0°), 
X 
m 
f 
5 
g 
g 
into 
x 1 cosa X 
T (3) 
y 0 sina y 
Formula (3) is the formula with which 
the coordinates of a point in plane 
rectangular coordinate system can be got 
from the corresponding coordinates in 
plane «-angle coordinate system. 
Similarly,if 41-90 anda»-e,formula (2) 
becomes into 
x' 1 —ctga fx 
= (4) 
y 0 esca y 
Formula (4) is the formula with which 
the coordinates of a point in plane 
4 -angle coordinate system can be got 
from the corresponding coordinates in 
plane rectangular coordinate system. 
In fact, formula (2) can be infered 
from formulas (3) and (4) .Firstly , the 
coordinates in plane o —angle coordinate 
system are transformed into rectangular 
coordinates by using formula (3). 
Xr 1 cosa: X 
2 (5) 
yr 0 sina; y 
And then the rectangular coordinates 
are be transformed into ones in plane 
œ -angle coordinate system by using 
formula (4). 
x' 1 —ctga2 Xr 
Il 
(6) 
y 0 esca 2 yr 
Considered formulas (5) and (6) , the 
following formula is obtained: 
x' 1 —ctga2z 1 cosa; X 
y 0 esca 2 0 sina; y 
i sin (222—241) X 
sina 2 
= sina 1 
0 E y 
sina 2 
It is evident that the above formula is 
as same as formula (2). 
3.Translation and Rotation of Plane 
c -angle coordinate System 
i)Translation 
Suppose XOY is a plane 4-angle coordinate 
system with zero point O and X'O'Y'is the 
same coordinate system but with zero 
point O',then the relation between (x,y) 
and (x',y') is 
x' X Xo 
= = (7) 
y y Yo 
See Fig.3. Here (x,y) and (x',y') are the 
coordinates of point M in XOY and X'O'Y' 
respectively, and(xo, yo)are the coordinates 
of O' in XOY. 
   
Fig.3 
ii)Rotation 
Suppose ¥'0'Y' is a plane c -angle 
coordinate system got by rotating the XOY 
with a angle of 68 and the coordinates of 
point M in XOY and X'O'Y' are (x,y) and 
(x',y') respectively.In  Fig.4, 
x =0Mi=ON+NM', 
InAONMi; - : r 
ZM10N-z6 , ZOMiN-180 —a , ZONM;, =a —6 
Then the following formulas can be 
obtained by using the theorem of sine. 
  
  
  
OM X 
ÓNZ nl. sin (180? — a ) 2————————— sina 
sin (a —6) sin (a —6) 
OM 
M Neo! inei. sing 
sin (a —6) sin (a — 6) 
MN=MM, —M,N=y ——! sine 
sin (a —6) 
MN : MN 
NMiz———-— ——— - sino —— — —- sino 
sin (180° —a) sina 
finally fomula (8) is obtained. 
in sin? « —sin? 0 sind 
^ Sinesin (a —6) sina 
sind sin (a —80) 
Xx 
  
"e 
sina sina 
Formula (8) can also be expressed 
in the form of matrix as 
  
  
x’ sin? a — sin? 0 sing 
"sinasin(a —6) sina x (33 
= sind sin (a —6) 9 
y sine sina y 
In formula (9),the matrix 
  
sin?a —sin?6 sin6 
= sinasin (a — 0) sina 
A= sind sin (a —6) 
^ sin« sina 
is the rotation trasformation 
matrix of plane d -angle coordinate 
system.It is easily proved that 
  
| Al =1 
sin (a —6) sin6 
sina sina 
A-!= : : 
sind sin? « — sin? 6 
^ sine sinasin (« —0) J 
Specially , if a =90°, formula (9) 
becomes into 
x' cosó sinó X 
- (10) 
y — sind cosû y