restriction of θ in parametric equation??? (1 Viewer)

[wori]

ellipse:

x= a cos θ
y= b sin θ

(0<=θ<2 pi) y???

hyperbola:

x= a sec θ
y= b tan θ

(-pi < θ<= pi) y???

 

[who_loves_maths]

for the ellipse:

the ellipse is enclosed in the region: -a <= x <= a ; and -b <= y <= b ;

hence, -a <= acos(theta) <= a ; and -b <= bsin(theta) <= b

combine them you get 0 <= theta <= 2pi, which is the same as 0 <= theta < 2pi

the reason why we don't go beyond ONE PERIOD of the sinusoidal curves is that all other values of (theta) outside one oscillation will yield the same results as one or more of the values inside one period ---> otherwise, the domain for (theta) is infinite. the restriction is just for therapeutic reasons.

for the hyperbola:

APPLY SAME LOGIC/DEDUCTION...


Edit: if you know the derivation of the parametrisations for the ellipse and hyperbola (which is in textbooks), then you can prove/show this GEOMETRICALLY too.

Edit (again): hope this helps

 

[wori]

thanks iron wjx~~

I thought they are different because one is (0<=θ<2 pi)and the other is (-pi < θ<= pi) but really they are the same, I checked some other books and find out all sorts of domains ...