What is the equation of a cylinder with an angled axis and variable intercepts?

[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

thanks for your help!
Xishan

 

[Integral]

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

 

[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!

 

[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.

 

[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!