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...Differential geometry revisited bybiquaternioncliffordalgebra patrick r girard clarysse romaric pujol liang wang and philippe delachartre universit e de lyon creatis cnrs umr inserm u insa france b at blaise pascal avenue jean capelle f villeurbanne p ole math ematiques l eonard vinci abstract in the last century dierential has been expressed within various calculi vectors tensors spinors exterior forms recently cliord algebras yield an excellent repre sentation of rotation group lorentz which are cornerstones theory moving frames though all related to quaternions via theorem a biquaternion formulation does not seem have formu lated so far paper develops d euclidean space calculus having associative product applies it dif ferential formalism being new approach is intended be pedagogical since methods similar other dimensions hoped that might open perspectives for hyperbolic presented here can easily implemented algebraically on mathematica numeri cally matlab examples matrix representa...