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The tire-force ellipse and tire-force circle (more frequently referred to as the friction ellipse and the friction circle, respectively) have been used for many years to qualitatively illustrate the concept of tire-road force interaction, particularly the force-limiting behavior for combined braking and steering (combined tire forces). Equations of the tire-force circle/ellipse, or, more specifically, the force limit envelope, in its idealized form have also been used in the development of quantitative models of combined tire forces used in vehicle dynamic simulation software. Comparisons of this idealized tire-force circle/ellipse using a simple bilinear tire force model and using actual tire data show that it provides only a limited, simplified notion of combined tire forces due to its lack of dependence upon the slip angle and traction slip.

The force distributed over the contact patch between a tire and a road surface is typically modeled in component form for dynamic simulations. The two components in the plane of the contact patch are the braking, or traction force, and the steering, or side or cornering force. A third force distributed over the contacts patch is the normal force, perpendicular to the road surface. The two tangential components in the plane of the road are usually modeled separately since they depend primarily on independent parameters, wheel slip and sideslip. Mathematical expressions found in the literature for each component include exponential functions, piecewise linear functions and the Bakker-Nyborg-Pacejka equations, among others. Because braking and steering frequently occur simultaneously and their resultant tangential force is limited by friction, the two components must be properly combined for a full range of the wheel slip and sideslip parameters.