Laguerre's differential geometry and kinematics

summary:In this paper the plane Laguerre's geometry in the augmented plane of dual numbers is presented. Basic integral and differential invariants of $\cal L$-curves in the plane are deduced, i.e. the $\cal L$-curve arc, $\cal L$-curvature, $\cal L$-minimal curves, $\cal L$-circle. Furthermore the contact of $\cal L$-curves, $\cal L$-osculating circle, $\cal L$-evolute of a curve and some special $\cal L$-motions are studied from the point of view of $\cal L$-Differential geometry