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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B5. Beijing 2008 
about 30 % increase of its determination. 
Possibility of such an approach, gives a construction of this test. 
It guarantees location of the test points in one vertical axis with 
0.5 mm accuracy. 
2. THEORY 
Ta king into consideration arc G with class C created from 
image points laying on the plumb line of the test, we can say 
that: if tangent in a point M (ordinary point) in not parallel to 
Oy axis, arc G in a neighborhood of the point M can be created 
using equation (1), where function y(x) has a second continuous 
derivative. 
if curves has mathematical form: x=x(t), y=y(t), and taking into 
account differential calculus and variations of variables, it is 
clear that: 
dy 
V'(x) = f. 
dt 
>”(*) 
dx d 2 y dy d 2 x 
dt dr dr dr 
i—Î 
{ dt, 
(6a) 
using formulas above we obtain equations of center coordinates 
(£, q) and radius of osculating circle R as parameterized 
formulations (Gdowski, 1997). 
v = v(x) 
(i) 
because equation of all circles series we can express by a 
formula (2), 
(x-*) 2 +(y-n) 2 -R-=0 
(2) 
(* ; +j V 
*y-x'y 
, (x ,2 +y'V 
/7 = V + : 
x’ v"-xy 
where parameters are center coordinates (£, q) and radius R, 
according to theory of osculating curves, we can claim, that we 
obtain osculating circle to arc G in point M(xo.yo) by creating 
function described below (3). 
,2 \3 '2 
R = 
(x +y ) 
xy’-x”v1 
(6b) 
<K x) = <x-f) 2 +0*to-77) 2 -* 2 
It fulfills conditions: 
4>(x 0 ) = 0 
<£•(*0) = ° 
3* ( x o) = 0 
If we replace xowith x, the conditions (4) can be written as: 
Because radius of curvature is equal to radius of osculating 
circle we can say that center of circle is a center of curvature. 
Set of curvature centers is an evolute, and its equation we can 
express the same as it is in formula (6) (first and second 
equation). 
Geometric interpretation says that: curvature center is a normal 
point to the curve in the point M, in which curve touches its 
envelope, and the same it has limited position. 
Intersection point of normal in point M with normal in 
neighboring point Mi approach this position (Biemacki, 1954). 
(x—£) 2 + (>' - tjŸ -R 2 =0 
x - £ + v’O - rj) = 0 
1+ v ,2 +.v ,, (v-/7) = 0 
from third condition equation (5), calculating q, and from 
second and first conditions ^and R, we obtain equations (6) 
expressing center coordinates (£, q) and R - radius of osculating 
circle (Biemacki, 1954). 
Taking into consideration that plumb lines of test (in case of 
fisheye lens) are conic sections (exactly fragments of ellipses) 
we can describe them by equation (7) or as parametric formulas 
(7a). 
.v = a cos t 
y = hsint 
(7) 
(1+v’ 2 ) 2 3 
= M 
0 < t < 2/r 
(7a) 
thus, as it was mentioned, that evolute is identical as envelope 
of normal, we can derive it from equation of normal (8). 
y(.v)[r->(x)] + .r-x = 0 (g) 
Now, we insert parametric expression of ellipse (7a) into 
(6) formulas (6b), we obtain: