Cylindrical Surface!

1. Apr 19, 2004

Xishan

What is the equation of a cylinder with its axis in the xy-plane and making an angle 'alpha' with the x-axis, the axis intersects the y-axis at a distance of 'k'?
Initially i thought this problem to be very simple but haven't got any success with it in last few days

Xishan

2. Apr 19, 2004

Integral

Staff Emeritus
Take the expression for a cylinder aligned with the axis, apply a rotation and translation of your coordinate system.
For a translation
$$x = x' + h$$
$$y= y'+k$$

for the rotation
$$x= x'\cos( \theta) + y'\sin( \theta)$$
$$y=x'\sin(\theta)+y'\cos(\theta)$$

Last edited: Apr 19, 2004
3. Apr 20, 2004

Xishan

No sir!

When the cylinder's axis lies in xy plane and is NOT PARALLEL to any of the axes, shouldn't the equation comprise of all the coordintes (i.e., x, y & z)?

What you've given here is OK for an in-plane rotation or translation but not for my case! or is it? This way the cylinder is rotated about its own axis which for a right circular cylinder doesn't need any axes transformation at all!

Last edited: Apr 20, 2004
4. Apr 20, 2004

jdavel

Xishan,

Your original question said the axis is in the xy plane, but not parallel to x or y. Integral's rotation will make it lie along the new x (or new y, I can never tell which until I've done the rotation!) axis.

5. Apr 21, 2004

Xishan

I've just managed to solve the problem, the equation of that cylindrical surface turns out to be,

x^2 + y^2 sin(a)^2 + z^2 cos(a)^2 -yz sin(2a) <= r^2

this cylinder has its axis in the yz plane and makes an angle 'a' with the y axis in the ccw direction. This can now be verified: putting a=0 gives the equation of a cylinder with its axis along y axis,
x^2 + z^2 <= r^2

and for a = 90,
x^2 + y^2 = r^2, a cylinder with its axis along z axis!

now if the axis is moved away from the origin, the respective intercepts may be subtracted from x, y or z.

Thanks everyone for considering this problem!