The Steering Law has long been a fundamental model in predicting movement time for tasks involving navigating through constrained paths, such as in selecting sub-menu options, particularly for straight and circular arc trajectories. However, this does not reflect the complexities of real-world tasks where curvatures can vary arbitrarily, limiting its applications. This study aims to address this gap by introducing the total curvature parameter K into the equation to account for the overall curviness characteristic of a path. To validate this extension, we conducted a mouse-steering experiment on fixed-width paths with varying lengths and curviness levels. Our results demonstrate that the introduction of K significantly improves model fitness for movement time prediction over traditional models. These findings advance our understanding of movement in complex environments and support potential applications in fields like speech motor control and virtual navigation.