Simple, low-cost, permanent magnet geometries are frequently used for field generation in cases where a constant high strength magnetic field is required. This is the situation across many emerging quantum platforms, primarily for bias fields in quantum computing or nitrogen vacancy centres. Finding optimal arrangements of primitive magnet geometries such as these can be difficult, often escalating in complexity as further higher order field harmonics are considered. This is further complicated by the need to adhere to strict system constraints, for instance those implied by geometric or optical access requirements. Here we illustrate a method which encodes loop and arc magnet geometries into an analytical form, attaining solutions for the field harmonics produced by both axial and radial magnetisations for each form. These harmonic descriptions then form the basis of a numerical design procedure. Our approach facilitates solutions which improve fidelity of the target field, by reducing error harmonic contributions, while satisfying constraints on magnet position. This allows for considerations, such as optical access or maximum system extents, to be accounted for. To demonstrate this optimisation procedure we produce four system designs, two axially magnetised loops to produce a uniform axial field, and two radially magnetised arcs, which respectively produce a radial uniform and radial gradient field. Their performance is then compared against standard designs, obtained from analytical solutions alone. In each case, when using our approach, the volume of field uniformity within 99% agreement of the target field increases by a minimum factor of 1.37. Designs generated by this procedure therefore enable improved field fidelity and reduce magnetic field driven errors.