Hyperboloid equation types one sheet and two sheets with properties and solved examples
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.
[Image will be Uploaded Soon]
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 in Three Dimensional Coordinate Geometry
1. What is a hyperboloid in mathematics?
A hyperboloid is a three-dimensional quadric surface formed by revolving a hyperbola around one of its axes. It is defined by a second-degree equation in x, y, and z. There are two main types:
- Hyperboloid of one sheet (connected surface)
- Hyperboloid of two sheets (two separate surfaces)
2. What is the standard equation of a hyperboloid of one sheet?
The standard equation of a hyperboloid of one sheet centered at the origin is x²/a² + y²/b² − z²/c² = 1. In this equation:
- a, b, c are positive constants.
- The variable with the negative term determines the axis of symmetry.
- All cross-sections parallel to the xy-plane are ellipses.
3. What is the standard equation of a hyperboloid of two sheets?
The standard equation of a hyperboloid of two sheets centered at the origin is z²/c² − x²/a² − y²/b² = 1. In this form:
- The positive term determines the axis along which the surface opens.
- The graph consists of two separate surfaces (two sheets).
- Cross-sections parallel to the xy-plane exist only when |z| > c.
4. What is the difference between a hyperboloid of one sheet and two sheets?
The main difference is that a hyperboloid of one sheet is connected, while a hyperboloid of two sheets has two separate parts. Key differences:
- One sheet: equation has two positive terms and one negative term equal to 1.
- Two sheets: equation has one positive term and two negative terms equal to 1.
- One sheet forms a single continuous surface.
- Two sheets form two disconnected surfaces.
5. How do you identify a hyperboloid from its equation?
You identify a hyperboloid by checking the signs of the squared terms in its quadratic equation. Steps:
- If the equation equals 1 and has both positive and negative squared terms, it is a hyperboloid.
- Two positive and one negative term → hyperboloid of one sheet.
- One positive and two negative terms → hyperboloid of two sheets.
- If the equation equals −1, multiply both sides by −1 and recheck.
6. What are the cross-sections of a hyperboloid?
The cross-sections of a hyperboloid can be ellipses or hyperbolas depending on the slicing plane. For example, for x²/a² + y²/b² − z²/c² = 1:
- Horizontal slices (z = constant) → ellipses.
- Vertical slices (x = constant or y = constant) → hyperbolas.
7. Can you give an example of a hyperboloid of one sheet?
An example of a hyperboloid of one sheet is x²/4 + y²/9 − z²/16 = 1. Here:
- a² = 4, b² = 9, c² = 16
- The negative term is −z²/16, so the axis of symmetry is along the z-axis.
- The surface is connected and shaped like a cooling tower.
8. What are the real-life applications of a hyperboloid?
A hyperboloid is used in engineering and architecture because of its structural strength and stability. Common applications include:
- Cooling towers in power plants
- Hyperboloid structures and towers
- Reflector antennas
- Structural shell designs
9. How is a hyperboloid related to a hyperbola?
A hyperboloid is formed by revolving a hyperbola around one of its axes. Specifically:
- Rotating a hyperbola about its transverse axis produces a hyperboloid of two sheets.
- Rotating about its conjugate axis produces a hyperboloid of one sheet.
10. What is the general equation of a hyperboloid?
The general quadratic form of a hyperboloid is Ax² + By² + Cz² + D = 0, where the coefficients of x², y², and z² have different signs. To classify it:
- If two squared terms have the same sign and one is opposite → hyperboloid.
- After rearranging to standard form, compare with x²/a² + y²/b² − z²/c² = 1 or z²/c² − x²/a² − y²/b² = 1.
