1x-n_1y=d_1$, so one and only one of $x,y$ is even, and we can use the same argument when $m,n$ are odd, so $G=a^d+1$.  However, if $v_2(m) \neq v_2(n)$, I cannot find any solutions for this. I think that $G \in \{1,2\}$, but I cannot prove or disprove it. How can I find $G=\gcd(a^m+1,a^n+1)$ if $v_2(m) \neq v_2(n)$ ? Moreover, is there anything from my arguments that can be improved ? Why didn't the rebels on Hoth use X-Wings to fight AT-ATs? do a good job explaining why the Imperial forces didn't have fighter protection for their ground force on Hoth.  But nothing there explains why the Rebels didn't use X-Wings to easily dispatch the AT-ATs and buy more time. They obviously had them available because they were sending them off with transports and some of these were obviously not sent until after "defending" the base (since a few of the pilots of the X-Wings were the Rogue Squadron pilots fighting in the Battle of Hoth).  This happens in the EU in the book  (in fact, the cover picture shows this scene) with great success and Wedge actually comments, on page 120:     Back then we were in airspeeders - undergunned and overmatched... Not the case this time [referring to being in X-Wings against AT-ATs instead of speeders on Hoth]   This indicates X-Wings can perform in cold conditions - there was snow in this environment as described by Corran Horn after he is shot down. The temperature difference between day-time on Hoth and here is unknown however.  Additionally, the X-Wings were able to destroy the AT-ATs with their laser cannons only - no grappling was required.  I suspect the answer is basically "because that would have been boring to watch on-screen" but it seems there may have been an actual in-universe reason outside of this. How can I suspend/hibernate from command line? How can I suspend or hibernate my laptop using command line, without installing additional software? I'm trouble with exercise 1.8 of Carroll's Space-Time and Geometry:     If $\partial_\nu T^{\mu \nu} = Q^\mu$, what physically does the spatial vector $Q^i$ represent? Use the dust energy momentum tensor to make your case.    First of all, since the momentum energy tensor of dust $T^{\mu \nu}=\rho U^\mu U^\nu$ is symmetric, one has the energy-momentum conservation law $$ \partial_\nu T^{\mu \nu} = \partial_\nu T^{\nu \mu}  =0.$$ A vector $Q$ is constant in any frame related by a Lorentz transformation, while this isn't the case for the components $Q^\mu$. But here $Q=0$ and hence $Q^\mu=0$ also for every component $\mu =0,\cdots,4$?   Even if a straightforwardly apply the equation: $$ Q^i = \partial_\nu T^{i \nu} = (\partial_\nu \rho) U^i U^\nu + \rho(\partial_\nu U^i)  U^\nu + \rho U^i(\partial_\nu  U^\nu ),$$ doesn't seems to yield something...   Is my first reasoning right ? Otherwise: can someone give me a hint ?  Thanks Calculating a spherical polygon centroid I have been thinking of getting a flash to improve my low light photography (particularly indoors). However, there seem to be a number of different models available at similar price points (budget of INR 8,000-12,000 ~$150-250). What features and functionality should I look for to choose a suitable flash?  I'm using a Canon 550D with the 18-55mm IS &amp; 55-250mm IS lenses. Metz seems to be one of the brands easily available in India (beside the OEM flashes). Currently, I am considering the Metz 36 AF-5, 44 AF-1 and 50 AF-1, which seem to be considerably cheaper than their Canon counterparts. If $2a_{n+2} \le a_{n+1}+a_n$, then $\lim \sup a_n \le \frac23 a_2 + \frac13 a_1$ This is a reformulation of a deleted question:  If  $a_1 &gt; 0$ and $a_2 &gt; 0$ and $2a_{n+2} \le a_{n+1}+a_n$,  show that  $\lim \sup a_n \le \frac23 a_2 + \frac13 a_1$.  My proof involves showing that $a_{n+2} \le u_{n} a_{n+1} + (1-u_n)a_n $ where $u_n \to \frac23$, and I wondered if there is a simpler proof.  I will post my proof in a couple of days if no one does. Removing unwanted appearance of underlying mesh How could I define function with function such as Coefficient? Deep cloning objects What is the meaning, usage and formality of the greeting "What's up"? What is the difference between ubuntu-18.04-server and ubuntu-18.04-live-server? On the  there is a download link for ubuntu-18.04-live-server-amd64.iso, and on the  there is a download link for ubuntu-18.04-server-amd64.iso. Before 18.04, there is only a server version. What is the difference between these 2 images? Why do I get compiler errors with this Java code?  1  public List&lt;? extends Foo&gt; getFoos() 2  { 3    List&lt;? extends Foo&gt; foos = new ArrayList&lt;? extends Foo&gt;(); 4    foos.add(new SubFoo()); 5    return foos; 6  }  Where 'SubFoo' is a concrete class that implements Foo, and Foo is an interface.  