Solving an Equation: Is it a Paraboloid or Cone?

In summary, the conversation discusses an equation involving a square root that resembles a paraboloid or a cone. The equation is rewritten to reveal that it represents a sphere, specifically a hemisphere. The equation can be used to find the center and radius of the sphere. The conversation concludes with a clarification on the nature of the object described by the equation.
  • #1
Amaelle
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Homework Statement
what is the nature of this object?
Relevant Equations
z=sqrt(2-x^-y^2)
Good day
while solving some integral I met with the following equation
z=sqrt(2-x^-y^2) that looks like a paraboloid?!
I thought first that it might be a cone!
any insights?
thank you!
 
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  • #2
Try rearranging the equation to (2-x)=y2+z2 and look at it in the plane z=0 or y=0.
 
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  • Informative
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  • #3
you mean 2-x^2?
if I do as you said x=+-sqrt(2-z^2) ?!
 
  • #4
Your original equation does not have the square.

Rewrite the equation as
x2+y2+z2 = 2
Do you recognize it?
 
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  • #5
Amaelle said:
Homework Statement:: what is the nature of this object?
Relevant Equations:: z=sqrt(2-x^-y^2)

Good day
while solving some integral I met with the following equation
z=sqrt(2-x^-y^2) that looks like a paraboloid?!
I thought first that it might be a cone!
any insights?
thank you!
You have given your original equation as ##z = \sqrt {2 - x^{-y^2}}##

Was this a mistake? Maybe you meant ##z = \sqrt {2 - x^2 - y^2}##.
 
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  • #6
Steve4Physics said:
You have given your original equation as ##z = \sqrt {2 - x^{-y^2}}##
I didn‘t read that at all. I really need to cut back on the hallucinogens o_O
 
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  • #7
Steve4Physics said:
You have given your original equation as ##z = \sqrt {2 - x^{-y^2}}##

Was this a mistake? Maybe you meant ##z = \sqrt {2 - x^2 - y^2}##.
yes the last one was what I meant thank you!
 
  • #8
Amaelle said:
yes the last one was what I meant thank you!
Then my post #4 applies.
 
  • #9
caz said:
Your original equation does not have the square.

Rewrite the equation as
x2+y2+z2 = 2
Do you recognize it?
Now yes :smile:
it's an ellipsoid thank you very much
 
  • #10
Amaelle said:
Now yes :smile:
it's an ellipsoid thank you very much
True, but it is also a sphere.
 
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  • #11
Amaelle said:
it's an ellipsoid
Following up on @caz's comment, it's better to think of this as a sphere rather than an ellipsoid. A sphere is a special case of ellipsoids, in which both foci are located at the center of the sphere.

From the equation you can find the center of the sphere and its radius. Owing to the square root in the original equation, you get only part of the sphere.
 
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  • #12
It's a hemisphere,to be precise. As ##z## is only positive.

Just saw that's already been said.
 
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  • #13
Mark44 said:
Following up on @caz's comment, it's better to think of this as a sphere rather than an ellipsoid. A sphere is a special case of ellipsoids, in which both foci are located at the center of the sphere.

From the equation you can find the center of the sphere and its radius. Owing to the square root in the original equation, you get only part of the sphere.
thank you very much
 
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FAQ: Solving an Equation: Is it a Paraboloid or Cone?

1. What is an equation?

An equation is a mathematical statement that shows the equality between two expressions. It typically contains variables, constants, and mathematical operations.

2. What is a paraboloid?

A paraboloid is a three-dimensional surface that is created by rotating a parabola around its axis. It has a bowl-like shape and can be described by a quadratic equation.

3. What is a cone?

A cone is a three-dimensional shape that has a circular base and a curved surface that tapers to a point, known as the apex. It can be described by a conic equation.

4. How do you determine if an equation represents a paraboloid or cone?

An equation represents a paraboloid if it can be written in the form of z = ax^2 + by^2 + c, where a and b are constants. On the other hand, an equation represents a cone if it can be written in the form of z = ax^2 + by^2 - cz, where a, b, and c are constants.

5. What is the importance of identifying if an equation is a paraboloid or cone?

Knowing whether an equation represents a paraboloid or cone can help in understanding its properties and behavior. This information is useful in various fields such as physics, engineering, and computer graphics, where these shapes are commonly used to model real-world objects.

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