Three geometric groups commonly used in defining the body segments are semi-ellipsoids, elliptical solids and stadium solids. The semi-ellipsoid group consists of semi-ellipsoid, ellipsoid of revolution and sphere. The typical members of the elliptical solid group are the elliptical and circular cylinders, the truncated or normal circular and elliptical cones and the general elliptical and circular solids. The stadium solid group consists of the stadium solids and the stadium cylinders:
For each geometric group, equations for the mass (m), center-of-mass (CM) location from the proximal end (gz), and the three principal moments of inertia (Ixxx, Iyy & Izz) are provided. A uniform density of r is assumed here. The BSP equations for the stadium solid were adapted and modified from Yeadon (1990).
Let functions a(t), b(t), c(t) & d(t) be functions of t
where a, b, c, d > 0 and 0 < t < 1. Then,
Now, let functions a(t), b(t), c(t) & d(t) be
where, ao, a1, bo, b1, co, c1, do & d1 > 0 and 0 < t < 1. Then,
A set of trigonometric integral functions were also used in deriving the BSP functions:
The mass and principal moments of inertia of an elliptical plate (Figure 1) is as follows:
Note that  was used in deriving .
Let z in Figure 2 be a function of t (0 < t < 1):
 &  were used in deriving . The BSP equations for ellipsoid, ellipsoid of revolution, and sphere can be derived from :
Let z in Figure 3 be a function of t (where 0 < t < 1):
, ,  &  were used in deriving . The BSP equations for truncated circular cone, elliptical cylinder, and circular cylinder can be also derived from :
The mass and moment of inertia functions of the stadium plate (Figure 4) were obtained as follows:
 was used in deriving .
Figure 5 shows the stadium solid. Note that the directions of the axes are slightly different from Figures 2 & 3. Let z be a function of t (where 0 < t < 1):
, ,  &  were used in deriving . In the stadium cylinders, ao = a1 and ro = r1. If ao = a1 = 0, it becomes a circular solid (truncated circular cone). If a1 = r1 = 0, the stadium solid becomes a stadium cone. Note here that ,  &  are presented in a slightly different form from that in Yeadon (1990). Yeadon used a & b that were defined as
When ro = 0, or ao = 0,  causes divided-by-zero error. The equations presented in this page prevent this problem and, thus, are more flexible.
References & Related Literature
Hanavan, E. P. (1964). A mathematical model of the human body. AMRL-TR-64-102, AD-608-463. Aerospace Medical Research Laboratories, Wright-Patterson Air Force Base, Ohio.
Hatze, H. (1980). A mathematical model for the computational determination of parameter values of anthropometric segments. J. Biomechanics 13, 833-843.
Huston, R. L., and Passerello, C. E. (1971). On dynamics of a human body model. J. Biomech. 4, 369-378.
Jensen, R. K. (1978). Estimation of the biomechanical properties of three body types using a photogrammetric method. J. Biomechanics 11, 349-358.
Kwon, Y.-H. (1994). KWON3D Motion Analysis Package 2.1 User's Reference Manual. Anyang, Korea: V-TEK Corporation.
Kwon, Y.-H. (1993). The effects of body segment parameter estimation on the experimental simulation of a complex airborne movement. Doctoral Dissertation, Pennsylvania State University.
Miller, D. I. and Morrison, W. (1975). Prediction of segmental parameters using the Hanavan human body model. Med. Sci. Sports 7, 207-212.
Rodrigue, D. and Gagnon, M. (1984). Validation of Weinbach's and Hanavan's models for computation of physical properties of the forearm. Res. Q. Exercise Sport 55, 272-277.
Sady, S., Freedson, P., Katch, V. L. and Reynolds, H. M. (1978). Anthropometric model of total body volume for males of different sizes. Human Biol. 50, 529-540.
Weinbach, A. P. (1938). Contour maps, center of gravity, moment of inertia and surface area of the human body. Human Biol. 10, 356-371.
Yeadon, M. R. (1990). The simulation of aerial movement-II. A mathematical inertia model of the human body. J. Biomechanics 23, 67-74.
© Young-Hoo Kwon, 1998-