Errors I get with this code:   On Line 3: "Cannot instantiate ArrayList&lt;? extends Foo&gt;" On Line 4: "The method add(capture#1-of ? extends Foo) in the type List&lt;capture#1-of ? extends Foo&gt; is not applicable for the arguments (SubFoo)"   Update: Thanks to Jeff C, I can change Line 3 to say "new ArrayList&lt;Foo&gt;();".  But I'm still having the issue with Line 4. Duplicate objects transformation applies to both objects I'm facing this frustrating "feature" that I can't seem to remedy; when I duplicate an object (and now, even when I create two completely new objects), their transformations/scaling is synchronized. This is extremely annoying when trying to use any sort of transformation keyboard shortcut.  It should be noted by going to the properties menu (N) and manually setting the properties works just fine.  For instance, here is just the camera selected (trust me, I made sure):   And when I try to transform by clicking+dragging on the X axis OR by hitting G->X, this happens:   They are not grouped (verified by going to the Groups list up in the Scene panel) and they do not have parents (verified by going to Object->Relations).  I have tried the unlink command as well, to no avail.  I have seriously run out of options; what did I change to have my G->X shortcuts transform multiple objects? It seems completely at random, too; certain objects seemed to be "linked" to other random objects, and this "linking" is guaranteed when duplicating even when I do a non-linking duplicate (Shift+D).  What gives? What is the concept of erasure in generics in Java? I cannot find enable-interactive-search in    gsettings set org.gnome.nautilus.preferences enable-interactive-search true.     I searched using    gsettings list-recursively | grep search     and    gsettings list-recursively | grep interactive     but no such option.  Can you help me disable this in Ubuntu 17.10 and later? I have been wanting to understand how to find the sum of this series.   $$1^p + 2^p + 3^p +{\dots} + n^p$$  I am familiar with Gauss' diagonalised adding trick for the sum of the first $n$ natural numbers.  I can prove the formulas for  $$\begin{align}   \sum_{1}^{n} k^2 &amp;= \frac{n(n+1)(2n+1)}{6}\\  \sum_{1}^{n} k^3 &amp;= \frac{n^2(n+1)^2}{4}\\ \sum_{1}^{n} k^4 &amp;= \frac{n(n+1)(2n+1)(3n^2+3n-1)}{30} \\ \end{align}$$  With mathematical induction. But, beyond that even proofs with mathematical induction are difficult.  I'm interested in learning the theory and the proof behind Faulhaber's formula. What is the knowledge required to understand this proof ? The use of nominative "whom" (as in “persons whom it is foreseeable are likely to...”) From  page 48 of Law: A Very Short Introduction, by Raymond Wacks:     In other words, you owe a duty to persons whom it is foreseeable are likely to be harmed by your conduct.   To try to parse this, temporarily overlook 'it is foreseeable'. Then persons is the subject of the verb phrase are likely to be harmed. So , should the relative pronoun for persons be who instead? Is my parse wrong?   Are the (deleted) comments right: that this style predominates in law texts? If this is correct grammar, what is the general rule? Is it commonly used in other contexts? How can I run a script that starts before my login screen? I was calibrating my touch screen, and saw that the best tool around was xinput_calibrator. So I used it. It have two options (one of which did not work), so I am here for the second. It says I should execute this command "in a script that starts with your X session":  xinput set-int-prop "3M 3M USB Touchscreen - EX II" "Evdev Axis Calibration" 32 14410 2146 14574 2115  So I tried ~/.xinitrc, ~/.xsession and ~/.xsessionrc, all of which did not exist. So I created them and the exact content was this command. The first two files made my logins fail (after I login, I fall back to the login screen).  With the last file, the calibration was functional, but only after logging in...  I need that command to run before the login dialog shows up. I thought of adding this command to the end of /etc/X11/xinit/xinitrc with no result (nothing changed). Also, I tried to add it to the end of /etc/X11/Xsession.d/40x11-common_xsessionrc (after inspecting some of the files), but the result was exactly the same as adding it to ~/.xsessionrc.  How can I make this command run before the login screen shows (is this before the window manager starts, or before the X session starts)?  (I am running Kubuntu with the default window manager, if that matters)  UPDATE As I am using Kubuntu, my display manager is kdm. As the accepted answer suggests, I edited the file /etc/kde4/kdm/Xsetup, and as mentioned  I added the command before the command that is there by default. And it works like a charm :) How can the pixel values of a raster image be extracted under a polygon fishnet?