We introduce and study a random non-compact space called the bigeodesic Brownian plane, and prove that it is the tangent plane in distribution of the Brownian sphere at a point of its simple geodesic from the root (for the local Gromov-Hausdorff-Prokhorov-Uniform topology). We also show that it is the local limit of the Brownian plane rerooted further and further on its unique infinite geodesic. Furthermore, we discuss various properties of this space, such as its topology, the behavior of its geodesic rays, and its invariance in distribution under several natural transformations.