Chapter 10: Problem 5
Consider the hyperbola \(x^{2}-y^{2}=1\) in the plane. If this hyperbola is rotated about the \(x\) -axis, what quadric surface is formed?
Short Answer
Expert verified
A hyperboloid of one sheet.
Step by step solution
01
Identify the Original Equation
The given hyperbola is \(x^2 - y^2 = 1\). This equation represents a hyperbola centered at the origin, opening along the x-axis in the xy-plane.
02
Understand Rotation and Quadric Surfaces
Rotating a curve around an axis can transform the curve into a surface. In this case, rotating the hyperbola \(x^2 - y^2 = 1\) about the x-axis produces a quadric surface. Quadric surfaces can be classified into types based on their equations.
03
Write Equation of Quadric Surface
To find the equation of the surface formed by the rotation, replace \(y^2\) in the equation with \(y^2 + z^2\), since rotation around the \(x\)-axis extends the hyperbola into three dimensions. The new equation becomes: \[x^2 - (y^2 + z^2) = 1\].
04
Identify the Quadric Surface
The equation \(x^2 - y^2 - z^2 = 1\) represents a hyperboloid of one sheet. This quadric has a hyperbolic trace in the x-direction and circular traces in planes perpendicular to the x-axis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Hyperbola
A hyperbola is an important type of conic section. It is characterized by its unique open curve, which is made up of two disconnected parts called branches. Mathematically, it is described by an equation of the form \(x^2 - y^2 = 1\), which indicates that the hyperbola is centered at the origin.
In a hyperbola, every point is determined by a relationship between two fixed points called foci. The difference of the distances to these foci is constant for any point on the hyperbola. This is quite different from circles or ellipses where the sum or the single distance to focal points determines the shape.
When you see a hyperbola in the XY-plane, it opens towards the direction of the variable with the positive coefficient. In our example, this is along the x-axis. Each branch approaches as straight lines known as asymptotes which the hyperbola will get infinitely close to but never touch.
In a hyperbola, every point is determined by a relationship between two fixed points called foci. The difference of the distances to these foci is constant for any point on the hyperbola. This is quite different from circles or ellipses where the sum or the single distance to focal points determines the shape.
When you see a hyperbola in the XY-plane, it opens towards the direction of the variable with the positive coefficient. In our example, this is along the x-axis. Each branch approaches as straight lines known as asymptotes which the hyperbola will get infinitely close to but never touch.
Hyperboloid of One Sheet
A hyperboloid of one sheet is a fascinating three-dimensional surface. It emerges when a hyperbola is rotated around one of its axes.
Imagine taking our initial hyperbola, \(x^2 - y^2 = 1\), and spinning it around the x-axis. As it rotates, every point on the hyperbola traces out a path in space. The resulting surface is modeled by the equation \(x^2 - y^2 - z^2 = 1\).
This equation shows that the hyperboloid has two hyperbolic sections and one circular section. In this form, it appears as a smooth, continuous shape that wraps around similar to a distorted cylinder. This is why hyperboloids of one sheet are useful in architecture and engineering, creating strong, stable structures like cooling towers or water towers.
Imagine taking our initial hyperbola, \(x^2 - y^2 = 1\), and spinning it around the x-axis. As it rotates, every point on the hyperbola traces out a path in space. The resulting surface is modeled by the equation \(x^2 - y^2 - z^2 = 1\).
This equation shows that the hyperboloid has two hyperbolic sections and one circular section. In this form, it appears as a smooth, continuous shape that wraps around similar to a distorted cylinder. This is why hyperboloids of one sheet are useful in architecture and engineering, creating strong, stable structures like cooling towers or water towers.
Rotation of Conic Sections
Rotation of conic sections is a technique that helps transform two-dimensional curves into three-dimensional surfaces.
By taking a flat shape like our hyperbola \(x^2 - y^2 = 1\) in the xy-plane and spinning it around an axis (in this case, the x-axis), a complex surface forms.
This manipulation translates simply in equations: the variable being rotated stays the same, while new variables are added for every point that the original curve could sweep through. In our example, \(y^2\) becomes \(y^2 + z^2\).
This technique can not only create hyperboloids, but also spheres and ellipsoids depending on the starting conic section and axis of rotation.
By taking a flat shape like our hyperbola \(x^2 - y^2 = 1\) in the xy-plane and spinning it around an axis (in this case, the x-axis), a complex surface forms.
This manipulation translates simply in equations: the variable being rotated stays the same, while new variables are added for every point that the original curve could sweep through. In our example, \(y^2\) becomes \(y^2 + z^2\).
This technique can not only create hyperboloids, but also spheres and ellipsoids depending on the starting conic section and axis of rotation.
Three-Dimensional Geometry
Three-dimensional geometry provides the tools to explore shapes and surfaces beyond flat, two-dimensional planes.
In 3D geometry, shapes like hyperboloids arise when we give height or depth to two-dimensional curves. With an extra dimension, we can explore a wider range of surfaces not possible on a flat plane.
These concepts are crucial in fields like physics, computer graphics, and any discipline requiring a spatial understanding of shape and form. Understanding quadric surfaces like hyperboloids provides insight into the behavior of light, the structure of molecules, and much more.
In 3D geometry, shapes like hyperboloids arise when we give height or depth to two-dimensional curves. With an extra dimension, we can explore a wider range of surfaces not possible on a flat plane.
- In 3D, we define points with (x, y, z) coordinates.
- Shapes can have volume as well as area.
- Rotating shapes around axes creates new dimensions which are described with surface equations.
These concepts are crucial in fields like physics, computer graphics, and any discipline requiring a spatial understanding of shape and form. Understanding quadric surfaces like hyperboloids provides insight into the behavior of light, the structure of molecules, and much more.
