The following link has the Paper: http://www.wisdom.weizmann.ac.il/~vision/courses/2010_2/papers/OnGraphsAndRigidity.pdf
In the definition 2.2 Laman defined the plane skeletal structure of a graph consisting of a map $\chi$ that takes the vertices of a graph to some points on euclidean plane.
next in definition 2.3 he is defining "Length Preserving Displacement" which consist of
a) A segment $[\beta , \gamma]$ of real numbers with 0.
b) for every $a \in $Vertex Set and for every $\tau \in [\beta,\gamma]$ a point $\chi_\tau(a) \in E^2$ satisfying the following conditions:
i)$\chi_0(a) = \chi(a), \forall a \in $Vertex Set.;
ii)$\chi_\tau(a)$ is differentiable for every $a$;
iii)$(\Gamma,\chi_\tau)$ is a Plane Skeletal Structure for every $\tau$;
iv) $|\chi_\tau(a) - \chi_\tau(b)| = |\chi(a)-\chi(b)| , \forall (a,b) \in $ Edge Set
So we can interpret it physically that images of all vertices on the euclidean plane is moving from without changing the length between the adjacent vertices, That is for every $\tau$ we are getting a Plane Skeletal Structure.
Now Laman defines in 2.4 that an "Infinitesimal Displacement" of a Plane Skeletal Structure is a map $\mu$ from Vertex set to $\mathbb R^2$.
So my questions are
1) as each $\mu(a)$ is a vector what it is signifying physically?
is it the velocity due to the motion of plane skeletal structure?
2) in the section 3 of this paper there are several example of non rigid skeletal structure.
for at least one (say) 3.1 can you please explain why the non rigid skeletal structure doesn't have any trivial infinietesimal displacement?
1) Yes, it is velocity.
2) Every skeletal structure has trivial infinitesimal displacement. However, non-rigid structures have other displacements. You can think of non-trivial displacements as those: if you fix two vertices of skeletal structure, can some other vertex move? For non-degenerate triangle the obvious answer is no. However, if three vertex of triangle lie on the same line, if you fix two vertices, the infinitesimal displacement of a third vertex in the perpendicular direction doesn't change any distances. So this skeletal structure has one non-trivial infinitesimal displacement conserving distances.