

What Is a Hyperboloid? Definition, Real-World Uses & Key Formulas
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.
FAQs on Hyperboloid: Meaning, Types & Applications
1. What is a hyperboloid in three-dimensional geometry?
A hyperboloid is a three-dimensional quadric surface, which means it's a surface described by a second-degree equation in three variables (x, y, z). Visually, it is the surface created by rotating a hyperbola around one of its axes of symmetry. Its general equation, when centred at the origin, is a variation of (x²/a²) + (y²/b²) - (z²/c²) = 1, where a, b, and c are the semi-axes that define its dimensions.
2. What are the two main types of hyperboloids and their standard equations?
The two primary types of hyperboloids are distinguished by their shape and mathematical equation:
Hyperboloid of One Sheet: This is a single, connected surface that looks like an infinitely extended tube or hourglass. Its standard equation has one negative term, such as (x²/a²) + (y²/b²) - (z²/c²) = 1.
Hyperboloid of Two Sheets: This consists of two separate, disjoint surfaces that open away from each other, like two bowls facing opposite directions. Its standard equation has two negative terms, such as -(x²/a²) - (y²/b²) + (z²/c²) = 1 or (x²/a²) - (y²/b²) - (z²/c²) = 1.
3. What is the fundamental difference between a hyperboloid of one sheet and two sheets?
The fundamental difference lies in their connectivity and geometric structure. A hyperboloid of one sheet is a single, continuous surface, meaning you can travel between any two points on it without leaving the surface. In contrast, a hyperboloid of two sheets is composed of two completely separate pieces. This difference is determined by the number of negative signs in their standard equations; one negative sign results in a connected surface, while two negative signs result in a disconnected one.
4. What are some real-world examples and applications of hyperboloids?
Hyperboloid shapes are used in various fields due to their unique structural properties. Key examples and applications include:
Architecture and Engineering: The most famous example is the use of hyperboloid structures for cooling towers at power plants. Their shape provides exceptional structural stability with minimal material.
Mechanical Engineering: Hyperboloid gears, or hypoid gears, are used in vehicle transmissions to transfer torque between non-intersecting shafts.
Everyday Objects: Some lampshades, decorative vases, and modern furniture designs utilise the elegant curves of a hyperboloid.
5. How is a hyperboloid of revolution formed?
A hyperboloid of revolution is formed by taking a two-dimensional hyperbola and rotating it 360 degrees around one of its principal axes. The choice of axis determines the type of hyperboloid created:
Rotating a hyperbola around its conjugate axis (the axis that does not intersect the hyperbola) generates a hyperboloid of one sheet.
Rotating a hyperbola around its transverse axis (the axis that passes through the vertices) generates a hyperboloid of two sheets.
6. Why are hyperboloid shapes used for building structures like cooling towers?
The primary reason is its property as a 'doubly ruled surface'. This means that a hyperboloid of one sheet, despite its curved appearance, can be constructed entirely from a lattice of straight beams. This makes it possible to build a very strong, stable, and tall curved structure using simple, straight materials like steel or concrete girders, which is more cost-effective and structurally sound than other methods for creating such large, curved forms.
7. How can you tell the difference between a hyperboloid and a hyperbolic paraboloid?
While both are quadric surfaces with 'hyperbolic' in their names, they are structurally very different. The easiest way to distinguish them is by their shape and cross-sections:
A hyperboloid has cross-sections that are either ellipses or circles in one plane and hyperbolas in the other two.
A hyperbolic paraboloid is a saddle-shaped surface. Its cross-sections are hyperbolas in one orientation and parabolas in another. It does not have any circular or elliptical cross-sections.
Their equations are also different; a hyperbolic paraboloid typically involves a linear term (like z) rather than a squared term (z²).

















