(untitled) (European Journal of Pure and Applied Mathematics, 2025)
This study investigates the theoretical basis of the quasi-frame in three-dimensional Galilean geometry. We derive mathematical expressions for the position vectors of curves defined in relation to this quasi-frame and establish the quasi equations that govern their behavior. Our findings demonstrate the absence of normal curves in Galilean 3-space, challenging existing theories in the field and providing new insights into the geometric structure of the Galilean 3-space. We explore the geometric properties of quasi-rectifying and quasi-osculating curves, establishing the necessary and sufficient conditions for their classification. A curve is identified as quasi-rectifying if its position vector can be represented as a linear combination of its tangent and quasi-binormal vectors. In contrast, a curve is classified as quasi-osculating if it remains entirely within its quasi-osculating plane, determined by its tangent and quasi-normal vectors. The quasi-frame serves as a generalization of the classical Frenet frame, particularly useful in scenarios where the curvature vanishes and the Frenet frame becomes undefined. By introducing the quasi curvatures, we provide a robust framework for analyzing curves in Galilean 3-space. We derive explicit expressions for the position vectors of curves with respect to the Quasi frame and solve for their components under specific conditions. Furthermore, we prove that normal curves cannot exist in Galilean space, a result that clarifies the limitations of certain geometric classifications in this context.