Surface of revolution

A surface of revolution is a specific type of surface in Euclidean space. It is created by taking a single curve and rotating it one full revolution around a fixed line. This line is known as the axis of rotation. The original curve used to create the surface is called the generatrix. Usually, the axis does not intersect the generatrix, except perhaps at its endpoints.

When this rotation creates a three-dimensional object, the space inside is called a solid of revolution.

Different types of curves produce very different geometric results. If the generatrix is a straight line parallel to the axis, the result is a cylindrical surface. If the straight line is not parallel to the axis, it creates a conical surface. A circle can generate a sphere if it is rotated around its own diameter. However, if you rotate a circle around an axis that does not intersect its interior, you create a torus. A torus is a ring-shaped object, often compared to a donut.

If you rotate a square around an axis parallel to one of its edges, you produce a hollow square-section ring. These shapes with holes in the middle are called toroids.

We can understand these surfaces by looking at their cross sections. If you slice a surface of revolution with a plane that passes through the axis, the slice is called a meridional section. These sections are actually identical to the original generatrix. If you instead slice the surface with a plane perpendicular to the axis, the cross sections are always circles. This circular property is very important in geometry. For example, certain quadratic surfaces like elliptic paraboloids and specific hyperboloids are surfaces of revolution. We identify them because all their cross sections perpendicular to the axis are perfect circles.

Calculating the surface area of these shapes requires the tools of calculus. If a curve is described by parametric functions, the area can be found using an integral. This method is the calculus equivalent of Pappus's centroid theorem. The formula uses a small segment of the arc length, which comes from the Pythagorean theorem. It also considers the path of the centroid, or the center, of that small segment.

For a continuous curve described by a function, we can use specific integrals depending on whether we rotate around the x-axis or the y-axis. For instance, a sphere with a unit radius of one has a surface area of exactly four times pi.

Mathematicians also study a special category called minimal surfaces of revolution. A minimal surface is one that minimizes the surface area between two given points. Finding the specific curve that produces this minimum area is a classic problem in the calculus of variations. There are only two types of surfaces that are both surfaces of revolution and minimal surfaces. The first is a simple flat plane. The second is a more complex shape known as a catenoid.

Geometry also explores how paths move across these surfaces. These paths are called geodesics. On any surface of revolution, the meridians are always geodesics. This means they are the shortest paths between points along those specific slices. Other geodesics on the surface are governed by a mathematical rule called Clairaut's relation. This helps scientists and mathematicians understand how objects might move or how light might travel across curved shapes.

Finally, we can describe these surfaces using coordinate expressions. If we rotate a curve around the x-axis, we can use parameters to define the x, y, and z coordinates. This allows us to map the surface precisely in three-dimensional space. We can also use multivariable integration or surface integrals to derive these areas. By using these different mathematical perspectives, we can fully describe the complex geometry of everything from simple cylinders to complex toroids.