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Question: Trying to generalize parametrization for radius not centered at origin

Question:Trying to generalize parametrization for radius not centered at origin

hitstudent 50 Maple
center circle
December 02 2016
0

Hi.

In some cases when dealing with vectofields an such the are integral has to be expressed in terms of r(t).

the general form for r is r^2=(r*cos(t)-a)^2+(r*sin(t)-b)^2, When I solve this in maple it seems like I get the inverse of the desired result.

If I knew that was always the case I could just inverse my result to get the right expression for r, but im not sure if it only applies for this particular cas or all cases.

I would be happy if anyone took a quick look and suggested a way to obtain the desired solution for any center (a,b) for the circle.

 


 

Expression for radius, circle centred at (a,b)

> 

RA := r^2 = (r*cos(t)-a)^2+(r*sin(t)-b)^2

r^2 = (r*cos(t)-a)^2+(r*sin(t)-b)^2

 (1)
> 

isolate(r^2 = (r*cos(t)-a)^2+(r*sin(t)-b)^2, r)

r = (1/2)*(a^2+b^2)/(cos(t)*a+b*sin(t))

 (2)
> 

eval(%, [a = -1, b = 0])

r = -(1/2)/cos(t)

 (3)
> 

plot3d([-2*x, x^2+y^2], y = -sqrt(-x^2-2*x) .. sqrt(-x^2-2*x), x = -2 .. 0, color = [green, red], orientation = [0, 0, 0])

 
> 

(1/2)*Pi <= t and t <= 3*Pi*(1/2)

(1/2)*Pi <= t and t <= (3/2)*Pi

 (4)
> 

0 <= r and r <= -2*cos(t)

0 <= r and r <= -2*cos(t)

 (5)
> 

Area_off_center = int(r, [r = 0 .. -2*cos(t), t = (1/2)*Pi .. 3*Pi*(1/2)]); 1; Area_at_center = int(r, [r = 0 .. 1, t = 0 .. 2*Pi])

Area_off_center = Pi

  

Area_at_center = Pi

 (6)
> 

``
> 

``


 

Download parametrization_of_r_not_centred_at_orgin.mw
 

I would also happily like to know how I can solve for the range r can take, obviously in the example i´m working with here r starts at 0, but that is not always the case i guess.

 

Thank you, your help is much apperciated
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