Extended Camus Theory and Higher Order Conjugated Curves

According to Camus’ theorem, for a single degree-of-freedom (DOF) three-body system with the three instant centers staying coincident, a point embedded on a body traces a pair of conjugated curves on the other two bodies. This paper discusses a fundamental issue not addressed in Camus’ theorem in the context of higher order curvature theory. Following the Aronhold–Kennedy theorem, in a single degree-of-freedom three-body system, the three instant centers must lie on a straight line. This paper proposes that if the line of the three instant centers is stationary (i.e., slide along itself) on the line of the instant centers, a point embedded on a body traces a pair of conjugated curves on the other two bodies. Another case is that if the line of the three instant centers rotates about a stationary point, the stationary point embedded on a body also traces a pair of conjugated curves on the other two bodies. The paper demonstrates the use of instantaneous invariants to synthesize such a three-body system leading to a conjugate curve-pair generation. It is a supplement or extension of Camus’ theorem. Camus’ theorem may be regarded as a special singular case, in which all three instant centers are coincident.