Is there a general method for trying to finding parametric equations for polynomial equations?  The classic example of $x^2 + y^2 = 1$ having multiple parameterizations:  \begin{array}{rcl} (x,y) &amp;=&amp; (\sin t, \cos t) \\ (x,y) &amp;=&amp; (\cos t, \sin t) \\ (x,y) &amp;=&amp; \left(\frac{1-t^2}{1+t^2}, \frac{2t}{1+t^2}\right) \end{array}  which are easily seen to work by plugging in. Note that the first two involve elementary functions, but the last one only rational functions.  I'm curious about higher degree polynomials. Is there a general method to find a parameterized set of equations in rational functions? That is, is there an algorithm for extracting the parameter functions for $x$ and $y$ from the equation (or arbitrary sets of equations over $v\ge1$)? And if not, is there a characterization of which implicit sets can be parametrized?  Take the (random off the top of my head) example:  $$x y^3 + x^2 y + x y = 1$$  Is there a set $x(t), y(t)$ for this that can be found algorithmically? (I may well be naive here, not realizing something obvious, or maybe the entire field of algebraic geometry is geared towards possible solutions to this).   One can do the reverse process of creating an implicit set of equations via Gröbner basis calculation, trying to eliminate $t$. Or likewise in the forward direction if you can eliminate variables one-by-one with GB that would suffice to parameterize via one of the original variables. But in general if elimination isn't possible, is there still an algorithm to (even partially) parameterize?
We all know some curves can be described by $y=f(x)$ and some surfaces can be described by $z=f(x,y)$ However, there exists curves and surfaces which cannot be described by those, such as a circle and a sphere. Therefore, we introduce parameterized vector equations, which can describe them. For example, circle: $\vec r(t)=r\cos(t)\hat i+r\sin(t)\hat j$ sphere: $\vec r(u,v)=\rho\cos(u)\sin(v)\hat i+\rho\sin(u)\sin(v)\hat j+\rho\cos(v)\hat k$ However, all curves described by $y=f(x)$ and  $z=f(x,y)$ can be parameterized. For curves, $$\vec r(t)=t\hat i+f(t)\hat j$$ For surfaces, $$\vec r(u,v)=u\hat i+v\hat j+f(u,v)\hat k$$ Therefore, I think this suggests the set of all parameterized surfaces (or curves) is the super-set of the set of all surfaces (or curves) described by $z=f(x,y)$ (or $y=f(x)$). Is that correct? Now, here comes the the real challenge. A curve can also be described by an implicit function $f(x,y)=0$ and a surface can also be described by an implicit function $f(x,y,z)=0$ I have 3 questions regarding this.   Can all surfaces (curves) described by an implicit function be parameterized? (If yes, then what is the general way?) Can all surfaces (curves) described by parametric vector equations be represented using implicit function? (If yes, then what is the general way?) Compare the set of all parameterized surfaces (curves) and the set of all surfaces (curves) represented by implicit function. (which is which super-set?)   Sorry for the use of nontechnical terms. I use them because I don't know the technical ones. I have only started learning vector calculus last year in university.  EDIT: I think my question is not too clear, so I will give an example of writing the surface $f(x,y,z)=0$ into $\vec r(u,v)$ We want to parameterize a sphere. $$x^2+y^2+z^2-\rho^2=0$$ Let $x=\rho\cos(u)\sin(v)$, $y=\rho\sin(u)\sin(v)$, $z=\rho\cos(v)$, $$\rho^2\cos^2(u)\sin^2(v)+\rho^2\sin^2(u)\sin^2(v)+\rho^2\cos^2(v)-\rho^2$$ $$=\rho^2\sin^2(v)(\cos^2(u)+\sin^2(u))+\rho^2\cos^2(v)-\rho^2=\rho^2-\rho^2=0$$ I want to know if there is a general way of finding $x=x(u,v)$, $y=y(u,v)$ and $z=z(u,v)$ for any given $f(x,y,z)=0$
