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Polar Coordinates Let P = (x,y) be a point of the plane different from the origin. Let (r, θ) be an ordered pair) be an ordered pair of real numbers such that
x r cos θ , y r sin θ .
We say that (r, θ) be an ordered pair) is a set of polar coordinates for P. Example Consider the curve defined in polar coordinates by
1 cos 0
e ed r where e and d are positive real numbers, and where ^0 is a real number. Note that cos 0 cos cos 0 sin sin 0 . In the special case where (^02) we have that sin . 2 cos (^) We have that
1 cos cos 0 sin sin 0
e ed r Then cos sin .
cos sin.
cos sin.
cos cos sin sin.
1 cos cos sin sin.
0 0 0 0 0 0 0 0 0 0
r e d x y
r ed e x y
r e x y ed
r e r r ed
r e ed
Next we get that y x P = (x,y) r θ) be an ordered pair 0
2 cos sin sin . 2 cos 2 sin cos cos sin sin sin cos sin cos sin cos cos cos sin cos sin cos sin 2 0 2 0 0 2 0 2 0 0 2 2 0 2 0 0 0 0 0 2 0 2 0 0 0 2 0 0 0 0 2 0 0 x y y d d x d y x x y d y x y y d d x d y d x x d x y d x y d x y
Thus 2 sin . cos 2 cos sin sin 2 cos 2 0 0 2 0 2 0 0 2 0 2 2 2 d y d r e x x y y d x
Hence 2 cos 2 sin . cos 2 cos sin sin 2 0 0 2 0 2 0 0 2 0 2 2 2 2 d x d y d x y e x x y y
Therefore sin 1 2 cos 2 sin 0.
cos 1 2 cos sin
2 2 0 2 0 2 2 0 2 2 0 0 2 2 0 2 2
e y e d x e d y e d
e x e x y
Now let sin 1. cos 1 , 2 cos sin , 0 2 2 0 0 2 0 2 2
C e A e B e Note that cos^ ^ ^ sin ^ ^1. cos sin cos sin 1 2 0 2 0 4 2 0 2 2 0 2 2 0 2 0 4 2
e e AC e e e
Hence B^2 4 AC 4 e^2 1 . We now consider the possible cases for e. Case 1 e=1. Then B^2 4 AC 0. Now the curve is a parabola or a limiting form.
^ ^ ^ ^ ^ ^ ^ ^ ^ ^
2 cos sin 2 cos sin sin cos 0.
sin 1 sin 2 sin cos cos
2 cos sin cos sin cos sin sin cos
cos 1 cos 2 cos sin sin
2 2 0 0 0 0 2 0 2 0 2 2 0 2 2 0 0 0 2 2 2 2 0 0 2 0 2 0 2 0 0 2 0 0 2 0 2 2 0 0 0 2 2 2 2
e du edv e d
e u uv v
e u uv v
e u uv v
Collecting coefficients we get
1 - e 2.
Thusfinally we get e cos sin 1 0 1 2 0.
cos sin sin 2 cos sin sin cos cos ( 2 sin cos 2 sin cos (( 2 cos sin 2 cos sin( )) 2 cos sin (cos sin ( )) cos cos 2 cos sin sin sin 2 2 2 2 2 2 (^222222) 0 2 0 2 2 2 2 2 2 0 2 0 2 0 2 2 0 2 0 2 2 0 2 0 2 0 2 2 0 0 0 0 2 3 0 2 0 2 0 0 2 0 0 0 0 2 3 2 0 2 0 2 4 0 2 0 2 2 0 2 0 2 4 u v edu e d u uv v edu e d so edu e d e e e v e uv e e e e e u
