Intersection of surfaces

In summary: The other is the sphere x^2+ y^2+ z^2= 2 so this is a sphere of radius [tex]\sqrt{2}[/tex] centered at the origin. The intersection is the circle where z= 1 in that sphere.
  • #1
astrokat11
7
0
z=x^2+y^2 and x^2+y^2+z^2=2...I need to find the intersection of these two surfaces. Would I just substitute z=x^2+y^2 into the equation of the sphere to find the curve of intersection? But when I do that I get an equation with fourth powers and I don't know what kind of curve that makes.
 
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  • #2

Homework Statement


z=x^2+y^2 and x^2+y^2+z^2=2...I need to find the intersection of these two surfaces. Would I just substitute z=x^2+y^2 into the equation of the sphere to find the curve of intersection? But when I do that I get an equation with fourth powers and I don't know what kind of curve that makes.



Homework Equations





The Attempt at a Solution

 
  • #3
I am not sure what you are looking for, but let u=x^2 and v=y^2, and you will get an equation of second degree in u and v, so the curve is a conic in (u,v) space. How that looks in the original I can only guess.
 
  • #4
Is that so bad, that you do not know how to draw the curve? You will have found the answer nonetheless.

Interesting remark: if you set x'=x^2, y'=y^2, you get the equation of a conic. So the curve is "a conic squared"!
 
  • #5
I would suggest to make a sketch of the problem first. Then you would "see" that the intersection is a circle. Try to find the properties of the circle, i.e. the radius and the center point. This can be done by inserting the equation of the parabolic cone into the one of the sphere.

If you are looking for an equation of the circle, there are several. Firstly, stating that the curve is the intersection of the two equations you gave is a valid one. Secondly a circle in space can be presented also as an intersection of a cylinder and a plane perpendicular to the axes of the cylinder. In this case it would be a cylinder with it's axis the Z-axes and a radius you have obtained and a plane z=1 if I'm not mistaken. There are possibilities in using cylinder coordinates etc. Up to you to see what's most appropriate. If anything is not clear just post...

[Edit] It seems that you have posted this question twice. It is not the intention to do this astrokat11. Anyway, I seems I was right on z=1 plane.
 
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  • #6
astrokat11 said:

Homework Statement


z=x^2+y^2 and x^2+y^2+z^2=2...I need to find the intersection of these two surfaces. Would I just substitute z=x^2+y^2 into the equation of the sphere to find the curve of intersection? But when I do that I get an equation with fourth powers and I don't know what kind of curve that makes.

Yes, that gives [itex]x^2+ y^2+ (x^2+ y^2)^2= 2[/itex]. Do not multiply out that last square! As astrokat11 suggested, if you let [itex]u= x^2[/itex] and [itex]v= y^2[/itex] you get [itex](u+ v)^2+ u+ v= 2[/itex]. Now let p= u+v, and we have simply [itex]p^2+ p= 2[/itex] so p= u+ v= -2 or p= u+ v= 1. That's a "degenarate" conic- two straight lines. Going back to x and y, [itex]x^2+y^2= -2[/itex], which, of course, is impossible or [itex]x^2+y^2= 1[/itex], a circle.
One of your equations is z= [itex]x^2 +y^2[/itex]= 1 so this is a circle in the z= 1 plane with center (0, 0, 1) and radius 1.
 
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What is the intersection of surfaces?

The intersection of surfaces is the set of points where two or more surfaces meet or overlap.

How is the intersection of surfaces calculated?

The intersection of surfaces is calculated by finding the common points between the equations of the surfaces. This can be done by solving the equations simultaneously or by graphing the surfaces and finding the points of intersection.

What are some real-life applications of the intersection of surfaces?

The intersection of surfaces is used in many fields such as architecture, engineering, and computer graphics. It can be used to design and analyze structures, create 3D models, and determine the path of objects in motion.

Can the intersection of surfaces be a line or a curve?

Yes, the intersection of surfaces can result in a line or a curve, depending on the shapes and orientations of the surfaces. For example, the intersection of two planes can be a line, while the intersection of a plane and a cylinder can be a curve.

Are there any special cases of the intersection of surfaces?

One special case of the intersection of surfaces is when the two surfaces are parallel, resulting in no intersection. Another special case is when the two surfaces are the same, resulting in an infinite number of points of intersection.

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