Contracting de Sitter and anti-de Sitter spaces into a Poincaré space and interpolating Poincaré generators between the instant form and the light-front form

De Sitter and anti-de Sitter spaces are the maximally symmetric vacuum solutions of Einstein's field equation with positive and negative cosmological constants, respectively. Considering the contraction of groups and algebras 'a la Inonu and Wigner in 1953 as well as the fact that the geometry of the spacetime is deeply connected with the corresponding groups and algebras, we present the contraction of de Sitter and anti-de Sitter groups (SO(4,1) and SO(3,2)) into Poincaré group (ISO(3,1)) in the limit of cosmological constants of their spaces go to zero making the curvature of the spaces vanish. From this result, we may understand the Minkowski space as the tangential space of the de-Sitter and anti-de Sitter spaces. For the illustration, we show our calculation in the usual vector representation as well as in the matrix representation using the operators and gamma matrices from the spinor representation. To further understand the Poincaré group more extensively, we interpolate the Poincaré generators between the instant form dynamics (IFD) and the light-front dynamics (LFD) and discuss their kinematic and dynamic properties presenting the algebra of the interpolating Poincaré group.