I am seeking theory that would help answer these questions.
How many pairs of curve points and tangents at those points (point-tangent pairs) determine an ellipse? How many point-tangent pairs determine a super ellipse of degree $d$? How many point-tangent pairs determine a closed convex curve $C$ defined as the zero-set of a specific polynomial of degree $d$?
For example, If $C$ is a circle $x^2+y^2=r^2$, then one point $p$ and tangent vector $t$ at $p$ do not suffice, because $C$ could lie on either side of that tangent at $p$. But two (distinct) point-tangents suffice to pinpoint the circle's location.
Super ellipse $x^4/2 + y^4 = 1$, rotated.
In a sense, I am seeking variants of the theorem that $5$ points determine a conic section, but in my case, I have point-tangent pairs. I am expecting that the number of point-tangent pairs needed increases with $d$.
Hint: Just like in the case of conic sections, take $x^n+y^n=r^n$, and apply a $2D$ rotation and a $2D$ translation to it, so as to obtain the most general formula possible. How many terms or coefficients does it possess ? In the case of quadratics, we had $$Ax^2+Bxy+Cy^2+Dx+Ey=F,$$ implying $5$ unknowns for any given F. Here, the role of F is played by $r^n$.