DARBOUX FRAME OF VIVIANI’S CURVE

Year 2025, Volume: 26 Issue: 4, 416 - 425, 25.12.2025

Emre Öztürk

https://doi.org/10.18038/estubtda.1729355

Abstract

This study investigates the differential geometric properties of Viviani's curve using the Darboux frame apparatus. Viviani's curve, a classical space curve arising from the intersection of specific surfaces, is examined from two distinct geometric perspectives: first as the intersection of a sphere and a circular cylinder, and second as the intersection of a circular cone and a parabolic cylinder. For each representation, the Darboux frame field consisting of the tangent vector, surface normal, and their cross product is explicitly constructed. The geodesic curvature, normal curvature, and geodesic torsion are derived and analyzed in detail. It is proven that Viviani's curve becomes a geodesic on the circular cylinder at specific parameter values (s=2kπ,k∈Z), while on the circular cone, the curve exhibits asymptotic behavior at s=kπ/2 and principal curve characteristics at s=kπ. The relationship between Darboux curvatures and the Frenet curvature is established, providing an alternative computational approach to classical Frenet-Serret formulas. Several illustrative examples demonstrate the Frenet and Darboux frames at specific points on the curve, revealing geometric insights about frame coincidence and orthogonality properties. Additionally, a double helix-like structure is constructed using two Viviani curves. This work contributes to the geometric understanding of Viviani's curve through the lens of surface-curve interaction theory and extends the theoretical framework for analyzing curves lying on classical surfaces.

Keywords

Viviani’s curve , Darboux Frame , Cone

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