Help with basic multivariable problem.

[Vampire]

Help with basic multivariable problem. [SOLVED]

Homework Statement

Two surfaces intersect at a space curve C.
The two surfaces are 4y² + 9z² = 36 and x = 2y² - 3z²

Find a vector parametrization for C. (r(t) = ( f(t) , g(t) , h(t) )
Find a set of values for the parameter t over which C is traced once.

Homework Equations

None needed.

The Attempt at a Solution

I've solved for a vector curve r(t) = ( t , $$\sqrt{3t/10+36/10}$$, $$\sqrt{2t/15-36/15}$$)

I used x=t and solved the rest, but I'm not sure that it even correct. Additionally, I have no idea how I will find bounds for exactly one trace of C.

 

[lanedance]

hi vampire, your paramterisation may not help as i don't think it traces the whole curve, only a piece of it

have a think about your two sufaces, maybe try drawing some pieces along the axis planes in 3D...

The first equation represents a elliptic cylinder along the axis, the 2nd is harder to draw & looks like some kind of saddle type thing...

However its likely the 2nd surface cuts completely through the cylinder... giving a distorted ellipse over which C will retrace itself

With this mind... an idea could be to first try and parameterise z & y in terms of the projected ellipse on the y-z plane.

Choose cylindrical coordinates, with cylinder along the x axis, then z & y should be easy to parameterise in term of the angle. This should solve the first equation, use your 2nd to get x.

Not 100% it will work, but worth a crack...

 

[Vampire]

Thank you for your help! Looking at it facing the y-z plane helped me solve it; thank you, lanedance.

I've solved it. I parametrized the elliptical cylinder using trigonometric functions:

z= 2sin(t) y= 3cos(t) and solved for x. x= 18cos^2(t) - 12sin^2(t)

t (0,2$$\pi$$)