Chapter 11: Problem 13
Name and sketch the graph of each of the following equations in three-space. $$ x^{2}-z^{2}+y=0 $$
Short Answer
Expert verified
The graph is a saddle-shaped hyperbolic paraboloid in 3D space.
Step by step solution
01
Identify the Type of Surface
The equation given is \( x^2 - z^2 + y = 0 \). Rewrite it as \( y = z^2 - x^2 \), which resembles the equation of a hyperbolic paraboloid. This is a type of saddle-shaped surface in three-dimensional space.
02
Understand the Surface Shape
Surface \( y = z^2 - x^2 \) represents a hyperbolic paraboloid. Such surfaces curve upwards in some directions and downwards in others, thus forming a saddle shape. The axis passing through the origin where parabola cross sections form can be discerned from the variables involved.
03
Sketch in the Coordinate Planes
- In the \(xy\)-plane, fix \(z=0\) to get \(y = -x^2\), a parabola opening downwards.- In the \(yz\)-plane, fix \(x=0\) to get \(y = z^2\), a parabola opening upwards.- Along the \(xz\)-plane by setting \(y=0\), we have \(z^2 = x^2\), showing the lines \(z = x\) and \(z = -x\).
04
Combine Information to Sketch 3D Graph
The graph is symmetric with respect to the \(y\)-axis. From step 3, draw a saddle shape by using the behavior along the coordinate planes: upward parabolas in \(yz\)-plane and downward in \(xy\)-plane. Lines in the \(xz\)-plane indicate where the surface flattens.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Hyperbolic Paraboloid
When you hear the term "hyperbolic paraboloid," it may sound complex, but it's actually a fascinating surface in 3D geometry. This surface gets its name because it takes on a shape resembling both a hyperbola and a parabola. Think of a hyperbolic paraboloid as a saddle that curves up in one direction and down in another.
The general equation for a hyperbolic paraboloid can be given as \( y = z^2 - x^2 \). Notice that unlike other paraboloids, the equation involves subtraction between squared terms. That's what gives us the hyperbolic shape. If you imagined slicing through the surface along distinct planes, you'd see cross-sections that look like either parabolas or hyperbolas, depending on the direction.
Because of its unique shape, hyperbolic paraboloids are quite common in architectural designs, from roofs to bridges, due to their stability and aesthetic form.
The general equation for a hyperbolic paraboloid can be given as \( y = z^2 - x^2 \). Notice that unlike other paraboloids, the equation involves subtraction between squared terms. That's what gives us the hyperbolic shape. If you imagined slicing through the surface along distinct planes, you'd see cross-sections that look like either parabolas or hyperbolas, depending on the direction.
Because of its unique shape, hyperbolic paraboloids are quite common in architectural designs, from roofs to bridges, due to their stability and aesthetic form.
Coordinate Planes
In three-dimensional geometry, coordinate planes are essential for understanding and visualizing surfaces like the hyperbolic paraboloid. These planes interact to form a grid in 3D space, which includes:
- The **xy-plane**, where the value of \(z\) is zero.
- The **yz-plane**, where \(x\) is zero.
- The **xz-plane**, where \(y\) is zero.
In the context of our surface, exploring the shape in these planes helps us understand its full form:
- In the **xy-plane**: Set \(z = 0\) in the equation \(y = z^2 - x^2\), and you get \(y = -x^2\). Here, you'll see a downward-opening parabola.
- In the **yz-plane**: By setting \(x = 0\), the equation becomes \(y = z^2\), showing an upward-opening parabola.
- In the **xz-plane**: With \(y = 0\), the equation becomes \(z^2 = x^2\), which forms lines along \(z = x\) and \(z = -x\).
Each of these planes provides a "slice" of the 3D space, aiding us in visualizing and sketching the full surface. Understanding these intersections helps in combining all sections to see the entire surface in our minds.
- The **xy-plane**, where the value of \(z\) is zero.
- The **yz-plane**, where \(x\) is zero.
- The **xz-plane**, where \(y\) is zero.
In the context of our surface, exploring the shape in these planes helps us understand its full form:
- In the **xy-plane**: Set \(z = 0\) in the equation \(y = z^2 - x^2\), and you get \(y = -x^2\). Here, you'll see a downward-opening parabola.
- In the **yz-plane**: By setting \(x = 0\), the equation becomes \(y = z^2\), showing an upward-opening parabola.
- In the **xz-plane**: With \(y = 0\), the equation becomes \(z^2 = x^2\), which forms lines along \(z = x\) and \(z = -x\).
Each of these planes provides a "slice" of the 3D space, aiding us in visualizing and sketching the full surface. Understanding these intersections helps in combining all sections to see the entire surface in our minds.
Saddle Surfaces
Saddle surfaces are a special kind of surface that include both upward and downward curves like a saddle on a horseback. The hyperbolic paraboloid is a quintessential example of a saddle surface, with its distinct shape and unique geometry.
What's fascinating about saddle surfaces like the hyperbolic paraboloid is their symmetry and curvature. They are minimal surfaces, which means that they represent the smallest possible area for a surface bordering a closed loop. That's why you find them in nature and architecture where efficiency is crucial.
The saddle surface curves along more than one axis. On the one hand, the surface descends as it moves away from a central axis (like along the \(x\)-direction in our equation). On the other, it ascends away from the axis in perpendicular directions (like along the \(z\)-direction).
Visualizing this in 3D can be like seeing a saddle on a horse or even the iconic shape of a Pringles chip. By incorporating these natural concepts, you can better imagine the entire form of a hyperbolic paraboloid and understand how mathematicians and architects use it in various fields.
What's fascinating about saddle surfaces like the hyperbolic paraboloid is their symmetry and curvature. They are minimal surfaces, which means that they represent the smallest possible area for a surface bordering a closed loop. That's why you find them in nature and architecture where efficiency is crucial.
The saddle surface curves along more than one axis. On the one hand, the surface descends as it moves away from a central axis (like along the \(x\)-direction in our equation). On the other, it ascends away from the axis in perpendicular directions (like along the \(z\)-direction).
Visualizing this in 3D can be like seeing a saddle on a horse or even the iconic shape of a Pringles chip. By incorporating these natural concepts, you can better imagine the entire form of a hyperbolic paraboloid and understand how mathematicians and architects use it in various fields.
