# Translation and Rotation of Axes

Document Sample

```					                                                                                            1

Translation and Rotation of Axes

Consider a seconddegree equation
(1)             Ax 2  Bxy  Cy 2  Dx  Ey  F  0
in x and y. If B  0, we may use translation of axes if necessaryto put the equation in a standard form for
a parabola or an ellipse or a hyperbola.
Example. Consider the seconddegree equation
(2)             9 x 2  16 y 2  54 x  64 y  127  0
in x and y. We begin by completing the squares in x and y.
9 x 2  54x   16 y 2  64 y   127.
                      
9 x 2  6 x  16 y 2  4 y  127.   
9x   2
                   
 6 x  9  16 y 2  4 y  4  127  81  64.
9( x  3) 2  16( y  2) 2  144.
( x  3) 2 ( y  2) 2
            1.
16         9
We then use translation of axes to introduce new (u,v)-coordinate axes defined by
(3)                   u = x + 3, v = y - 2
or equivalently by
(4)                   x = u - 3, y = v + 2.
In the new (u,v)-coordinate system the curve here has the equation
u2 v2
    1,
16 9
which is an equation of a hyperbola with center at the origin, vertices at (±4,0), foci at
                 
 16  9 ,0   5,0  with eccentricity e  , with the directrices the straight lines
5
4
16
defined by the equations u   , and with asymptotes the straight lines defined by the
5
3
the equations v   u . In the original (x,y)- coordinate system, the center is at the point
4
(-3,2), the vertices are at the points (1,2) and (-7,2), the foci are at the points (2,2) and
16
(-8,2), the directrices are the straight lines defined by the equations x  3  , and the
5
asymptotes are the straight lines defined by the equations (y – 2) =   x  3 .
3
4
Example. Consider here the second degree equation
(5)             4 x 2  16 x  16 y  32  0
2

in x and y. We begin by completing the square in x.
             
4 x 2  4 x  16 y  2  0.
4x   2

 4 x  4  16 y  2  16.
4  x  2  16   y  1.
2

 x  2 .
1
( y  1) 
2

4
We then use translation of axes to introduce new (u,v) –coordinate axes defined by
(5)            u = x + 2, v = y – 1
or equivalently by
x = u – 2, y = v + 1.
In the new (u,v) – coordinate system the curve has the equation
1
v  u2,
4
which is an equation of a parabola with the vertex at the origin. We now discuss finding
the focus and the directrix of this parabola. Put the above equation in the form
u 2  4v  4 pv
where p=1. In the (u,v) – coordinate system, the focus of this parabola is the point (0,p)
= (0,1), and the directrix is the straight line defined by the equation v= -p = -1. In the
original (x,y) – coordinate system, the vertex of this parabola is at the point (-2,1), the
focus is at the point (-2,2), and the directrix is the straight line defined by the equation
y=0. We now discuss this in terms of our previous study of a parabola defined by an
equation
y  f ( x)  ax 2  bx  c
where a ≠ 0. Here we have
16 y  4 x 2  16 x  32
or
1
y  x 2  x  2.
4
Let
1
a  , b  1, c  2.
4
The vertex is the point
 b 4ac  b 2 
 ,
 2a                .
            4a    
Note that
b          1
                2
2a           1
2
4
and
3

1
4   2  12
4ac  b2
 4             1,
4a              1
4
4
so the vertex is the point (-2,1). The focus is the point

 b 4ac  b 2 1 1 

 2a , 4a  4  a        
                         
               
               
    2,1  1  1 
         4 1
               
             4
 ( 2,2).
The directrix is the straight line defined by the equation
4ac  b 2 1 1         1 1
y                 1    0.
4a        4 a      4 1
4
_____________________________________________________________________
Consider again the second degree equation
(1)              Ax 2  Bxy  Cy 2  Dx  Ey  F  0
in x and y. Suppose now that B≠0. We discuss here use of rotation of axes to get rid of
the xy-term. Consider a new (u,v) – coordinate system. Let  be an angle from the
positive x-axis to positive u-axis. Let O be the point which is the origin for both
coordinate systems. Then, let P be some given point other than O. Let OP denote the
vector with the initial point O and with terminal point P. Let r denote the length of this
vector OP , and let  be an angle from the positive u-axis to OP . Let (x,y) be the
coordinates of P in the original coordinate system, and let (u,v) be the coordinates of P in
the new coordinate system.

y-axis
v-axis
P
r               u-axis



O                                                 x-axis
4

We have that
x  r  cos   
 r  cos  cos   sin  sin  
 r cos   cos   r sin    sin  
 u  cos   v  sin  
and
y  r  sin    
 r  sin   cos   cos sin  
 r cos   sin    r sin    cos 
 u  sin    v  cos .
Thus we have the equations
 x  u cos   v sin  
(6)      
 y  u sin    v cos 
for rotation of axes. We now make the substitutions (6) in the equation (1). We get:

A  u cos   v sin    B  u cos   v sin    u sin    v cos 
2

 C u sin    v cos   D  u cos   v sin  
2

 E  u sin    v cos   F  0.

