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A family of rectangular hyperbolas of the form $xy = k$ (e.g. $xy = 2$, $xy = 3$ etc) consists of curves that are neither parallel nor intersect. $k$ is any constant.

Similar families seem to be $xy^2 = k$, $y = e^{(k/x)}$ and so on

$1)$ Is there some other property common to these curves?

$2)$ Can they be regarded as non-Euclidean curves in a plane?

TIWARI
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phil342
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  • Parallelism is defined for lines. For curves what is your definition? For any function $f(x,y)$, the level curves $f(x,y)=k$ for different values of $k$ are disjoint, pairwise. If your definition of parallel means simply no intersection then level curves for any function of two variables will provide a rich collection of examples. There nothing non-eculidean here. – P Vanchinathan Nov 17 '16 at 08:19
  • Are these called level curves? I am thinking of them in connection with economics and wondering whether they have a common property. – phil342 Dec 04 '16 at 07:21

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