Light Presets .exr I would like to install light presets for Blender to automate my lightning in my scenes. I have some .exr files (from Greyscalegorilla) but I don't know how to do it on Blender to make it work. Can someone help me? Thank you I have edited a script that creates some nodes, but the only problem is, it always created a new material. I want it to either add the nodes on the existing material, or a pre-defined material. Here is the script base I am using.    Thanks for the help in advance! :D I've seen lots of examples of making an entire regular expression case-insensitive.  What I'm wondering about is having just part of the expression be case-insensitive.  For example, let's say I have a string like this:  fooFOOfOoFoOBARBARbarbarbAr   What if I want to match all occurrences of "foo" regardless of case but I only want to match the upper-case "BAR"s?  The ideal solution would be something that works across regex flavors but I'm interested in hearing language-specific ones as well (Thanks )  Edit  The link Espo provided was very helpful.  There's a good example in there about turning modifiers on and off within the expression.  For my contrived example, I can do something like this:  (?i)foo*(?-i)|BAR   which makes the match case-insensitive for just the foo portion of the match.  That seemed to work in most regex implementations except Javascript, Python, and a few others (as Espo mentioned).    The big ones that I was wondering about (Perl, PHP, .NET) all support inline mode changes. How do I achieve this kind of light star effect? I'm kinda new to Blender. Currently playing around with version 2.8 to practice some rendering. Is there a way to achieve this 'light-star' effect for the blooming objects with the emission? If there is an alternative way then it's fine too. See these images below for references. Thanks in advance: Prove that $i^i$ is a real number I understand that unlocking the bootloader will wipe my Android phone, but have been looking around for why.  Seems to be by design, but what is the reasoning for that design?  Is it some security concern, some obscure technical reason, or just for lulz?  I'm looking for something solid to chew on here, something more than because "that's how it is". I'm from the State of Kuwait and hold an Eritrean passport. I'm planning to visit UK in October but my visa will expire in November.    Is it OK to make a journey in the last month of my visa, provided that I make sure that I book my plane back before expiring date? The problem of instant velocity The concept of velocity is by definition the movement divided by the interval of time between initial position and final position.  If $f(t)$ is the position of a particle at time $t$; the velocity in the interval $[t_0;t_1]$ is $\dfrac{f(t_1)-f(t_0)}{t_1-t_0}$  The problem is that in a single instant there is no movement and the time is not changed; so no velocity.  I can consider $\lim_{t_1 \to t_0} \dfrac{f(t_1)-f(t_0)}{t_1-t_0}$, but mathematically it is only the limit of average velocity function and doesn't represent velocity at instant $t_0$  What are your views about this problem ? How to measure a qubit Hamiltonian in qiskit Adding values to a picklist using apex How to install HPLIP on my Ubuntu to support my HP printer and/or scanner? Let's assume that one has some HP printer and/or scanner and checked its support status on the corresponding page of HPLIP site - .  So one have the following information: the device Model name, Min. HPLIP Version and Driver Plug-in need.   What should one install to make Ubuntu fully support the HP printer and/or scanner? I know that $\operatorname{Spec} \mathbb{C}[x]$ can be identified with the set $\mathbb{C}\cup *$, where $*$ is a generic point via the correspondence  $$ \prod_{i}(x-a_i) \leftrightarrow \{a_i\}_i , \ \ \ \ (0)\leftrightarrow *. $$ This correspondence holds for any $\operatorname{Spec} k[x]$ with algebraic closed field $k$.   When $k$ is not algebraic closed, how should one understand $\operatorname{Spec} k[x]$? Add Appendix with correct page to ToC Why does the weighing balance restore when tilted and released I'm talking about a Weighing Balance shown in the figure:    Press &amp; Hold on onside of the horizontal beam and then release it. It makes some oscillations and comes back to equilibrium like shown in the figure.  Both the pans are of equal equal masses. When the horizontal beam is tilted by an angle using external force, the torque due to these pan weights are equal in magnitude &amp; opposite in direction. Then why does it come back to it position? What's making it to come back? $\mathbb{F}_p[X]/(X^2+X+1)$ is a field iff $p \equiv 2 \bmod 3$ Let $p$ be a prime. Prove that $\mathbb{F}_p[X]/(X^2+X+1)$ is a field iff $p \equiv 2\bmod3$.    