This paper presents a mathematical model for kernel-based projection pursuit regression, allowing explicit formulas for bias and variance of estimators. It shows that the bias of an orientation estimate dominates error about the mean, with the latter being asymptotically negligible. However, bias and error about the mean are of the same order in projection pursuit curve estimates. The paper discusses implications of these results for projection pursuit regression. The model is constructed to analyze how an estimated projection tracks a theoretical projection. The main results include explicit formulas for bias and error about the mean in orientation and curve estimates. It is shown that kernel-based projection pursuit regression achieves convergence rates identical to one-dimensional problems, though higher smoothness is required. The paper also discusses a two-stage procedure for improving convergence rates of orientation estimators, though it does not alter the convergence rate of curve estimates. The paper introduces a tractable mathematical model for projection pursuit regression, develops essential calculus for projective approximation, and analyzes the model to understand how an estimated projection tracks a theoretical projection. It provides explicit formulas for bias and error about the mean in orientation and curve estimates, showing that orientation estimates have most of their error in the form of bias. The paper also discusses the use of kernel-based projection pursuit regression, showing that it can achieve convergence rates similar to one-dimensional problems, though higher smoothness is required. It introduces a two-stage procedure for improving convergence rates of orientation estimators, though it does not alter the convergence rate of curve estimates. The paper concludes by discussing the implications of these results for projection pursuit regression.This paper presents a mathematical model for kernel-based projection pursuit regression, allowing explicit formulas for bias and variance of estimators. It shows that the bias of an orientation estimate dominates error about the mean, with the latter being asymptotically negligible. However, bias and error about the mean are of the same order in projection pursuit curve estimates. The paper discusses implications of these results for projection pursuit regression. The model is constructed to analyze how an estimated projection tracks a theoretical projection. The main results include explicit formulas for bias and error about the mean in orientation and curve estimates. It is shown that kernel-based projection pursuit regression achieves convergence rates identical to one-dimensional problems, though higher smoothness is required. The paper also discusses a two-stage procedure for improving convergence rates of orientation estimators, though it does not alter the convergence rate of curve estimates. The paper introduces a tractable mathematical model for projection pursuit regression, develops essential calculus for projective approximation, and analyzes the model to understand how an estimated projection tracks a theoretical projection. It provides explicit formulas for bias and error about the mean in orientation and curve estimates, showing that orientation estimates have most of their error in the form of bias. The paper also discusses the use of kernel-based projection pursuit regression, showing that it can achieve convergence rates similar to one-dimensional problems, though higher smoothness is required. It introduces a two-stage procedure for improving convergence rates of orientation estimators, though it does not alter the convergence rate of curve estimates. The paper concludes by discussing the implications of these results for projection pursuit regression.