Visualizing 3D Graph of x^2+y^2-z^2=1

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In summary, the equation x^2+y^2-z^2=1 creates a 3D surface with cross sections that are circles in the x-y plane, and both ellipses and hyperbolas in the x-z and y-z planes. The size and shape of the cross sections vary with the value of z, with the smallest circle occurring at z=0. The hyperbola traces in the x-z and y-z planes are due to the increasing radius of the circles as z increases.
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This is just a coursework question ( I didn't know where to post this). When I find the traces of an equation ( let's say x^2+y^2-z^2=1 for the sake of argument). How does it affect my graph if one part of the equation is an ellipse and the other is an hyperbola? I mean in this case I would expect to be like this

When z=0

x^2+y^2=1 --> ellipse (well a circle but a circle is an ellipse)

when x=0

y^2-z^2=1 --> hyperbola

when y = 0

x^2-z^2=1 --> hyperbola

How does this affect the graph in 3d? I mean if it is both an ellipse and hyperbola, I can't visualize it geometrically.

Sorry if this is the wrong section
 
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MarcL said:
This is just a coursework question ( I didn't know where to post this). When I find the traces of an equation ( let's say x^2+y^2-z^2=1 for the sake of argument). How does it affect my graph if one part of the equation is an ellipse and the other is an hyperbola? I mean in this case I would expect to be like this

When z=0

x^2+y^2=1 --> ellipse (well a circle but a circle is an ellipse)

when x=0

y^2-z^2=1 --> hyperbola

when y = 0

x^2-z^2=1 --> hyperbola

How does this affect the graph in 3d? I mean if it is both an ellipse and hyperbola, I can't visualize it geometrically.

Sorry if this is the wrong section
This is the right section, but you need to include the problem template, not just discard it. It's there for a reason.

The traces are just the cross sections of the surface in the x-y, x-z, and y-z coordinate planes. You are not limited to just those cross sections. It might be useful to calculate the cross-sections in some other planes, such as z = 1 and z = -1. [STRIKE]You might also note that the surface doesn't extend above the plane z = 1 or below the plane z = -1.[/STRIKE]
Edit: Deleted a sentence that resulted from misreading the problem.
 
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It wasn't an actual problem, I was reading in my book, there wasn't a problem per se. However, how do you notice it doesn't go above 1 and -1? I mean is it for this equation?
 
  • #4
I steered you wrong on that, by misreading a sign. Write the equation of the surface as x2 + y2 = z2 + 1, and look at what happens for cross sections that are perpendicular to the z-axis (i.e., horizontal cross sections).

Each horizontal section is a circle whose radius increases as z increases. Due to symmetry, the cross sections below the x-y plane look the same as those above it. The minimum circle comes when z = 0.

Do the hyperbola traces in the x-z and y-z planes start to make sense now?
 
  • #5
Yeah in the y-z I can see it how they'll never touch ( correct me if I'm wrong) same goes for x-z ( again correct me if I'm wrong)
 

1. How do I plot a 3D graph of x^2+y^2-z^2=1?

To plot a 3D graph of x^2+y^2-z^2=1, you will need to use a graphing software or program that supports 3D graphing. Some examples include Desmos, GeoGebra, or MATLAB. Once you have opened the software, you can input the equation x^2+y^2-z^2=1 and the program will generate the 3D graph for you.

2. What do the axes represent in a 3D graph?

In a 3D graph, the x and y axes represent the horizontal and vertical planes, respectively, while the z axis represents the depth or height. These axes allow us to plot points in three-dimensional space.

3. How can I interpret the shape of the graph x^2+y^2-z^2=1?

The graph x^2+y^2-z^2=1 represents a double cone or a cone that has been cut in half. This shape is also known as a hyperboloid of one sheet. The points on the graph satisfy the equation x^2+y^2-z^2=1, meaning that the squared values of x and y added together must be equal to the squared value of z plus 1.

4. Can I change the viewing angle of a 3D graph?

Yes, most graphing software or programs allow you to change the viewing angle of a 3D graph. This can help you see the shape of the graph from different perspectives. You can usually do this by clicking and dragging the graph or by using the software's designated tools.

5. How can I use a 3D graph to solve problems?

A 3D graph can be a useful tool for visualizing and understanding mathematical concepts. It can help you analyze relationships between variables and identify important points or patterns in an equation. In the case of x^2+y^2-z^2=1, the graph can help you understand the shape and properties of a hyperboloid of one sheet. It can also be used to solve equations and inequalities involving three variables.

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