Ellipsoid modeling is essential in a variety of fields, ranging from astronomy to medicine. Many response surfaces can be approximated by a hemi-ellipsoid, allowing estimation of shape, magnitude, and orientation via orthogonal vectors. If the shape of the ellipsoid under investigation changes over time, serial estimates of the orthogonal vectors allow time-sequence mapping of these complex response surfaces. We have developed a quantitative, analytic method that evaluates the dynamic changes of a hemi-ellipsoid over time that takes data points from a surface and transforms the data using a kernel function to matrix form. A least square analysis minimizes the difference between actual and calculated values and constructs the corresponding eigenvectors. With this method, it is possible to quantify the shape of a dynamic hemi-ellipsoid over time. Potential applications include modeling pressure surfaces in a variety of applications including medical.
 D. A. Turner, I. J. Anderson, J. C. Mason and M. G. Cox, “An Algorithm for Fitting an Ellipsoid to Data,” 1999.
 E. S. Maini, “Enhanced Direct Least Square Fitting of Ellipses,” International Journal of Pattern Recognition and Artificial Intelligence, Vol. 20, No. 6, 2006, pp. 939-953.
 I. Markovsky, A. Kukush and S. Van Huffel, “Consistent Least Squares Fitting of Ellipsoids,” Numerische Mathematik, Vol. 98, No. 1, 2004, pp. 177-194. http://dx.doi.org/10.1007/s00211-004-0526-9
 A. Gefen, “Reswick and Rogers Pressure-Time Curve for Pressure Ulcer Risk. Part 1,” Nursing Standard, Vol. 23, No. 45, 2009, pp. 64-74. http://dx.doi.org/10.7748/ns2009.07.23.45.64.c7115
 S. Portnoy, N. Vuillerme, Y. Payan and A. Gefen, “Clinically Oriented Real-Time Monitoring of the Individual’s Risk for Deep Tissue Injury,” Medical & Biological Engineering & Computing, Vol. 49, No. 4, 2011, pp. 473-483.
 L. Agam and A. Gefen, “Toward Real-Time Detection of Deep Tissue Injury Risk in Wheelchair Users Using Hertz Contact Theory,” Journal of Rehabilitation Research and Development, Vol. 45, No. 4, 2008, pp. 537-550.
 A. Gefen, “Bioengineering Models of Deep Tissue Injury,” Advances in Skin and Woundcare, Vol. 21, No. 1, 2008, pp. 30-36.
 E. L. Ganz, N. Shabshin, Y. Itzchak and A. Gefen, “Assessment of Mechanical Conditions in Sub-Dermal Tissues during Sitting: A Combined Experimental-MRI and Finite Element Approach,” Journal of Biomechanics, Vol. 40, No. 7, 2007, pp. 14431454.
 “Matrices and Linear Algebra,” Reference Guide for Matrix.xla, 2006.