Biomechanical Modeling of the Shoulder - Fundamentals and Assessment Tools for Occupational Ergonomics - page 564

glenoid fossa), the geometric relations were:

z) 2 E ¼ (R

z) 2

R 2 B ¼ (R

C ¼ (R z)R

R
L 2 sin a

G ¼

(27 : 9)

D ¼ GE þ C

p

4D 2 I ¼ 4BGD þ B 2

B 2GD I

J ¼

2G 2 1

p

J

y ¼

For the third conditions (Figure 27.2c and e) where f ¼ 10 8 to 170 8 (a tangent to the glenoid fossa was

a line crossing the Z-axis in the range from 10 0 mm (the deltoid lateral part was

attached to the glenoid fossa which was an inclined articular surface), the conditions were represented
by both an acute angle (Figure 27.2c) and an obtuse angle (Figure 27.2e). The geometric relations were:

8 to 170 8 ) and z

¼

s

L 2 sin a þ R sin f

y ¼ R sin f (27 : 10)

L 2 sin a R sin f

For the fourth conditions (Figure 27.2d and f) where f 10 8 to 170 8 (a tangent to the glenoid fossa

¼

was a line crossing the Z-axis in the range from 10 ) and z ranged from 0 to 32 mm (the deltoid

lateral part was attached at a certain distance from the glenoid fossa), the conditions are represented by
both an acute angle (Figure 27.2d) and obtuse angle (Figure 27.2f). The geometric relations were as
presented in Equation (27.7).

8 to 170 8

27.2.2 Method of Geometrical Description
27.2.2.1 Concavity of the Glenoid Fossa
The problem of defining the smallest number of tangents which can be used to determine the planar
shape of the glenoid fossa was resolved. Assuming that a glenoid fossa can be described as a convex
function and defined as a polyhedral model, the mathematical task was formulated. 4 Reference axes
Y-Z were fixed in the scapula at the center of the lateral margin of the acromion (the middle deltoid
attachment), where the Y-axis was parallel to the longitudinal axis of the thorax. The concavity of the
glenoid fossa, as a convex function, was considered as a polyhedral model of f based on linearizations
f i , i 1, ... , k with an error of

¼

1 i (z) ¼ f (z) f i (z) ¼ f (z) k g i , z-z i l f (z i ) (27 : 11)

where g i is a subgradient of f in the point z i (if f is differentiable). The nodes z i , , z k on the curve of the

glenoid in the frontal plane were determined when the maximal linearization error 1(z) was minimal.
The tangents to a given glenoid fossa in the frontal plane f (z i ) are the straight lines at the points
(z i , f (z i )) which cross the Z-axis at angles f 1 ,

...

... , f i . Applying the smallest error of the polyhedral

model of a convex function, 2 it was found that the minimum number of tangents necessary to