Alternatives to the existing axial distances of the Central Composite Design (CCD) in spherical design using three axial distances were studied. The aim of this study is to determine a better alternative to already existing axial distances whose prediction properties are more stable in the spherical design regions. Using the concepts of the three Pythagorean means, the arithmetic, harmonic and geometric axial distances for spherical regions were developed. The performances of the alternative axial distances were compared with the existing ones using the D and G optimality criteria. The study shows that the alternative axial distances are better using the D and G optimality criteria.

Central Composite Design (CCD) is often used in Response Surface Methodology (RSM) to fit second-order models. However, little is known about their behavior as the number of factors increases. The study develops a cross-dimensional evaluation methodology to investigate the behavior of three types of CCDs as the number of factors increases from $k=3$ to $k=10$. The CCD families considered in this study include Face-Centered CCD (FCCD), Rotatable CCD (RCCD), and Spherical CCD (SCCD). The designs are evaluated using the alphabetic optimality criteria of D-, A-, and G-optimality. A total of 1080 design configurations are generated by varying the replication levels of factorial points, axial points, and center points. The results show that classical CCDs encounter structural limitations in high-dimensional settings. When $k\ge 7$, the quadratic model matrix becomes rank deficient due to the fixed factorial core of 32 runs, which takes on a budget-constrained approach and uses regularization as a diagnostic tool to evaluate the stability of CCDs in the rank-deficient regime. In lower dimensions ($k\le 6$), RCCD consistently provides the highest efficiency, followed by SCCD and FCCD. In higher-dimensional settings, fixed regularization becomes less effective as the scale of the information matrix increases. To address this limitation, scaled regularization is introduced by adjusting the regularization parameter according to the average eigenvalue of the information matrix. The results indicate that rotatable designs provide greater efficiency and stability in high-dimensional settings compared with spherical and face-centered designs.