I’ve been trying to convince myself of the veracity of this for a while and I can’t seem to do it.   Given two planes in R3, will an ellipse on one plane necessarily project to an ellipse on the other?  If the planes are parallel, it’s trivially true. Similarly, if the ellipse is a circle, I know it projects to an ellipse. But if they’re not either of those cases? Assuming an orthographic projection, it seems to me the projection could end up some skewed, egg-like shape where the ellipse axes’ intersections point has been shifted away from the center of the ellipse.   Can someone please shed some light on this?
While studying the problem of determining the orbit of a binary star system, my astronomy notes say: "the projection of an ellipse onto a plane is still an ellipse (except the special case in which it's a segment)". How could I prove this?  EDIT: As suggested i'm adding what I know about ellipses. I know the basics of ellipses in the plane, the equation of an ellipse in a plane in cartesia coordinates, the equation of an ellipse in polar coordinates, the fact that ellipses can be obtained by the intersection of planes with a cone. In my geometry course we defined ellipses as the the points of the plane such that the ratio between the distance from a point and a line is constant and less than 1. I also know that the sum of the distances of the points of the ellipse from the foci is constant.
Query Feature and Usage based entitlements license How is light affected by gravity? Why is my StackOverflow with an indent and doesn't have reputation Well everything is in the title. Why is StackOverflow showing up with a indent and why don't I see my rep next to it like on the other websites. How to determine the amount of experience needed for leveling up Related questions under this topic might include:   How should I base how much experience points it takes to get to the next level? What factors are involved in this? How do I keep the player not entertained long enough till he levels up? How can I vary the way the player gets experience points? Objects vs arrays in Javascript for key/value pairs True\False?  For every matrix $A\in M_{2}( \mathbb{C}) $ there's  $X\in M_{2}( \mathbb{C})$ such that $X^2=A$.  I know that every complexed matrix has a Jordan form matrix $J$ such that $P^{-1}CP=J$, But it's not diagonalizable for sure.  Thanks I am confused. I don't understand the difference a ARMA and a GARCH process.. to me there are the same no ?  Here is the (G)ARCH(p, q) process  $$\sigma_t^2 =      \underbrace{     \underbrace{      \alpha_0       + \sum_{i=1}^q \alpha_ir_{t-i}^2}     _{ARCH}        + \sum_{i=1}^p\beta_i\sigma_{t-i}^2}     _{GARCH}$$  And here is the ARMA($p, q$):  $$ X_t = c + \varepsilon_t +  \sum_{i=1}^p \varphi_i X_{t-i} + \sum_{i=1}^q \theta_i \varepsilon_{t-i}.\,$$  Is the ARMA simply an extension of the GARCH, GARCH being used only for returns and with the assumption $r = \sigma\varepsilon$ where $\varepsilon$ follows a strong white process? Mismatch Reputation in the thumbnail and the actual Reputation The Higgs boson and gluons have no electric charge and photons couple to charge, so there is no tree level interaction between them and photons.  But what prevents higher order diagrams from contributing a non-zero mass term to the photon, for instance    where a photon couples to some fermion (say an electron, or a top quark) which can interact with the Higgs field.   Or consider that same diagram but with quarks and a gluon interacting between them? Or any higher diagram with even more loops?  I have heard charge conservation depends on gauge invariance, which in turn depends on photons being massless.  So it appears the photon has no mass, and these diagrams must all cancel somehow.  So I'm hoping there is a very nice symmetry explanation for why they all disappear, but if I say "because of gauge invariance" that would be circular logic, so there must be another symmetry at stake here?     What prevents photons from obtaining a mass from high-order self-energy loop diagrams? Sole minimal element: Why not also the minimum? A minimal element (any number thereof) of a partially ordered set $S$ is an element that is not greater than any other element in $S$.  The minimum (at most one) of a partially ordered set $S$ is an element that is less than or equal to any other element of $S$.    Let's consider the power set $\mathcal P (\{x,y,z\})$ together with the binary relation $\subseteq$.  The Hasse diagram shows what element(s) we're looking for:    It's easy to see that:   $\emptyset$ is a minimal element $\emptyset$ is the minimum   Now if we remove $\emptyset$ and consider $\mathcal P (\{x,y,z\})\setminus \emptyset$ instead, we get the following:   $\{x\}$, $\{y\}$ and $\{z\}$ are minimal elements there is no minimum   (1) We know that a minimum is unique and it is always the only minimal element.  (2) And from the example above, it seems that, if a sole minimal element exists, it is always the minimum.  But I read that (2) is false. Why? What my thought was that the bijection was a piecewise function:  $f(x) = 1/n+1$ if $x = 1/n$ for some $n \in \Bbb N$ and $x=1/n$ if $x \neq1/n$ for some $n \in \Bbb N$.  However, the textbook says this is incorrect. I don't actually see why. Was it a matter of brackets or the open variable $(0,1]$, which is what I tend to think. How to put a variable in anchor tag induction proof: $\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$ I encountered the following induction proof on a practice exam for calculus:  $$\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$$  I have to prove this statement with induction.  Can anyone please help me with this proof? Is $29$ the only prime of the form $p^p+2$? What is the "wheel" user in OS X? I just made a minor , and  the grammar correction was unnecessary.  The original text was     Another factor--sometimes life happens and you can't fly. In the old days you could simply sell your ticket to someone else, now you either have to eat a hefty change fee or lose it outright. That's money in their pockets that they didn't used to get.   It was corrected to:     Another factor--sometimes life happens and you can't fly. In the old days you could simply sell your ticket to someone else, now you either have to eat a hefty change fee or lose it outright. That's money in their pockets that they didn't use to get.    Was this grammatical correction an improvement? Was the original text grammatically correct? Is the edited text grammatically correct? How do I manage double-faced cards in Innistrad (MTG) tournament play? My system freezes completely at random, frequent intervals. I started to have the same problem in Ubuntu 14.04 but after recent upgrade to 16.04 there is no improvement, in fact it seems worse.  When it happens, it's impossible to do anything. I've tried everything in this thread:  but nothing works, I have to hard reset. I have read all the system logs and journalctl but there is never any information that could help diagnose the problem.  This is a dual-boot system with Windows 10 and there's no problem there, so it's not defective hardware.  My laptop has an Intel Bay Trail processor (Pentium N3540) Err http://archive.canonical.com natty InRelease     Err http://security.ubuntu.com oneiric-security InRelease                Err http://extras.ubuntu.com natty InRelease                             Err http://security.ubuntu.com oneiric-security Release.gpg   Temporary failure resolving ‘security.ubuntu.com’ Err http://archive.canonical.com natty Release.gpg   Temporary failure resolving ‘archive.canonical.com’ Err http://extras.ubuntu.com natty Release.gpg   Temporary failure resolving ‘extras.ubuntu.com’ Err http://gb.archive.ubuntu.com oneiric InRelease Err http://gb.archive.ubuntu.com oneiric-updates InRelease Err http://gb.archive.ubuntu.com natty-backports InRelease Err http://gb.archive.ubuntu.com oneiric Release.gpg   Temporary failure resolving ‘gb.archive.ubuntu.com’ Err http://gb.archive.ubuntu.com oneiric-updates Release.gpg   Temporary failure resolving ‘gb.archive.ubuntu.com’ Err http://gb.archive.ubuntu.com natty-backports Release.gpg   Temporary failure resolving ‘gb.archive.ubuntu.com’  Reading package lists... Done W: Failed to fetch http://gb.archive.ubuntu.com/ubuntu/dists/oneiric/InRelease       W: Failed to fetch http://gb.archive.ubuntu.com/ubuntu/dists/oneiric-updates/InRelease   W: Failed to fetch http://gb.archive.ubuntu.com/ubuntu/dists/natty-backports/InRelease   W: Failed to fetch http://security.ubuntu.com/ubuntu/dists/oneiric-security/InRelease   W: Failed to fetch http://archive.canonical.com/ubuntu/dists/natty/InRelease   W: Failed to fetch http://extras.ubuntu.com/ubuntu/dists/natty/InRelease   W: Failed to fetch http://archive.canonical.com/ubuntu/dists/natty/Release.gpg  Temporary failure resolving ‘archive.canonical.com’ W: Failed to fetch http://security.ubuntu.com/ubuntu/dists/oneiric-security/Release.gpg  Temporary failure resolving ‘security.ubuntu.com’ W: Failed to fetch http://extras.ubuntu.com/ubuntu/dists/natty/Release.gpg  Temporary failure resolving ‘extras.ubuntu.com’ W: Failed to fetch http://gb.archive.ubuntu.com/ubuntu/dists/oneiric/Release.gpg  Temporary failure resolving ‘gb.archive.ubuntu.com’ W: Failed to fetch http://gb.archive.ubuntu.com/ubuntu/dists/oneiric-updates/Release.gpg  Temporary failure resolving ‘gb.archive.ubuntu.com’ W: Failed to fetch http://gb.archive.ubuntu.com/ubuntu/dists/natty-backports/Release.gpg  Temporary failure resolving ‘gb.archive.ubuntu.com’ W: Some index files failed to download. They have been ignored, or old ones used instead.   This is what I'm seeing when I try to run sudo apt-get update. I did an update on my instance yesterday and am now experiencing this. Total space of a finite rank locally free sheaf, Vakil's 17.1.4 & 17.1.G