The exact joint multifractal distribution for the scaling and spiraling of electrostatic potential lines near any conformally invariant scaling curve is de-rived in two dimensions. Its spectrum f(α,λ) gives the Hausdorff dimension of the points where the potential scales with distance r as H～r~α while the curve spirals logarithmically with a rotation angle l = λlnr. It obeys the scaling law f(α,λ) = (1 + A2)/ (a) - &A2 with α-bar = α/(l + λ~2) and b = (25 - c)/12, and where f(α) = f(α,λ= 0) is the pure harmonic measure spectrum, and c the conformal central charge. The results apply to O(N) and Potts models, as well as to SLEK.
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