From my understanding (and research) focal length affects how a face will end up looking on a photograph.  Here's an example of focal length distortion on a full-frame sensor:    My question is: will a photo taken with a 28mm focal length on an APS-C sensor look identical to a photo taken with a 40mm (28mm equivalent on a full frame sensor) on a full frame sensor? or will the levels of distortion remain the same as with a 28mm - full-frame sensor combo?
I see many posts on both this forum and elsewhere the discuss the use of a wide angle on a crop sensor.  For example, they discuss how at 14mm lens is 14mm on a 35mm film or full frame, but with a 1.6x crop sensor it is an effective 28mm.  My question is, is a 14mm on a crop sensor the same image as a 28mm on a full frame.  Or does "effective" include some other conotation.  I read that a 14mm can create distortion on a full frame and some vignette, so people seem to say using it on a crop takes out some of those edge problems. My confusion is how the lens is actually taking in the extra image.  Reading kenrockwells blog I learned that wide angle keeps lines straight, while fish eye actually skews the lines.    So with this as my understanding, I am still a bit confused how a lens that is curved in order to get a greater angle of view, will be the same as a cropped version of a less curved lens.  Or does "effective" only refer to the objects to the sides but not to the distortion of the image.  Or, if it is different, is it so small that for photography it does not matter?  A technically description would be welcome as long as a basic understanding of the situation.  Thanks!  Edit: To be more clear after the comments,  I am specifically talking about the end resulting 2d projection from 3d space.  I have read both those previous answers.  the first one is closer to what I am talking about.  However, it is still confusing for me.  I have experience with 3d modeling and projection matricies, which might be confusing me with this.  But for example, i dont understand the images having a 50 mm lens being drawn Starting from the sensor, so it shows a different fov for crop and full frame . If the lens is the same, it takes the same amount of information from the world, which makes a ray from a certain degree off the last lens projected into a smaller space, there is no projection from the sensor, so drawing the line from the sensor doesnt make sense to me.  The distance of the sensor to the rear of the lens must have some impact on the light projected but is not represented in the images.  Further, again from that first answer it says cropping is the same as zoom, but from what i understand from how perspective works, it is different, since a wide angle will project lines differently than a small fov to the same size, so cropping the center of a wide angle and zooming into something is very different.  This is easily recreated in an opengl application with varying fov projection matrix, i imagine in painting class people learn this as well.  A fisheye would be a good example, since if cropping was the same as zooming, a fish eye lens would keep the center exactly like a normal lens and then a gradient weighted towards the outside would rapidly create a warped perspective, but from what I see, it is even.  To me those images just look like its comparing cropped with full frame in regards to orthogonal projections.
