Surface Function of Tilted & Rotated Circle Around Z-Axis

[zs96742]

My first idea is this will result in a elliptic torus. The horizontal semi-axis a=R and the vertical semi-axis b=R*cos(beta). assuming the titled or inclined angle is beta. The distance away from the z-axisis c and it is a constant. But it looks not when I plot the surface in 3D using the elliptic equation given on wolfram elliptic.

The green dot points are generated by rotating the red line around the z axis and then plot the corresponding circle in 3D space.

The surface I created using the torus equation is somehow like the bottom one:

 

[jvicens]

Hello,

Thank you for sharing your idea! I can offer some insights and feedback on your proposed elliptic torus.

Firstly, an elliptic torus is a valid geometric shape that can be created by rotating an ellipse around an axis. In this case, the ellipse has a horizontal semi-axis of R and a vertical semi-axis of R*cos(beta). This means that the shape of the torus will change depending on the value of beta, which represents the angle at which the ellipse is tilted or inclined.

However, I noticed that you mentioned a constant distance c away from the z-axis. In order for this to be a valid torus, the distance from the center of the torus to the z-axis should be the same at every point around the torus. This means that c should be a function of beta, rather than a constant.

Additionally, when plotting the surface in 3D using the elliptic equation, it is important to note that the resulting shape may not look exactly like a torus. This is because the equation used by Wolfram is a general equation for an elliptic surface, and not specifically for a torus.

I also noticed that you mentioned generating the green dot points by rotating a red line around the z-axis. In order to create a true torus, the red line should be an ellipse, rather than a straight line. This will ensure that the resulting surface is smooth and continuous.

Overall, your idea of creating an elliptic torus is valid, but some adjustments may need to be made in order to accurately represent this shape in 3D space. I encourage you to continue exploring and experimenting with different parameters to see how the shape changes.

Best of luck with your project!