So:   If $p \equiv 2 $ mod $ 3$, I have to show that every element of $\mathbb{F}_p[X]/(X^2+X+1)$ has an inverse.  If $\mathbb{F}_p[X]/(X^2+X+1)$ is a field I have to show $p \equiv 2 $ mod $ 3$. How to remove Firefox's "inspect element" button? This is driving me crazy! When FF 12 came out, Firefox's own 'Inspect Element' button on the right-click menu took the place of there the Firebug 'Inspect Element' button used to be.  As I've been doing web development in Firefox forever, my muscle memory still clicks there when I want to use Firebug.    Is there any way to remove the Firefox internal 'Inspect Element' and replace the button on the menu with Firebug's, returning it to its original position (marked NO in the screenshot)? How do I Insert a carriage return on a card in Trello? braces on the sides of an easybmat matrix I created the following matrix with , and I would like to add at the bottom (respectively right) something like an underbrace (resp. "rightbrace") under each of the two blocks, together with a number specifying the number of columns (resp. rows) of the block. This amounts to two braces on the bottom and two on the right, and each brace should have a number below (resp. at the right) of it.  I did not find anything about that on the easybmat manual. I think I see how to do the braces with the  package, but then I am not able to create the block structure of the matrix in the first place.  \documentclass{article} \usepackage{easybmat} \begin{document} \[  M(z)= \left[ \begin{BMAT}[5pt]{c|c}{c} \begin{BMAT}[5pt]{c:c:c}{ccccccc} &amp; &amp; \\ &amp; &amp; \\ &amp; &amp; \\ v_1 &amp; \dots &amp; v_m \\ &amp; &amp; \\ &amp; &amp; \\ &amp; &amp; \end{BMAT} &amp; \begin{BMAT}{c}{c|c} \begin{BMAT}[5pt]{c:c:c}{ccc} &amp; &amp; \\ w_1(a) &amp; \dots &amp; w_n(a) \\ &amp; &amp; \end{BMAT} \\ \begin{BMAT}[10pt]{c}{c} B(z) \end{BMAT} \end{BMAT} \end{BMAT}  \right] \] \end{document}     This is more or less what i wish: Imported 3D scan displays texture in editor, but renders without texture I am new to blender and I have a 3D scan of a stone tablet (which I am unfortunately not able to share, so I will try to be as detailed as possible) which was imported as OBJ and MTL files.  The editor displays the texture and mesh perfectly, but when I render the scene (pressing F12) the texture is not there.  I am trying to use blender to reduce the file size of the tablet for use in a three.js visualizer for a website.  I have tried exporting the file as a JSON file (as well as several other file formats to the same effect) for use in my three.js webapp and thought it was an issue with three.js, but after seeing that blender did not render the tablet with the texture I now believe it is an issue with my blender configuration. I am unable to find within blender or online any way to check what current resources I have displayed and what options I have with them.  I have not applied a mesh or texture to the object, but when I do so it renders with the generated texture applied in the default pink.  I am confused as to where the disconnect is between the editor and the renderer. I am still figuring out what is important and where to find all the relevant information, so any help on that regard would be very much appreciated.  Here is what blender looks like for me with the sensitive parts censored (can't be too careful these days):   I have looked at this question:  And there is no material overwriting the texture - I have not assigned a material to the tablet at all. EDIT: In regards to the duplicate answer, I looked at that one as well and the issue I have is that it doesn't have an actual texture file that I can just put on it.  The OBJ MTL files come with their own file that can't be applied directly to the model.  Blender read the OBJ MTL files and recreated the object inside blender algorithmically - and now I need to know how to get blender to tell me where that texture is and how I can get it to show up in the render.  The texture is not on my hard drive, but entirely within blender which is the issue I am having. Now that Unity is gone, the usual way of setting a Compose Key (via Keyboard Preferences > Layout) no longer works (because in Ubuntu 18.04, Devices > Keyboard has no Layout tab).  How do we set a Compose Key in Ubuntu 18.04 Bionic Beaver?