Sidewinding is an efficient translational gait used by biological snakes to locomote over flat ground. Prior work has identified the fact that it is possible to steer the moving direction of sidewinding. The previously proposed virtual tread model reveals the working principal of sidewinding from a geometric point of view. Unfortunately, the implementation of the virtual tread model relied on a computationally expensive numerical fitting algorithm that impeded online applications. Motivated by this limitation, in this work we propose a novel approach to develop analytical expressions for snake robot gaits based on the study of the corresponding geometric model. This approach is rooted in the identification of dominant frequency components afforded by the two-dimensional Fast Fourier Transformation (FFT). Applying this method to the virtual tread model for conical sidewinding, we derive an analytical expression between the parameters that describe the gait's motion and the turning radius of the system moving in the world. This analytical expression, which we call the extended gait equation, is verified by experimental results.