  Janus particles with the ability to move phoretically in self-generated
chemical concentration gradients are model systems for active matter. Their
motion typically consists of straight paths with rotational diffusion being the
dominant reorientation mechanism. In this paper, we show theoretically that by
a suitable surface coverage of both activity and mobility, translational and
rotational motion can be induced arbitrarily in three dimensions. The resulting
trajectories are in general helical, and their pitch and radius can be
controlled by adjusting the angle between the translational and angular
velocity. Building on the classical mathematical framework for axisymmetric
self-phoretic motion under fixed-flux chemical boundary condition, we first
show how to calculate the most general three-dimensional motion for an
arbitrary surface coverage of a spherical particle. After illustrating our
results on surface distributions, we next introduce a simple intuitive patch
model to serve as a guide for designing arbitrary phoretic spheres.
