the lemniscate revolves about a tangent at the pole find sufaces of the solid generated?
Take a google for "lemniscate rotate around tangent at pole" and select the reference to
books google com
which shows content from the book
Advanced Engineering Mathematics, Volume 1 By H.C. Taneja
The article there shows the solution in great detail.
To understand the surfaces generated by a lemniscate revolving about a tangent at the pole, we first need to consider the properties of a lemniscate.
The lemniscate is a plane curve with a characteristic figure-eight shape (∞). It has a single point called the "pole" at which the tangent will be revolving around.
When the lemniscate revolves about a tangent at the pole, it generates a solid called a "lemniscate torus." A torus is a doughnut-shaped object formed by revolving a circular curve around an axis.
To visualize the surfaces of the lemniscate torus, imagine taking the lemniscate and rotating it around the tangent line at the pole. This rotation will create a hollow shape with a tube-like appearance, where the circular curve of the lemniscate forms the outer surface of the torus, and the tangent line becomes the axis of rotation.
The lemniscate torus has two surfaces: an outer surface and an inner surface. The outer surface is the curved outer boundary of the torus, while the inner surface is the curved inner boundary, which is connected to the outer surface. Both surfaces are continuous and meet at the point where the lemniscate intersects the tangent line.
In summary, when a lemniscate revolves about a tangent at the pole, it creates a lemniscate torus with an outer surface and an inner surface.