My First question on here so be kind...  I am writing an aerodrome simulator in as2, I need to calculate a turn rate between 2 points so that the aircraft turns nicely. I have tried a few things but the maths forums on this type of subject use complex calculations and I struggle to understand them. I need an answer in a more variable specific format.  I have the XY of both tile and target, the speed of tile and the distance of target from the tile.    I use vector math, with frame time calculations to move the tile. I already have a good calculation for shortest turn i.e. left right.  I need to calculate the turn rate required to go from point a to point b in a nice fluid motion and turn the tile/aircraft in a nice fashion.  Any help would be much appreciated.
I have a sprite that moves from waypoint to waypoint. It turns to face its velocity vector, but not instantly (there's a rotation speed).  The problem I'm having is, from one direction to another, the sprite will often turn in wild, unnecessary ways instead of appearing like you might expect a car to appear (turn such that the minimum delta in angle occurs).  Right now I'm using the following formula to calculate the rotation float for the current velocity:  MathHelper.PiOver2 + -Math.Atan2(anotherVector.Y - vector.Y, -(anotherVector.X - vector.X));   And then I lerp the current rotation by the new rotation by the rotation speed (10f):  rotation = MathHelper.Lerp(rotation, targetRotation, 10f * (float)gameTime.ElapsedGameTime.TotalSeconds);   But unfortunately, it's causing the sprite to spin in wild directions before reaching the correct angle. Any suggestions?
Prove that the $\lim_{(x,y) \to (0,0)}h(x,y)$ Does not Exist using Polar Coordinates Let $h(x,y)=\frac{x^5y}{2x^{10}+y^2}$. How would I prove that the $\lim_{(x,y) \to (0,0)}h(x,y)$ Does Not Exist?  I think that we might use polar coordinates, but I am not sure. How to Expand , Resize "extended partition"? Covariance of binary and continuous variable Suppose $y$ is a continuous random variable and $d$ is a binary random variable that takes the value $1$ with probability $p$ and $0$ with probability $1-p$.   How do I show that $\text{Cov}(y,d)=(E[y|d=1]-E[y|d=0])p(1-p)$? Energy conservation and interference I have a problem with energy conservation in case of interfering waves.  Imagine two harmonic waves with amplitudes $A$. They both carry energy that is proportional to $A^2$, so the total energy is proportional to $2A^2$. When they interfere, the amplitude raises to $2A$, so energy is now proportional to $4A^2$ and bigger than before.  The equivalent question is what happens to the energy with the superposition of two waves that interfere destructively.  Also, if someone could comment on the statement about this problem in my physics book (Bykow, Butikow, Kondratiew):      The sources of the waves work with increased power during the interference because they feel the wave from the other source. "need to do" vs "need do" Consider:  I need to do this. I need do this.  My English grammar knowledge tells me that &quot;need&quot; doesn't have the same status as the modal verbs &quot;may&quot;, &quot;can&quot;, &quot;should&quot; and what not. Hence the second usage where two verbs appear consecutively is incorrect. But yesterday, my native English speaking friends (Americans) told me that &quot;I need compute this.&quot; is a perfectly grammatical sentence, and one is simply omitting the &quot;to&quot;. How can this be? Is it a colloquial usage but grammatically incorrect, or is it grammatically correct? If it is grammatically correct, is it because &quot;need&quot; is a semi-modal verb? EDIT: in particular, is it okay to use &quot;need compute&quot; in a scientific paper? EDIT2: the exact phrase that raised the question was &quot;The advantage of this representation is that we need only compute sums and products&quot; Why are my sauces splitting? Recently all my attempts at sauces are splitting and I'm not sure what's going wrong. Tonight I tried a balsamic vinegar reduction / chicken stock / pan juices recipe to go with my lamb chops, but once again the dreaded split occurred. My wife is patient and forgiving, but I feel like I must be missing something fundamental to get this wrong so often.  Two questions.   What could I be doing wrong to cause these sauces to split? What can I do to recover when it happens again?   UPDATE: The Pan juices contained a tbsp or two of olive oil, and some of the fresh herb rub I'd coated the lamb chops in (thyme, rosemary, and mint). I did not drain the pan before starting the sauce. It has crossed my mind that perhaps there was too much oil / fats in the pan at that point?  For completeness, I should point out that I also sauteed some chopped shallots in the pan juices immediately before adding the vinegar. How to protect website from bulk scraping /downloading? Use a rectangular contour to evaluate the integral $$\int_{-\infty}^{\infty} \frac{\cos(mx) dx}{e^{-x}+e^x} = \frac{\pi}{e^{m\pi /2}+e^{-m\pi /2}}$$  I need to evaluate the above integral specifically using a rectangluar contour and at some point applying the residue theorem. I should be ok with the residue part, but I'm confused at to how to get this into the example form shown in the book:  $$I =\int_{-\infty}^{\infty} \frac{e^{ax}dx}{1+e^x}=\frac{\pi}{\sin(a\pi)} $$  I tried using the identity $\cos (mx) = \frac{e^{ix}+e^{-ix}}{2}$ but I end up with $\frac{1}{2} \int_{-\infty}^{\infty}\frac{e^{ix}(e^{imx}+e^{-imx})dx}{1+e^{2ix}}$, which almost resembles the example but not quite... Alternatively, why does the force created by blowing out air feel so much stronger than the force created by sucking in air?  Ok, so forget the human factor involved in blowing out candles. Consider a vacuum cleaner with a suction end and a blower end. Anyone who has tried it out notices the blower end creates a much stronger force than the suction end, despite the discharge being (more or less) equal at both ends.  Why does this happen? for (i=0 ; i&lt;=10; i++) {     ..     .. }  i=0; while(i&lt;=10) {     ..     ..     i++; }   In for and while loop, which one is better, performance wise? This works perfectly:  $ inotifywait --event create ~/foo Setting up watches. Watches established. /home/ron/foo/ CREATE bar   However, this just sits there when directory tun0 is created under /sys/devices/virtual/net.  $ inotifywait --event create /sys/devices/virtual/net Setting up watches. Watches established.   Since those folders are world readable, I'd expect inotifywait to work.  So, what am I doing wrong?  Thanks How to overwrite only part of a file in c++ sequence $x_{n+1}=x_n+e^{-2018x_n}$ In a tic-tac-toe implementation I guess that the challenging part is to determine the best move to be played by the machine.  What are the algorithms that can pursued? I'm looking into implementations from simple to complex. How would I go about tackling this part of the problem? Batch files: How to read a file? About $\lim \left(1+\frac {x}{n}\right)^n$ A correct preposition - account closed due/by/at 12th of May Where [if anywhere] should the apostrophe go in "three days work"? Which is correct?     1 Three day's work      2 Three days' work      3 Three days work   I would probably guess (2) is right, since the work belongs to the three days ("three days of work"). But I'm not sure. Would it be different if it was only one day? Word for sentences that make sense when read backwards? Is there a word for sentences that make sense when read normally or backwards (not necessarily by reversing the words)? Example:     "Are you as bored as I am?" How to make different materials for different objects? When I change one material, it changes it on another object. Why and how do I fix it? is there some link between the objects?    Here is my blender file of Iron Man's faceplate.  Here is my whole screen with the main portion of the face selected. On the right, you can see the material, and I can tell you that that is the same as the eyes part: