# Finding traces

MarcL
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

## Answers and Replies

Mentor
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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MarcL
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?

Mentor
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?

MarcL
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)