Parametize equation of ellipse

[markraz]

In this example of parameterizing an ellipse, how do they get this final parametric equation? what are the steps?
It's not documented in any books I have. What happens to the 4 after the equal sign?
any ideas? thanks in advance

 

[skeeter]

[MATH]y^2+4z^2=4[/MATH]
standard form for this ellipse ...

[MATH]\dfrac{y^2}{a^2} + \dfrac{z^2}{b^2} = 1[/MATH]
[MATH]\dfrac{y^2}{2^2} + \dfrac{z^2}{1^2} = 1[/MATH]
[MATH]y = a\cos{t}[/MATH]
[MATH]z = b\sin{t}[/MATH]

 

[Deleted member 4993]

In this example of parameterizing an ellipse, how do they get this final parametric equation? what are the steps?
It's not documented in any books I have. What happens to the 4 after the equal sign?
any ideas? thanks in advance

View attachment 22532

Standard & simplified 2-D equation of ellipse is:

(z/a')^2 + (y/b')^2 = 1

Two terms - squared and added - results in 1! So:

z/a' = sin(t) .... and .... y/(2b') = cos(t)

 

[markraz]

thanks everyone appreciate it

 

[Steven G]

In this example of parameterizing an ellipse, how do they get this final parametric equation? what are the steps?
It's not documented in any books I have. What happens to the 4 after the equal sign?
any ideas? thanks in advance

View attachment 22532

If y=2cost, then y^2 = 4cos^2 t
If z =sin t, then 4*z^2 = 4*sin^2 t.

Now what do you get when you add y^2 and 4z^2? I get y^2 + 4z^2 . What do you get when you add 4cos^2 t and 4sin ^2 t? I get 4 !!!!

So using the two parametric equations I do get y^2 + 4z^2 = 4cos^2 t + 4*sin^2 t =4. Do you see where the 4 comes from???