Can't select and delete an object What counts as "entering a zone"? There are many actions that create persistent zones and have a trigger for their effect of among others: "...enters the zone..." What are the things that qualify as entering the zone?  Things that would seem to fit this would be:   Willing movement into the zone. Forced movement into the zone. Teleporting into the zone. Appearing inside the zone. Having the zone moved to include the square you occupy.   Do any of those listed not count as having entered the zone and why? And are there any that are not listed that would count as having entered the zone?  This is for D&amp;D 4e. Can photons be detected without being absorbed? I am thinking about a detector that would beep if light passes through it. Is it possible? How prove this $\lim_{n\to \infty}\sin{n^m}$ divergent. show that $$\lim_{n\to \infty}\sin{(n^m)}$$  divergent,where $m\in N^{+}$  I have kown $$\lim_{n\to\infty}\sin{n^2}$$  to be divergent and dense in $[-1,1]$. This is very  famous problem,the problem is first is post $AMM$( The American Mathematical Monthly,1970-1975) problem,and after some years  $AMM$ post $\sin{(n^{14})}$ in $[-1,1]$ was dense? and this is open problem.so I Guess  $$\sin{(n^m)}$$ is dense in $[-1,1]$. Center image element in parent div What is a socket? Why do adult insects have 6 legs? Apparently, there is an advantage to having 6 legs in the insect world. What is that advantage, if anything? Why would such an advantage exist for insects, but not for other, larger land animals?  What do those middle legs do that the front and hind legs cannot, if anything? Are they really essential, or do they just generate more muscle power? How can standing waves be produced in an open organ pipe if the ends are open? How can standing waves be produced in an open organ pipe even though both of its ends are open? Can someone explain with more clarity? In my book, it proves that an infinite subset of a coutnable set is countable. But not all the details are filled in, and I've tried to fill in all the details below. Could someone tell me if what I wrote below is valid?  Let $S$ be an infinite subset of a countable set $T$. Since $T$ is countable, there exists a bijection $f: \mathbb{N} \rightarrow T$. And $S \subseteq T = \{ f(n) \ | \ n \in \mathbb{N} \}$. Let $n_{1}$ be the smallest positive integer such that $f(n_{1}) \in S$. And continue where $n_{k}$ is the smallest positive integer greater than $n_{k-1}$ such that $f(n_{k}) \in S$. And because $S$ is infinite, we continue forever. Now consider the function $\beta : \mathbb{N} \rightarrow S$ which sends $k \rightarrow f(n_{k})$.   So in order for $S$ to be countable, $\beta$ would have to be a bijection. $\beta$ is injective since if $f(n_{k}) = f(n_{j})$, then $n_{k} = n_{j}$ because $f$ is injective. We also can conclude $k = j$ because the various $n_{i}$ chosen were strictly increasing.   Edit: $\beta$ also needs to be surjective. So for every $r \in S$ there exists a $q \in \mathbb{N}$ such that $\beta(q) = f(n_{q}) = r$. We know that $f^{-1}(r) \in \{n_{1}, n_{2}, ..., n_{k} ... \}$, so we can let $f^{-1}(r) = n_{d}$. Thus $\beta(d) = f(n_{d}) = r$. How to set a JavaScript breakpoint from code in Chrome? Data inconsistency prohibition if a table refers to another via two many-to-many relationships I have following database design (by =&gt; foreign key constraints are depicted):   Company[id] CompanyRealm[id, company_id=>Company.id]many2many  Project[id, company_id=>Company.id] ProjectRealm[id, project_id=>Project.id, company_realm_id=>CompanyRealm.id]many2many   Problem is that provided database design allows inconsistent data. For example:   Company1(id=1) Company2(id=2) CompanyRealm(id=11, company_id=1) Project(id=33, company_id=2) ProjectRealm(id=44, project_id=33, company_realm_id=11)   (Company can have many Realms, Project belong to Company and  may concern any number of company's Realms)  ProjectRealm refers to two different companies:   Company1 (via CompanyRealm); and Company2 (via Project).    