# Localizing a plane in 3-D using distance geometry

Assume that I have a set of coplanar points $P = \{p_1, p_2, ... , p_n\}$ The equation of the plane is unknown. $\forall p_i,p_j \in P$, pairwise euclidian distance $d(p_ip_j)$ is known.

And I have a set of points of whose coordinates are known $Q = \{q_1, q_2, ..., q_m\}$

If I know some of the pairwise distances $d(pq)$ $p \in P$, $q \in Q$;

At least how many $d(pq)$ distance measurements do I need to find the equation of the plane that points in $P$ lie on?

• Can you give more information about the distance function d? How do you define it? Is it Euclidean distance? I think you need to define it to solve the question. And, can you clarify what do you mean by connections? – Sayan Bandyapadhyay Apr 30 '14 at 22:08
• @SayanBandyapadhyay I have edited the question. Thanks for your notice. – padawan Apr 30 '14 at 22:18
• It's still not clear. Did you mean how many (p,q) distances we need to know to find the equation of the plane? – Sayan Bandyapadhyay Apr 30 '14 at 22:27
• Assuming general position, the plane is the affine span of $p_1$, $p_2$, $p_3$. So how many measurements do you need to work out the coordinates of these points? – Louis May 1 '14 at 17:59