Problem 27
Question
Sketch the surfaces HYPERBOLOIDS $$x^{2}+y^{2}-z^{2}=1$$
Step-by-Step Solution
Verified Answer
The surface is a hyperboloid of one sheet symmetric about the \(z\)-axis.
1Step 1: Identify the Surface Type
The given equation is \(x^{2} + y^{2} - z^{2} = 1\). This is identified as the equation of a hyperboloid. To determine the type, observe the signs of the variables. Since we have two positive squares and one negative square, it is a hyperboloid of one sheet.
2Step 2: Understanding the Axis of the Hyperboloid
The variable with the negative coefficient, which is \(z\) in this case, indicates that the hyperboloid is symmetric about the \(z\)-axis.
3Step 3: Analyzing Cross-sections
For any fixed \(z = c\), the cross-section in the \(xy\)-plane is a circle: \(x^{2} + y^{2} = c^{2} + 1\). As \(z\) varies, the radius of the circle changes, indicating that the hyperboloid expands as it moves away from \(z = 0\).
4Step 4: Sketching the Hyperboloid
Begin by sketching the \(z = 0\) plane where the section is a circle with radius 1 (since \(x^{2} + y^{2} = 1\)). As \(z\) increases or decreases, draw larger concentric circles, maintaining the symmetry about the \(z\)-axis. The hyperboloid will resemble a continuous surface, opening along the \(z\)-axis, expanding outward as \(z\) moves away from zero.
Key Concepts
Equation of hyperboloidCross-sectionsSymmetry about z-axis
Equation of hyperboloid
The equation of a hyperboloid is a mathematical representation of a 3D surface. In the equation \(x^2 + y^2 - z^2 = 1\), we notice a specific arrangement of terms. Here, you have two positive terms \(x^2\) and \(y^2\), and one negative term \(-z^2\). This indicates a hyperboloid of one sheet.
- The balance between these terms determines the shape of the surface.
- The constant on the right side of the equation, \(1\), helps define the scale and size of the shape.
Visualizing this, think of the hyperboloid as a smooth, continuous object that opens outward from a central point. It has a double-cone-like structure that forms one continuous surface. The mathematical expression not only tells us what kind of surface we're dealing with but also provides details about its geometry.
- The balance between these terms determines the shape of the surface.
- The constant on the right side of the equation, \(1\), helps define the scale and size of the shape.
Visualizing this, think of the hyperboloid as a smooth, continuous object that opens outward from a central point. It has a double-cone-like structure that forms one continuous surface. The mathematical expression not only tells us what kind of surface we're dealing with but also provides details about its geometry.
Cross-sections
Cross-sections help us understand what the shape of the hyperboloid looks like at particular levels. Imagine cutting through the object at different heights.
- For a hyperboloid given by \(x^2 + y^2 - z^2 = 1\), the cross-sections tell us a lot.
- If we fix \(z = c\) (a specific height), our equation becomes a circle: \(x^2 + y^2 = c^2 + 1\).
This means at any constant \(z\)-level, the intersection with the \(xy\)-plane is a circle. As you move up or down the \(z\)-axis, these circles change in size.
- At \(z = 0\), the circle's radius is 1 since \(x^2 + y^2 = 1\).
- As \(z\) increases, the radius expands, illustrating how the hyperboloid stretches as it moves along the \(z\)-axis.
- For a hyperboloid given by \(x^2 + y^2 - z^2 = 1\), the cross-sections tell us a lot.
- If we fix \(z = c\) (a specific height), our equation becomes a circle: \(x^2 + y^2 = c^2 + 1\).
This means at any constant \(z\)-level, the intersection with the \(xy\)-plane is a circle. As you move up or down the \(z\)-axis, these circles change in size.
- At \(z = 0\), the circle's radius is 1 since \(x^2 + y^2 = 1\).
- As \(z\) increases, the radius expands, illustrating how the hyperboloid stretches as it moves along the \(z\)-axis.
Symmetry about z-axis
Symmetry in geometric terms often means that a shape remains the same when flipped or rotated around a particular axis. For the hyperboloid \(x^2 + y^2 - z^2 = 1\), this symmetry is about the \(z\)-axis.
- This means if you rotate the hyperboloid around the \(z\)-axis, the object looks identical.
- The \(z\)-axis serves as a central line around which the entire object is consistently shaped.
- Another indication of this symmetry is how the cross-section circles remain centred around the \(z\)-axis across all heights.
This symmetry contributes to the object’s aesthetic and structural balance, making the hyperboloid a fascinating subject in mathematics and architecture alike.
- This means if you rotate the hyperboloid around the \(z\)-axis, the object looks identical.
- The \(z\)-axis serves as a central line around which the entire object is consistently shaped.
- Another indication of this symmetry is how the cross-section circles remain centred around the \(z\)-axis across all heights.
This symmetry contributes to the object’s aesthetic and structural balance, making the hyperboloid a fascinating subject in mathematics and architecture alike.
Other exercises in this chapter
Problem 26
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