Lesson Transformations of coordinates

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In the following lesson we will deal with different kinds of the Transformation of axis and will deal with the issues related to them.

Transformations of <A HREF=Coordinates_(elementary_mathematics).wikipedia>coordinate</A>

To transform from one <A HREF=http://www.algebra.com/algebra/homework/Coordinate-system/Coordinate_system.wikipedia>coordinate system</A> to another. Consider ( x, y ) & ( {{{x[1]}}}, {{{y[1]}}} ), these are coordinates of arbitrary point P in old and new coordinate systems correspondingly.

Translation of axes :

Let's move the coordinate system  XOY in a plane so, that the axes OX and  OY  are parallel to themselves, and the origin of coordinates  O  moves to the point   {{{O[1]}}} ( a, b ). We'll receive the new coordinate system  {{{X[1]}}}{{{O[1]}}}{{{Y[1]}}}

{{{drawing( 160, 160, -15, 15, -15, 15,locate(0,15,Y[1]),locate(-6.5,13.5,Y),locate(12,-7.5,X),locate(13.5,-0.5,X[1]),line( -7, -7, 12,-7),line( -7,-7,-7,12),line(-1, -1, 15,-1),line( -1, -1, -1,15),circle(7,7,0.1),locate(6.5,-7.5,m),locate(3.5,0,m[1]),locate(-9,0,B),locate(-8,-8,O),locate(-4,-1,O[1]),locate(-1.2,-7.5,A),line(-1,-1,-0.5,-1),line(-1.5,-1,-2,-1),line(-3,-1,-3.5,-1),line(-4.5,-1,-5,-1),line(-6,-1,-6.5,-1),line(-1,-1,-1,-0.5),line(-1,-1.5,-1,-2),line(-1,-3,-1,-3.5),line(-1,-4.5,-1,-5),line(-1,-6,-1,-6.5),line(7,7,7,6.5),line(7,5.5,7,5),line(7,4,7,3.5),line(7,2.5,7,2),line(7,1,7,0.5),line(7,-0.5,7,-1),line(7,-2,7,-2.5),line(7,-3.5,7,-4),line(7,-5,7,-5.5),line(7,-6.5,7,-7),line(7,7,6.5,7),line(5.5,7,5,7),line(4,7,3.5,7),line(2.5,7,2,7),line(1,7,0.5,7),line(-0.5,7,-1,7),line(-2,7,-2.5,7),line(-3.5,7,-4,7),line(-5,7,-5.5,7),line(-6.5,7,-7,7),locate(-9,8.1,n),locate(-4,9,n[1]),locate(7.5,7.5,P))}}}

Now,

{{{Pm=x}}} and {{{Pm[1]=x[1]}}}

{{{Pn=y}}} and {{{Pn[1]=y[1]}}}

{{{AO[1]=a}}} and {{{BO[1]=b}}}

Coordinates of the point  P  in the new and old coordinate systems are tied by the equations:

{{{Pm[1]=(Pm-AO[1])}}}

=>{{{x[1]=(x-a)}}}

and,

{{{Pn[1]=(Pn-BO[1])}}}

=>{{{y[1]=(y-b)}}}

Another form of transformation is by rotating the axis about the origin or a fixed point.

Turning around origin of coordinates.

Let's turn the coordinate system  XOY in a plane by an angle Theta in anticlockwise direction.

{{{drawing(160,160,-15,15,-15,15,line( -7, -6, 13.38,-0.95),line( -7,-6,-11.06,8.41),line(-7,-6, 12,-6),locate(-7.5,-6.5,O),line(-7,-6,-7,12),locate(-7,15,Y),locate(12,-6,X),locate(13.38,0,X[1]),locate(-11.06,13,Y[1]),locate(6,7.5,P),locate(-11.8,3.6,n[1]),locate(7.5,-0.7,m[1]),locate(-9,7,n),locate(4.5,-6.8,m),line(5,6,6.91,-3.63),line(5,6,-8.76,2.86),line(5,6,4.5,6),line(3.5,6,3,6),line(2,6,1.5,6),line(0.5,6,0,6),line(-1,6,-1.5,6),line(-2.5,6,-3,6),line(-4,6,-4.5,6),line(-5.5,6,-6,6),line(5,6,5,5.5),line(5,4.5,5,4),line(5,3,5,2.5),line(5,1.5,5,0.5),line(5,-0.5,5,-1),line(5,-2,5,-2.5),line(5,-3.5,5,-4),line(5,-5,5,-5.5))}}}

And,

{{{Pm=x}}} and {{{Pm[1]=x[1]}}}

{{{Pn=y}}} and {{{Pn[1]=y[1]}}}

Now coordinates of the point  P  in the new and old coordinate systems are tied by the equations:

{{{x[1]=x*cosTheta + y*sinTheta}}}     ...................(1)

{{{y[1]= - x*sinTheta + y*cosTheta}}}     ...................(2)

If the axis is rotated in the clockwise direction then the change coordinates will be obtained by replacing angle {{{Theta}}} by {{{-Theta}}}.

The change coordinates will be,

{{{x[1]=x*cos(-Theta) + y*sin(-Theta)}}}

or, {{{x[1]=x*cosTheta +(-1*y*sinTheta)}}}

or, {{{x[1]=x*cosTheta - y*sinTheta}}}   ...............(3)

Now,

{{{y[1]= - x*sinTheta + y*cosTheta}}}

or, {{{y[1]=- x*sin(-Theta) + y*cos(-Theta)}}}

or, {{{y[1]=x*sinTheta + y*cosTheta}}}   ...............(4)

In the particular case when {{{Theta  = pi  }}} we'll receive a central symmetry relatively about the origin O :

Hence putting {{{Theta  = pi  }}} in equation 1 and 2, the coordinates will be,

{{{x[1]=x*cos(pi) - y*sin(pi)}}}

as <A HREF=Cosine.wikipedia>cos</A>{{{pi}}}=-1 and <A HREF=Sine.wikipedia>sin</A>{{{pi}}}=0

Hence, {{{x[1]=-x}}}

Similarly,

{{{y[1]=x*sin(pi) + y*cos(pi)}}}

or {{{y[1]=-y}}}

For a <A HREF=http://www.algebra.com/algebra/about/history/Homothetic_transformation&action=edit&section=2.wikipedia?pageview=dictionary>homothetic transformation</A> with a center O (a,b) and a coefficient {{{k<=0}}}

The change coordinates will be,

{{{x[1]-a = k(x-a)}}}

{{{y[1]-a = k(y-b)}}}

For a <A HREF=Affine-function.wikipedia>Affine Transformation</A> :

The change coordinates will be,

{{{x[1]=ax+by+c}}}

{{{y[1]=dx+ey+f}}}

where, {{{ae-bd!=0}}}