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 Mar15 answered Detailed diagram with mathematical fields of study Mar15 comment What are the formulas for topological transformations? How to obtain them? One more thing: There are studies of these formulas, right? I guess that W|A uses some method to convert a drawing into a parametric equation, which seems closely related to what I'm looking for. Do you have any idea of what is this method? Mar15 comment What are the formulas for topological transformations? How to obtain them? Yes. Thanks. I've imagined that the formula would be something very complicated. I remember that Wolfram Alpha is plotting equations for arbitrary drawings such as this. I guessed that there is an algorithm for that and that it would yield a very complicated formula, just as the one described in the given link. But I noticed that there might be a difference in both techniques, in the given link, there is a parametric equation. I guess that for that mapping, it would be something different. Mar13 revised What are the formulas for topological transformations? How to obtain them? added 216 characters in body Mar13 asked What are the formulas for topological transformations? How to obtain them? Mar10 comment What is this space with infinitely many different points with distance $1$ between any two different points? @WillJagy I don't get it. Mar10 accepted Is there a name for this kind of function? Mar10 accepted How to obtain all the rational numbers without repetitions? Mar10 accepted Is it true that there is no algorithm to approximate the least upper bound? Mar10 asked What is this space with infinitely many different points with distance $1$ between any two different points? Feb27 awarded Yearling Feb25 awarded Popular Question Feb23 comment Is it true that there is no algorithm to approximate the least upper bound? But isn't there another way to do such approximation? It baffles me that such algorithm can not exist. But I assume I have little knowledge in mathematics. Feb22 asked Is it true that there is no algorithm to approximate the least upper bound? Feb17 comment Why can't I obtain values of Sine/Cosine with $f(x)=\sqrt{1-x^2}?$ Then I'm confused. If sine is the vertical coordinate and cosine is the horizontal one, why can't I assume that $\sin 45=f(1/2)=\frac{\sqrt{3}}{2}$? Your answer and comments shown me that one needs to use this value in a reverse trig function to obtain the angle and then obtain the sine. Sorry, I'm confused. Feb17 revised When the continuum hypothesis settles the uniqueness (upto isomorphism) of the Hyper-reals doesn't it mean the hypothesis should be an axiom? edited title Feb16 comment Why can't I obtain values of Sine/Cosine with $f(x)=\sqrt{1-x^2}?$ There's just one more thing I still couldn't connect: Why do they say that the sine is the vertical coordinate and the cosine is the horizontal coordinate? I got stuck for a lot of time because I believed this was true. But actually - just as you said - the formulation is a little different. Feb10 comment Why can't I obtain values of Sine/Cosine with $f(x)=\sqrt{1-x^2}?$ Now it makes sense: $\sin \theta= \frac{Opp(\theta)}{Hyp(\theta)}$, the hypotenuse in this case was always 1. Then I was doing $\frac{f(x)}{1}=f(x)$. The idea of having to use the reverse trig function to get the angle is a revolution to me. Thanks. You taught me more trigonometry with the answer and comments than my school in 3 years. Feb9 accepted Why can't I obtain values of Sine/Cosine with $f(x)=\sqrt{1-x^2}?$ Feb8 comment Why can't I obtain values of Sine/Cosine with $f(x)=\sqrt{1-x^2}?$ One curiosity: If this gives the location of the point in the perimeter of the circle, why isn't this used to get values for the Sine and Cosine? In the image I provided, it says that the Sine is the vertical coordinate of the point, and the cosine is the horizontal coordinate. I've found both with my formula, no? I'm confused about what's the difference between them.