Following the analogy of higher order point-path generation, new concepts for tangent-line higher order envelope curvature theory are introduced. Mathematical relationships are derived to define, using the instantaneous invariants and the stretch rotation concepts, the characteristic numbers λ1 and λ2 for third and fourth-order contacts at the tangency point of a tangent-line generating an envelope. The newly developed tangent-line higher order envelope curvature theory is applied to demonstrate its application in synthesis of mechanisms. Examples involving circular, elliptic and involute are generation using a mechanism are presented.

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