Problem 6 - APEX Calculus

Question

Consider the hyperbola \(x^{2}-y^{2}=1\) in the plane. If this hyperbola is rotated about the \(y\) -axis, what quadric surface is formed?

Step-by-Step Solution

Answer

The surface formed is a hyperboloid of revolution.

Step 1: Identify the given equation

The given hyperbola is described by the equation \(x^2 - y^2 = 1\). This defines a hyperbola centered at the origin in the \(xy\)-plane.

Step 2: Determine the axis of rotation

We are tasked to rotate the hyperbola about the \(y\)-axis. This means every point on the hyperbola will trace a circular path around the \(y\)-axis, generating a surface in three-dimensional space.

Step 3: Establish the generated surface

When a curve defined by \(x^2 - y^2 = 1\) is rotated around the \(y\)-axis, a three-dimensional surface is formed. Specifically, each point \((x, y)\) on the hyperbola generates a circle of radius \(|x|\) when rotated around the \(y\)-axis. This leads to the generation of a hyperboloid of revolution.

Step 4: Verify the type of hyperboloid

The original equation \(x^2 - y^2 = 1\) does not involve the \(z\)-axis. However, upon rotation, for each \(z\) corresponding to \(x^2 = 1 + y^2 + z^2\), we recognize the structure of a two-sheeted hyperboloid. As \(y\) or \(z\) increases, the \(x\) dimension adjusts, suggesting a hyperboloid of two sheets.

Step 5: Confirm the resulting equation

To formally describe the resulting surface, replace \(x^2\) with the expression for a circle radius in the \(xz\)-plane by adding \(z^2\) as it accommodates the rotational aspects. Thus, the equation becomes \(x^2 = y^2 + z^2 + 1\), which is a hyperboloid of one sheet.

Key Concepts

Rotation of Curves

When curves are rotated in three-dimensional space, they generate surfaces, creating a fascinating geometric phenomenon. Rotating a two-dimensional curve about an axis transforms it into a three-dimensional surface. In the case of the hyperbola, which is defined by the equation \( x^2 - y^2 = 1 \), rotating it around the \( y \)-axis causes every point on the curve to trace out a circular path.

• This circular path forms a surface in three dimensions.
• Each point on the hyperbola expands into a circle, which is integrated into the overall structure of the surface.
• Such rotations are a prelude to understanding complex geometries and forms in higher-dimensional spaces.

This rotational motion is key to visualizing how simple curves generate complex surfaces in 3D. By understanding this concept, you'll gain insight into the creation of solids from flat shapes.

Hyperbola

A hyperbola is a distinct type of curve defined by its equation, such as \( x^2 - y^2 = 1 \). Hyperbolas take a characteristic shape that resembles two opposite-facing arcs. Let's delve into some of its unique properties:

• Hyperbolas consist of two disconnected curves called branches.
• It is symmetrical about its principal axes, which in this standard equation are the \( x \)-axis and \( y \)-axis.
• The equation defines a pair of intersecting lines at larger values, known as asymptotes, which the branches approach but never touch.

Understanding hyperbolas is essential for grasping the broader category of conic sections, which include other curves like ellipses and parabolas. These curves have various applications in physics, astronomy, and engineering, such as in optimizing satellite dish shapes.

Three-Dimensional Surfaces

Three-dimensional surfaces are forms that extend through space, having width, depth, and height. They're generated from simple curves by rotations or other transformations. When the two-dimensional hyperbola \( x^2 - y^2 = 1 \) is rotated about the \( y \)-axis, it expands into a surface known as a hyperboloid. Key points about these surfaces include:

• They are critical in fields like architecture, where their aesthetic and structural properties are utilized.
• Unlike flat surfaces, they can support complex interactions with light, shadow, and texture.
• Hyperboloids of one sheet, generated in this case, show up regularly in industrial designs such as cooling towers.

By forming such surfaces, one can visualize and apply mathematical concepts in tangible, real-world scenarios.

Quadric Surfaces

The term 'quadric surfaces' refers to a group of surfaces in three-dimensional space defined by second-degree algebraic equations. These surfaces, including spheres, ellipsoids, and hyperboloids, have unique characteristics based on their equations. For instance, our rotated hyperbola becomes a hyperboloid of one sheet when integrated into the equation \( x^2 = y^2 + z^2 + 1 \). Here's what makes quadric surfaces interesting:

• They display symmetry, which aids in their aesthetic appeal and structural capabilities.
• Quadric surfaces are employed in computer graphics for rendering realistic scenes due to their complex shapes yet computational simplicity.
• They provide insights into physical phenomena, such as gravitational potential surfaces.

Understanding quadric surfaces enhances one's ability to interpret complex spatial relationships and apply mathematical theories to diverse fields, such as engineering and art.