We define and study a family of generalized non-intersection exponents for planar Brownian motions that is indexed by subsets of the complex plane: For each $A\subset\CC$, we define an exponent $\xi(A)$ that describes the decay of certain non-intersection probabilities. To each of these exponents, we associate a conformally invariant subset of the planar Brownian path, of Hausdorff dimension $2-\xi(A)$. A consequence of this and continuity of $\xi(A)$ as a function of $A$ is the almost sure existence of pivoting points of any sufficiently small angle on a planar Brownian path.