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Computationally efficient tire models are needed to meet the timing and accuracy demands of the iterative vehicle design process. Axisymmetric, circumferentially isotropic, planar, discretized models defined by their quasi-static constraint modes have been proposed that are parameterized by a single stiffness parameter and two shape parameters. These models predict the deformed shape independently from the overall tire stiffness and the forces acting on the tire, but the parameterization of these models is not well defined. This work develops an admissible domain of the shape parameters based on the deformation limitations of a physical tire, such that the tire stiffness properties cannot be negative, the deformed shape of the tire under quasi-static loading cannot be dominated by a single harmonic, and the low spatial frequency components must contribute more than higher frequency components to the overall tire shape.