Joint confidence and likelihood regions for the parameters in nonlinear regression models can be defined using the geometric concepts of sample space and solution locus. Using a quadratic approximation to the solution locus, instead of the usual linear approximation, it is shown that these inference regions correspond to ellipsoids on the tangent plane at the least squares point. Accurate approximate inference regions can be obtained by mapping these ellipsoids into the parameter space, and measures of the effect of intrinsic nonlinearity on inference can be based on the difference between the tangent plane ellipsoids and the sphere which would be obtained using a linear approximation.