How to interpolate in 3-D non-euclidean space?

Assume, one has a 3-D non-euclidean space of points $p_i = \left(x_i, y_i, z_i\right) \in \mathcal{R}^2 \times \mathcal{R}_{> 0}$ with the following "distance" function $d\left(p_1, p_2\right) = \log \left( \frac{\frac{z_1}{z_2} + \frac{z_2}{z_1}}{2} \right) + \frac{(x_2 - x_1)^2 + (y_2 - y_1)^2}{2(z_1^2 + z_2^2)}$ that does not satisfy the triangle law. So step $\Delta r$ in $xy$-plane is not interchangable with step $\Delta z = \Delta r$ in z-direction.

Further, some table $T$ of points like $(p_i, g(p_i))$, where $g$ is a parabolic function (a function that is at most quadratic in the $x,y,z$ coordinates), is given. How can I find a maximum of $g$ given table $T$ with respect to "distance" function $d$?

Thank you!

• By 'parabolic function' do you mean a function that is at most quadratic in the $xyz$ coordinates? – Robert Bryant Nov 19 '12 at 15:44
• Yes, I do. I put a corresponding note to the question body. – spk Nov 19 '12 at 16:58
• I know what the 'maximum of $g$' would mean, but I don't know what 'maximum of $g$ given Table $T$ with respect to "distance" function $d$" means. Why is the distance function needed to figure out the maximum of $g$? If you have enough points in the table $T$, you can figure out the coefficients of $g$ as a quadratic polynomial in $x$, $y$, and $z$ and then you can compute the maximum in the usual way. How is your distance function relevant? – Robert Bryant Nov 19 '12 at 17:41
• Thank you for your interest. I mean the following. Function $d$ captures the fact that step $\delta r$ in xy-plane and step $\delta z$ in z-direction are not interchangeable, equivalent. So I suffer from that. I do not know how to process further. How should I figure out the coefficients of $g$ in this case? – spk Nov 19 '12 at 18:17
• @spk: I do not understand your answer to Robert Bryant. As he said, the max of $g$ does not depend on any extra structure such as $d$. Given the title of the question I think your question in fact lacks some context (e.g. do you want to interpolate "naturally" outside the points tackled by $T$ and search the max of the extended function?) – Benoît Kloeckner Nov 19 '12 at 20:26