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

In summary, the equation of a cylinder with its axis in the yz plane and making an angle 'alpha' with the x-axis, the axis intersects the y-axis at a distance of 'k'.
  • #1
Xishan
40
0
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
 
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  • #2
Take the expression for a cylinder aligned with the axis, apply a rotation and translation of your coordinate system.
For a translation
[tex] x = x' + h[/tex]
[tex] y= y'+k [/tex]

for the rotation
[tex] x= x'\cos( \theta) + y'\sin( \theta) [/tex]
[tex] y=x'\sin(\theta)+y'\cos(\theta)[/tex]
 
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  • #3
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!
 
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  • #4
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
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!
 

What is a cylindrical surface?

A cylindrical surface is a three-dimensional geometric shape that is formed by a straight line, called the generatrix, moving parallel to a fixed line, called the axis, while also passing through a fixed point, called the directrix.

What are the properties of a cylindrical surface?

The properties of a cylindrical surface include having a constant radius, a curved surface, and two flat circular bases. It also has a curved surface area, lateral surface area, and total surface area that can be calculated using specific formulas.

What are some real-life examples of cylindrical surfaces?

Some real-life examples of cylindrical surfaces include cans, pipes, straws, and columns. These objects have a circular cross-section and a constant radius, making them cylindrical in shape.

How is a cylindrical surface different from a cylinder?

A cylindrical surface is a curved shape that extends infinitely in both directions, while a cylinder is a solid object with two flat circular bases and a curved surface. A cylinder can only have a finite volume, unlike a cylindrical surface.

What are the applications of cylindrical surfaces?

Cylindrical surfaces have various applications in different fields such as engineering, architecture, and design. They are commonly used in construction for structures such as bridges, tunnels, and pipes. They also have applications in manufacturing, transportation, and everyday objects like cans and bottles.

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