determined solely through determination of space 
coordinates of several tens of RRT pasted on the 
hand bone surface; 
C. The purpose of changing the general equation 
to a standard equation is to creat a situation in 
which the contact surfaces of the two hand bones 
are in a comparative position; 
D. The determination of dynamic degree of 
coincidence of two contact surfaces of the hand 
bones is performed through translation and 
spinning of a hand bone to another. 
4.2 The shape of hand bone surface is 
assumed as following general quadratic 
equation: 
V. F(X, Y, Z)2 aX *az2 Y *asiZ^ 
*t2a1;XY *2a43XZ*2a23YZ (2) 
+2a14X+2a24Y+2a34Z+244=0 
Even if a44 is simplified , it requires at least nine 
points whose space coordinates are already known 
on the contact surface to define every coefficient ai; 
of the general equation. Moreover, the quadratic 
equation is assumed as a quadratic equation with a 
centre. 
It means that there are points on the quadratic 
curved surface and symmetrically located relating to 
the centre. 
Equation (2) can also be expressed as 
following: 
V. F(X, Y, Z)= (anata1zY+a13Z+A14)X+ 
(a24X*a22Y *ta23Z*a24)* Y * (3) 
(a31X+a32Y+a33Z+a34)Z+ 
(a41X+a42Y+a43Z+A44) 
so there is a matrix expression: 
F(X,Y,Z)=[XYZ1] A P | (4) 
| 
1 
where A is a symmetrical matrix(aij 7 aj): 
311 312 313 314 
321 az a23 a24 
A= | (5) 
331 332 333 334 
| 341 342 343 344 
Fax 
ed 
q(X, Y, Z) S[X Y Z] Aaa N Z “ (6) 
where : 
f a 212 213 | 
Ayu = az az 323 | (7) 
asi 232 ass | 
Any quadratic surface has three main directions, 
they are conjugate and perpendicular to each other. 
Moreover, the main diameter having main direction 
is the symmetrical axis of camber with centre. So to 
find out the main directions of the cambers is the 
key step in determining the degree of coincidence 
between the two cambers. 
The main direction V(X : Y Z)and its 
corresponding eigenvalue À can be derived from 
following formula: 
CIN X 
Austivy Nou SEN Sf (8) 
eZ Vv zZ 
where upon : 
a1 Äh a 413 
| 
| 
a) 43-7. 45 =0 (9) 
a31 a32 à33-A | 
So, the eigenvalue equation is 
  
  
  
2842245740320 (10) 
and 
|17a41*312t*213 
l= a1 an Mi a1 13 + a22 223 
a21 322 431 333 332 233 
(11) 
341 412 213 
= | an az az | |A] 
A31 a32 ass 
All eigenvalues have relationship as follows : 
l12A4*À2*7.3 
12244J.2*2.12.3* 4.25.3 (12) 
[3=A4A2A3 
According to the eigenvalues mentioned above 
three main directions of curved surface are derived, 
and their directions are regarded as coordinateaxes 
of a new coordinate system o' - x'y'Z' , so the 
equation of the quadratic surface is expressed as 
follows : 
X^ 2Y^ «132^ *2a'4X *2a'4Y'*2a'uZ^*a'u-0 — (13) 
After a translation of the origin of coordinate 
system, equation(13) can be simplified as follows : 
jaX 3225132732" 4420 (14) 
According to the numeral value symbols of 1, 
A2, A3, a"44 and whether a"44 equals zero or not , the 
quadratic surface can be judged as a kind of 
ellipsoid .pseudoellipsoid .hyperboloid . of one 
sheet. hyperboloid of two sheets or cone . Under 
special conditions, the curved surface may be a 
simple spin curved surface: 
5. Tests 
Test 1 
A cylinder lump made of aluminium with known 
shape are taken as the examination target. Thirty 
414 
RRT, 0. 
cylindei 
converc 
coordin 
frame à 
the spa 
through 
as follo 
The 
solving 
20.30m 
surveyi 
The 
system 
W 
relatioi 
be con 
Y") an 
spinnir 
an exa