Hyperboloid

A hyperboloid is a surface created by deforming a hyperboloid of revolution using directional scalings, or more broadly, an affine transformation. 

A hyperboloid of revolution, also known as a circular hyperbola, is a surface created by rotating a hyperbola around one of its primary axes in geometry.

A quadric surface, or a surface defined as the zero sets of a polynomial of degree two in three variables, is known as a hyperboloid. 

A hyperboloid is a quadric surface that is not a cone or a cylinder, has a centre of symmetry, and intersects numerous planes to form hyperbolas. Three pairwise perpendicular axes of symmetry and three pairwise perpendicular planes of symmetry make up a hyperboloid.

The hyperboloid grapher is used for graphing the hyperboloid of one sheet which is the most complicated of all the quadric surfaces.

Einschaliges hyperboloid is a German word for hyperboloid shapes.

The hyperboloid shape is shown in the figure below.

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Hyperboloid Formula

If one picks a Cartesian coordinate system whose axes are the hyperboloid's axes of symmetry and the origin is the hyperboloid's centre of symmetry, one can define the hyperboloid using the equations given below:

\[\frac{x^{2}}{a^{2}}\] + \[\frac{y^{2}}{b^{2}}\] - \[\frac{z^{2}}{c^{2}}\] = 1

\[\frac{x^{2}}{a^{2}}\] + \[\frac{y^{2}}{b^{2}}\] - \[\frac{z^{2}}{c^{2}}\] = -1

When both hyperboloid surfaces are asymptotic to the cone of the equation, then we get zero on the right-hand side of the equation as follow:

\[\frac{x^{2}}{a^{2}}\] + \[\frac{y^{2}}{b^{2}}\] - \[\frac{z^{2}}{c^{2}}\] = 0

If a2 = b2 then the surface will be a hyperboloid of revolution. Otherwise, the axes are defined uniquely until the x-axis and y-axis are switched.

Types of Hyperboloid

There are two types of the hyperboloid.

• One Sheet Hyperboloid or Hyperbolic Hyperboloid 

• +1 on the right side of the hyperboloid formula.

\[\frac{x^{2}}{a^{2}}\] + \[\frac{y^{2}}{b^{2}}\] - \[\frac{z^{2}}{c^{2}}\] = 1

• A hyperbolic hyperboloid is a linked surface with every point having a negative Gaussian curvature. This means that near any point, the intersection of the hyperboloid and its tangent plane at the point is made up of two curve branches with distinct tangents. 

• These branches of curves are lines on the one-sheet hyperboloid, making it a doubly ruled surface.

• Rotating a hyperbola around its semi-minor axis is the most popular way to make a one-sheet hyperboloid of revolution.

• A parabolic hyperboloid is projectively identical to a one-sheet hyperboloid. A parabolic hyperboloid is a doubly-curved surface with a convex form along one axis and a concave form along with the other, resembling the shape of a saddle.

• Two Sheet Hyperboloid or Elliptic Hyperboloid 

• -1 on the right side of the hyperboloid formula.

\[\frac{x^{2}}{a^{2}}\] + \[\frac{y^{2}}{b^{2}}\] - \[\frac{z^{2}}{c^{2}}\] = -1

• There are no lines in the hyperboloid of two sheets. Every point on the surface has a positive Gaussian curvature and two connected components. As a result, the surface is convex in the sense that the tangent plane crosses the surface only at this point at each point.

• Any two-sheet revolution hyperboloid comprises circles. This is also true in the broader case, but it is less clear.

• A two-sheet hyperboloid is projectively identical to a sphere.

• The Gaussian curvature of a one-sheet hyperboloid is negative, while that of a two-sheet hyperboloid is positive. The hyperboloid of two sheets with another correctly chosen metric can also be used as a model for hyperbolic geometry, despite its positive curvature.

Conclusion

In construction, one-sheeted hyperboloid are employed, and the structures are known as hyperboloid structures. Because a hyperboloid is a doubly ruled surface, it can be constructed with straight steel beams at a lesser cost than other approaches. Cooling towers, particularly those in power plants, and a variety of other structures are examples of hyperboloids.