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This paper proposes trajectory planning strategies for reconfiguration of a multi-agent formation on a Lissajous curve. In our earlier work (Borkar et al., 2016), a multi-agent formation with constant parametric speed was proposed in order to address multiple objectives such as repeated collision-free surveillance and guaranteed sensor coverage of the area with ability for rogue target detection and trapping. Here we address formation reconfiguration within this framework. Smooth parametric trajectories are designed for this purpose using calculus of variations. These are employed in conjunction with a local cooperation scheme to achieve collision-free reconfiguration between different Lissajous curves. A detailed theoretical analysis is provided. The claims are validated through simulations in MATLAB^Ⓡfor agents performing parametric motion along the curves, by Software-In-The-Loop simulation for quadrotors, and …