Summary: We define a new family of groups which generalize the classical braid groups on \(\mathbb{C}\). We denote this family by \(\{B^m_n\}_{n\geq m+1}\) where \(n,m\in\mathbb{N}\). The family \(\{B^1_n\}_{n\in\mathbb{N}}\) is the set of classical braid groups on \(n\) strings. The group \(B^m_n\) is related to the set of motions of \(n\) unordered points in \(\mathbb{C}^m\), so that at any time during the motion, each \(m+1\) of the points span the whole of \(\mathbb{C}^m\) in the sense of affine geometry. There is a map from \(B^m_n\) to the symmetric group on \(n\) letters. We let \(P^m_n\) denote the kernel of this map. In this paper we are mainly interested in finding a presentation of and understanding the group \(P^2_n\). We give a presentation of a group \(PL_n\) which maps surjectively onto \(P^2_n\). We also show the surjection \(PL_n\to P^2_n\) induces an isomorphism on first and second integral homology and conjecture that it is an isomorphism. We then find an infinitesimal presentation of the group \(P^2_n\). Finally, we also consider the analagous groups where points lie in \(\mathbb{P}^m\) instead of \(\mathbb{C}^m\). These groups generalize the classical braid groups on the sphere.