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I can find slices of a surface for any angle through the vertical axis, but I am not certain what the equation of the surface is.

Equation along x-axis: $z=1/(1+x^2)$

Equation along y-axis:$z=1/(1-y^2)$

Equation at 45 degrees: $z=1/(t^4+1)$, where t is the line in the x-y plane at 45 degrees.

Equation at 30 degrees: $z=1/((9/4*t^2)+1)$, where t is the line in the x-y plane at 30 degrees.

Feel free to ask for the equation of a slice through any angle. What is the equation of the surface?

enter image description here

User3910
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  • In the 45 degree version, what is $t$? Is it the distance along that line from the origin? Same question for the 30 degree case. – coffeemath Jun 19 '18 at 21:38
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    @coffeemath, yes – User3910 Jun 19 '18 at 21:48
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    Can you give a rule or algorithm by which we can figure out the equation for any slice without having to ask you for each angle? – David K Jun 22 '18 at 00:55
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    Is this related to https://math.stackexchange.com/questions/463174/how-to-slice-re1-1z-into-a-cartesian-function-for-any-angle? The equations for the $x$-axis, $y$-axis, and $45$ degrees match, but the $30$-degree equation seems different, which is all the more reason to need to know how that equation was obtained. – David K Jun 22 '18 at 12:18
  • @Dale Is the graphic made on any CAS for the actual surface? What is represented by the green and rotating vectors/arrows? a variable plane intersection around the normal as axis? – Narasimham Jun 28 '18 at 05:34
  • The surface passing through these 4 curves is by no means unique. You need to give more information about it. – Oskar Limka Jun 28 '18 at 19:57
  • @OskarLimka How many sliced Need I provide? – User3910 Jun 29 '18 at 02:19
  • @Dale the short answer is infinitely many (which is not very helpful, I guess). The longerish answer is to think of a two dimensional analogue: take $n+1$ points in the $(x_i,y_i)$, $i=0,\dotsc,n$ with $x_i=x_j\Leftrightarrow i=j$, there are infinitely many functions satisfying $f(x_i)=y_i$ for all $i=0,\dotsc,n$ but a unique one if $f$ is required to be a polynomial of degree $n$ or less. This example's "polynomial of degree $\leq n$" is what I mean by "more information". Other classes of functions are rational functions. – Oskar Limka Jun 29 '18 at 08:23
  • If you are interested in one function of many (maybe infinitely so) possible, then I'd look for rational functions of two variables. Roughly speaking rational functions of degree $(m,n)$ form a $p(m)+p(n)$ dimensional manifold where $p(k)$ is the number of linearly independent polynomials of degree $k$ in $\mathbb R^2$. From the examples you've given, I would look for $m=0$ and $n=4$. The interpolation problem could then be reduced to that of a polynomial of $2$ variables and total degree $n$, for which a finite number of (suitably chosen) points suffices. – Oskar Limka Jun 29 '18 at 08:30
  • In my comment above I was thinking of your unknown $f(x,y)$ to be of the form $S(x,y)/P(x,y)$ where $S$ and $P$ are polynomials of $\operatorname{deg} S=m$ and $\operatorname{deg}P=n$. – Oskar Limka Jun 29 '18 at 08:32
  • If $S\equiv c\in\mathbb R$, then your problem becomes of finding a polynomial $P$ coinciding with certain one-dimensional polynomials along the "spokes" that you've described. Since the maximal degree there was $4$, I think degree $4$ is more than enough for starters, possibly $P(x,y)=a+bx^2+cy^2+dx^2y^2$ could be enough... – Oskar Limka Jun 29 '18 at 08:37
  • @OskarLimka Perhapse we could write a lisp function to sort through the surfaces? – User3910 Jun 29 '18 at 19:28
  • Usually folks try to derive a linear system for the coefficients $a, b, c, d$ in this case by imposing the conditions. – Oskar Limka Jul 16 '18 at 08:53

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