Is something wrong with my DB design? If yes - which prescriptions are violated? If no - how to prevent incorrect data insertion (via constraint? trigger with exception?) I know that, whenever I want to see how some control sequence, say \mycontrolsequence, is defined, I can use \show\mycontrolsequence  then compile my code and look up the .log file. However, robust commands seem impervious to \show. For instance, if I compile the following code, \documentclass{article}  \newcommand\mycommand{foo} \show\mycommand  \DeclareRobustCommand{\myrobustcommand}{bar} \show\myrobustcommand  \begin{document} hello \end{document}  my log file contains  \mycommand=\long macro: -&gt;foo. l.4 \show\mycommand  \myrobustcommand=macro: -&gt;\protect \myrobustcommand  . l.7 \show\myrobustcommand   Note that, whereas the definition of \mycommand is exposed (foo), nothing transpires of \myrobustcommand's definition. Of course, in practice, I could always identify in which file a given robust command is defined and look up its definition there. However, finding the file in question may not be obvious. Hence my question: Is there an alternative way of peering into the definition of a robust command? Typsetting the Ponny Today, once again, the  came to my attention and I have decided to try to  using XeLaTeX. The result is far from perfect, worse even than Chrome's rendering.  Is there a way I can improve the result? Is XeLaTeX and LuaLaTeX incapable of handling the Ponny?  :  \documentclass[]{article} \usepackage{lmodern} \usepackage{amssymb,amsmath} \usepackage{ifxetex,ifluatex} \usepackage{fontspec} \defaultfontfeatures{Mapping=tex-text,Scale=MatchLowercase} \IfFileExists{upquote.sty}{\usepackage{upquote}}{} \IfFileExists{microtype.sty}{\usepackage{microtype}}{} \usepackage[normalem]{ulem} \setmainfont{Doulos SIL}  \title{html - RegEx match open tags except XHTML self-contained tags - Stack Overflow} \date{} \author{StackOverflow}  \begin{document} \maketitle  Rege̿̔̉x-based HTML parsers are the cancer that is killing StackOverflow \emph{it is too late it is too late we cannot be saved} the trangession of a chi͡ld ensures regex will consume all living tissue (except for HTML which it cannot, as previously prophesied) \emph{dear lord help us how can anyone survive this scourge} using regex to parse HTML has doomed humanity to an eternity of dread torture and security holes \emph{using rege}x as a tool to process HTML establishes a brea\emph{ch between this world} and the dread realm of c͒ͪo͛ͫrrupt entities (like SGML entities, but \emph{more corrupt) a mere glimp}se of the world of reg​\textbf{ex parsers for HTML will ins}​tantly transport a p\emph{rogrammer's consciousness i}nto a w\emph{orl}d of ceaseless screaming, he comes\sout{, the pestilent sl}ithy regex-infection wil​\textbf{l devour your HT}​ML parser, application and existence for all time like Visual Basic only worse \emph{he comes he com}es \emph{do not fi}​ght h\textbf{e com̡e̶s, ̕h̵i}​s un̨ho͞ly radiańcé de\emph{stro҉ying all enli̍̈́̂̈́ghtenment, HTML tags \textbf{lea͠ki̧n͘g fr̶ǫm ̡yo​͟ur eye͢s̸ ̛l̕ik͏e liq}​uid p}ain, the song of re̸gular exp​re\sout{ssion parsing} will exti\emph{​nguish the voices of mor​\textbf{tal man from the sp}​here I can see it can you see ̲͚̖͔̙î̩́t̲͎̩̱͔́̋̀ it is beautiful t​}he f\texttt{inal snuf}fing o\emph{f the lie​\textbf{s of Man ALL IS LOŚ͖̩͇̗̪̏̈́T A}}\textbf{LL I​S L}OST th\emph{e pon̷y he come}s he c̶̮om\sout{es he co}\textbf{\sout{me}s t\emph{he} ich​}or permeat\emph{es al}l MY FAC\emph{E MY FACE ᵒh god n\textbf{o NO NOO̼}}\textbf{O​O N}Θ stop t\emph{he an​*̶͑̾̾​̅ͫ͏̙̤g͇̫͛͆̾ͫ̑͆l͖͉̗̩̳̟̍ͫͥͨ}e̠̅s\texttt{ ͎a̧͈͖r̽̾̈́͒͑e} n\textbf{​ot rè̑ͧ̌aͨl̘̝̙̃ͤ͂̾̆ ZA̡͊͠͝LGΌ ISͮ̂҉̯͈͕̹̘̱ T}O͇̹̺ͅƝ̴ȳ̳ TH̘\textbf{Ë͖́̉ ͠P̯͍̭O̚​N̐Y̡ H̸̡̪̯ͨ͊̽̅̾̎Ȩ̬̩̾͛ͪ̈́̀́͘ ̶̧̨̱̹̭̯ͧ̾ͬC̷̙̲̝͖ͭ̏ͥͮ͟Oͮ͏̮̪̝͍M̲̖͊̒ͪͩͬ̚̚͜Ȇ̴̟̟͙̞ͩ͌͝}S̨̥̫͎̭ͯ̿̔̀ͅ  \end{document}   UPDATE  Using the Doulos SIL font gets me something closer to what Firefox renders but still far from perfect. The passive with "let" In thinking about Sturm-Liouville theory a bit I see I have no actual idea what is going on.  The first issue I have is that my book began with the statement that given  $$L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  the problem $L[y] \ = \ f$ can be re-cast in the form $L[y] \ = \ \lambda y$.  Now it could be a typo on their part but I see no justification for the way you can just do that!  More importantly though is the motivation for Sturm-Liouville theory in the first place. The story as I know it is as follows:  Given a linear second order ode   $$F(x,y,y',y'') = L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  it is an exact equation if it is derivable from a differential equation of one order lower, i.e.  $$F(x,y,y',y'') = \frac{d}{dx}g(x,y,y').$$  The equation is exact iff  $$a''(x) - b'(x) + c(x) = 0. $$  If $F$ is not exact it can be made exact on multiplication by a suitable integrating factor $\alpha(x)$.  This equation is exact iff  $$(\alpha(x)a(x))'' - (\alpha(x)b(x))' + \alpha(x)c(x) = 0 $$  If you expand this out you get the Adjoint operator  $$L^*[\alpha(x)] \ = \ (\alpha(x)a(x))'' \ - \ (\alpha(x)b(x))' \ + \ \alpha(x)c(x) \ = 0 $$  If you expand $L^*$ you see that we can satisfy $L \ = \ L^*$ if $a'(x) \ = \ b(x)$ &amp; $a''(x) \ = \ b'(x)$ which then turns $L[y]$ into something of the form  $$L[y] \ = \ \frac{d}{dx}[a(x)y'] \ + \ c(x)y \ = \ f(x).$$  Thus we seek an integratiing factor $\alpha(x)$ so that we can satisfy this &amp; the condition this will hold is that $\alpha(x) \ = \ \frac{1}{a(x)}e^{\int\frac{b(x)}{a(x)}dx}$  Then we're dealing with:  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \alpha(x)f(x)$$  But again, by what my book said they magically re-cast this problem as  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \lambda \alpha(x) y(x)$$  Then calling  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ ( \alpha(x)c(x)y \ - \ \lambda \alpha(x) )y(x) \ = \ 0$$  a Sturm-Liouville problem.  My question is, how can I make sense of everything I wrote above? How can I clean it up &amp; interpret it, like at one stage I thought we were turning our 2nd order ode into something so that it reduces to the derivative of a first order ode so we can easily find first integrals then the next moment we're pulling out eigenvalues &amp; finding full solutions - what's going on? I want to be able to look at $a(x)y'' \ + \ b(x)y' \ + \ c(x)y \ = \ f(x)$ &amp; know how &amp; why we're turning this into a Sturm-Liouville problem in a way that makes sense of exactness &amp; integrating factors, thanks for reading! php - Should I encrypt email addresses? If statement evaluating true no matter what. Python 3.3.0 Can a mathematically sound prediction interval have a negative lower bound? I have used R to form a 95% prediction interval for the number of endemic species on an island. My lower bound is negative – is that mathematically sound?  In the linear model used in the prediction interval, the data used are:  Area Surface area of island, hectares DiscSC Distance from Santa Cruz, kilometres Elevation Elevation of higher point in metres and it is coded as such:   selected.model &lt;- lm(ES ~ Area + Elevation + DistSC + I(Elevation^2)                      + (Elevation:DistSC) + (A‌​rea:Elevation))    and stepwise regression was performed to find this "best" model   I'm not exactly sure how a prediction interval works. I just want to make sure it is OK. Obviously a negative number of species is incorrect, but I know it takes into account the uncertainty of the mean as well as data scatter. How to describe arbitrary accelerations in special relativity Symbols instead of numbers as footnote markers How can I make LaTeX use symbols (*, †, ‡, and so on) instead of numbers to mark footnotes?  (The numbers are confusing because I use superscripted numbers for citations.)