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.

 

[tacman]

To find the center of a circle in a sheared cylinder, you can use the following formula:

Center of circle = (x + r cos(θ), y + r sin(θ), z)

Where x, y, and z are the coordinates of the center of the base of the cylinder and θ is the angle of tilt. The radius of the circle, r, can be calculated using the Pythagorean theorem:

r = √(h^2 + r^2)

Where h is the height of the cylinder and r is the radius of the base. Once you have calculated the radius, you can plug it into the formula above to find the center of the circle. This formula works because even though the cylinder is sheared, the cross-section is still circular and the center of the circle will be on the same plane as the center of the base of the cylinder. I hope this helps.