In this study, we introduce generalized osculating-type ruled surfaces of special singular curves. We give some theories and results about the geometric structure of the surface. In addition, the singular point classes of the surface are examined, and the conditions for being a cross-cap surface are expressed. Generalized osculating-type ruled surface is considered as a framed surface and its basic invariants are found and some results are given. Finally, we give some examples and figures to support the theories.
In this study, we investigate the Padovan (or Cordonnier) and Perrin generalized quaternions. We obtain the new identities for these special quaternions related to matrix forms. We also introduce Binet-like formulae, generating functions, several summation, and binomial properties concerning these quaternions.
In this study, we define Pauli–Leonardo quaternions by taking the coefficients of the Pauli quaternions as Leonardo numbers. We give the recurrence relation, Binet formula, generating function, exponential generating function, some special equalities, and the sum properties of these novel quaternions. In addition, we investigate the interrelations between Pauli–Leonardo quaternions and the Pauli–Fibonacci, Pauli–Lucas quaternions. Moreover, we create some algorithms that determine the terms of the Pauli–Leonardo quaternions. Finally, we generate the matrix representations of the Pauli–Leonardo quaternions and ℝ-linear transformations.
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