A  u 2 cos2    2uv cos sin    v 2 sin 2             

 B  u cos sin    uv cos    uv sin    v 2 sin   cos 
2                           2                   2

 C  u  sin    2uv sin   cos   v cos  
2   2                                   2           2

 D  u cos   v sin    E  u sin    v cos   F  0.
A cos    B cos sin    C sin   u
2                                   2               2

  2 A cos sin    B cos    sin   2C sin   cos uv
2               2

 A sin    B sin   cos   C cos    v
2                                   2               2

 D cos   E sin    u   D sin    E cos   v  F  0.

We want to have:
                       
 2 A cos sin    B  cos2    sin 2    2C  sin  cos   0.
C  A  sin2   B  cos2   0.
AC
(7)      cot2           .
B
5

rm)
Example. Use a suitable rotation of axes to eliminate the xy - term (the product te in the equation
34 x 2  24 xy  41y 2  250 y  325  0.
Let
A  34, B  24, C  41, D  0, E  250, F  325.
Let  be an angle of rotation such that
A - C 34  41

cot(2 )                    .
B      24
Hence
7

cot(2 )          .
24
Note that
(7) 2  (24) 2  625  25.
Suppose we take
7

cos(2 )          .
25
Then
24
sin(2 )  -      .
25
Thus we may take
3
 2  2.
2
Hence
3
   .
4
Therefore
7
1

1  cos(2 )         25  9  3
sin( )                
2             2     25 5
and
7
1

1  cos(2 )            25   16   4 .

cos( )                 
2               2        25     5
Then our equations for rotation of axes are
                              4     3

 x  u cos( )  v sin( )   5 u  5 v


 y  u sin( )  v cos( )  3 u  4 v.


                            5     5

We now make the substitutions in the given equation.
6

 16     24       9          12        7  12 
34   u 2  uv  v 2   24    u 2  uv  v 2 
 25     25      25          25       25  25 
 9       24     16           3   4 
 41   u 2  uv  v 2   250   u  v   325  0.
 25      25     25           5   5 
625 2            1250 2
u  0uv        v  150u  200v  325  0.
25               25
25u 2  50v 2  150u  200v  325  0.
u 2  2v 2  6u  8v  13  0.
u   2
                  
 6u  2  v 2  4v  13  0.
u   2
              
 6u  9  2  v 2  4v  4  13  9  8.
u  32  2  v  22  4.
u  32  v  22  1.
4           2
Next weuse translation of axes to introduce new (s, t) - coordinate axes defined by
s  u  3, t  v - 2
or equivalently by
u  s - 3, v  t  2.
In the new (s, t) - coordinatesystem the curvehas the equation
s2 t 2
  1,
4 2
which is an equation of an ellipse with center at the origin, verticesat the points  2,0, minor axis

               
with endpoints at the points (0, 2 ), foci at the points  4 - 2 ,0   2 ,0 , eccentrici e 
ty
2
2
,

4
and directrices the straight lines defined by the equations s        . In the (u, v) - coordinatesystem,
2
the center of the ellipse has coordinates (-3,2), the vertices have coordinates (-1,2)or (-5,2), the
endpoints of the minor axis have coordinates (-3,2 2 ) or (-3,2 2 ) , the foci have coordinates
4
(-3  2 ,2) or (-3 - 2 ,2), and the directrices are the straight lines defined by the equation u  3 
2
4
or the equation u  3     .
2
Return now to the original (x, y) - coordinatesystem.The coordinates of the center of the ellipse are given
by
7

       4     3       4        3       6
 x   5 u  5 v   5 (3)  5  2  5


 y  3 u  4 v  3 (3)  4  2   17 .

     5     5     5        5          5
ex
The coordinates of the first vert are given by
       4          3         2
 x   5  (1)  5  2   5


 y  3  (1)  4  2   11 ,

     5          5          5
ex
and the coordinates of the other vert are given by
       4          3      14
 x   5  (5)  5  2  5


 y  3  (5)  4  2   23 .

     5          5         5
The coordinates of the first endpoint of the minor axis are given by
       4          3

 x    (3)   2  2  
       5          5
6 3 2
5

5


     5          5
           
 y  3  (3)  4  2  2   17  4 2 ,
5      5
and the coordinates of the other endpoint of the minor axis are given by
       4          3

 x    (3)   2  2  
       5          5

6 3 2
5    5


     5          5
           
 y  3  (3)  4  2  2   17  4 2 .
5     5
8

y

•
•
•
3•
•
• 
•   •   •    • •   •   •   • • • •    •   •   •   •   •   x
-5                 •              5
•
•
•
•
•
•
-7 •           6 17 
Center   , 
5 5 

•
•
•
•
•
•
•
•
•
•
•
•
•

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 1541 posted: 4/8/2010 language: English pages: 8
Jun Wang Dr
About Some of Those documents come from internet for research purpose,if you have the copyrights of one of them,tell me by mail vixychina@gmail.com.Thank you!