Find Center of Sheared Cylinder (x, y, z, θ)

[isabella]

i am looking for a way to find the coordinate of the center of a circle in a sheared cylinder. the base of the cylinder has a center (x,y,z), the angle of tilt of the cylinder is theta.so i need a formula which allows me to get the center of any circle in the cylinder.however i can't seem to get the right formula.i've attached a file with the drawing of the sheared cylinder.(although the cylinder is sheared, it still have a circular cross-section)

 

[HallsofIvy]

Although they are not shown in the picture, can we assume that you are given the radius and height (measured along the axis) of the cylinder? Also, since this is a 3D problem, we would need to know in which direction the cylinder is tilted (over the x-axis, y-axis, or z-axis?). If you know those, this is a simple trig problem.

In order not to confuse it with general coordinates, I'm going to call the given point (x₀,y₀,z₀).

Assuming that the height of the cylinder, measured along the axis is h and that the axis lies above the x-axis, we get immediately that the coordinates of the "top" of the cylinder (the other end of the axis) are x₁= x₀+ h sin θ, y₁= y₀, and z₁= z₀+ h cos θ. You can get the coordinates of the center point, a, and point b (1/3 of the way from (x₀,y₀,z₀) to (x₁,y₁,z₁)?) from those.