To make it easier for the reader to imagine what a hyperboloid is - a three-dimensional object - first you need to consider the hyperbola curve of the same name that fits in two-dimensional space.
A hyperbola has two axes: the real axis, in this figure, coinciding with the abscissa axis, and imaginary - with the ordinate axis. If you mentally begin to rotate the hyperbola equation around its imaginary axis, then the surface “swept” by the curve will be a one-cavity hyperboloid.
If, however, we begin to rotate the hyperbola around its real axis, then each of the two "halves" of the curve will form its own separate surface, and together it will be called a two-sheeted hyperboloid.
Obtained by rotation of the corresponding planar curve, they are called, respectively, hyperboloids of revolution. In all directions perpendicular to the axis of rotation, they retain the parameters that belong to the rotated curve. This is generally not the case.
Hyperboloid equation
In general, a surface can be defined by the following equations in Cartesian coordinates (x, y, z):
In the case of a hyperboloid of revolution, its symmetry about the axis around which it was rotated is expressed in the equality of the coefficients a = b.
Hyperboloid Characteristics
He has a trick. We know that curves on a plane have tricks — in the case of a hyperbola, for example, the modulus of the difference of distances from an arbitrary point on a hyperbole to one focus and the second is constant by definition, in fact, focus points.
Upon transition to three-dimensional space, the definition remains practically unchanged: the foci are again two points, and the difference in the distances from them to an arbitrary point belonging to the surface of the hyperboloid is constant. As you can see, only the third coordinate appeared at the changes at all possible points, because now they are set in space. Generally speaking, the definition of focus is equivalent to identifying the type of curve or surface: speaking about how the surface points are located relative to the foci, we actually answer the question of what a hyperboloid is and how it looks.
It is worth remembering that the hyperbola has asymptotes - straight lines, to which its branches tend at infinity. If, when constructing a hyperboloid of rotation, we mentally rotate the asymptotes together with the hyperbola, then in addition to the hyperboloid, we also get a cone called asymptotic. Asymptotic cone is present in both single-celled and double-celled hyperboloids.
Another important characteristic that exists only in a single-sheeted hyperboloid is rectilinear generators. As the name implies, these are lines, and they completely lie on a given surface. Two rectilinear generators pass through each point of a single-sheeted hyperboloid. They belong respectively to two families of lines, which are described by the following systems of equations:
Thus, a single-sheeted hyperboloid can be composed entirely of an infinite number of straight lines of two families, with each line of one of them intersecting with all the lines of the other. Surfaces corresponding to such properties are called ruled; they can be built by rotating one straight line. The definition through the mutual arrangement of lines (rectilinear generators) in space can also serve as an unambiguous designation of what a hyperboloid is.
Interesting properties of a hyperboloid
The second-order curves and their corresponding surfaces of revolution each have interesting optical properties associated with the foci. In the case of a hyperboloid, this is formulated as follows: if a ray is emitted from one focus, then, reflected from the nearest “wall”, it will take such a direction as if it were coming from the second focus.
Hyperboloids in life
Most likely, most readers began their acquaintance with analytic geometry and second-order surfaces with Alexei Tolstoy’s science fiction novel “Engineer Hyperboloid Garin”. However, the writer either did not know well what a hyperboloid was, or sacrificed accuracy for the sake of artistry: the described invention is more likely a physical paraboloid that collects all the rays in one focus (while the optical properties of the hyperboloid are associated with ray scattering) .
In architecture, the so-called hyperboloid constructions are very popular: these are structures in the form of a single-cavity hyperboloid or a hyperbolic paraboloid. The fact is that only these second-order rotation surfaces have rectilinear generators: thus, a curved structure can be built only from straight beams. The advantages of such structures are the ability to withstand heavy loads, for example, from the wind: the shape of the hyperboloid is used in the construction of high structures, for example, television towers.