Help with Proving Ruled Surface Equation

In summary, the equation x^2-y^2+xy-1=0 represents a cylindrical surface, where for any fixed (x,y) satisfying the equation, the z variable can vary freely. This can be seen by considering the equation as a function of z, which results in a linear function. Hence, this equation is a ruled surface composed of straight lines.
  • #1
rulo1992
15
0
Hello, I am studying for an analytic geometry final but I am totally lost for this problem... We didn't even cover this topic in class (my prof didn't show up for class for two weeks) and I have no clue on how to do it. If anyone could help I would appreciate it.

Question: Prove that the equation: $$x^2-y^2+xy-1=0$$ is a ruled surface.

I understand that a ruled surface is a surface composed of straight lines but that is as far as my knowledge goes for this question... Again any help is appreciated.
 
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  • #2
rulo1992 said:
Hello, I am studying for an analytic geometry final but I am totally lost for this problem... We didn't even cover this topic in class (my prof didn't show up for class for two weeks) and I have no clue on how to do it. If anyone could help I would appreciate it.

Question: Prove that the equation: $$x^2-y^2+xy-1=0$$ is a ruled surface.

I understand that a ruled surface is a surface composed of straight lines but that is as far as my knowledge goes for this question... Again any help is appreciated.
You're close, but this article gives a more complete definition of ruled surfaces:

http://en.wikipedia.org/wiki/Ruled_surface
 
  • #3
rulo1992 said:
Hello, I am studying for an analytic geometry final but I am totally lost for this problem... We didn't even cover this topic in class (my prof didn't show up for class for two weeks) and I have no clue on how to do it. If anyone could help I would appreciate it.

Question: Prove that the equation: $$x^2-y^2+xy-1=0$$ is a ruled surface.

I understand that a ruled surface is a surface composed of straight lines but that is as far as my knowledge goes for this question... Again any help is appreciated.
Unrelated to your question -- the homework template is required. If you post a question again, please don't delete the three parts.
 
  • #4
rulo1992 said:
Hello, I am studying for an analytic geometry final but I am totally lost for this problem... We didn't even cover this topic in class (my prof didn't show up for class for two weeks) and I have no clue on how to do it. If anyone could help I would appreciate it.

Question: Prove that the equation: $$x^2-y^2+xy-1=0$$ is a ruled surface.

I understand that a ruled surface is a surface composed of straight lines but that is as far as my knowledge goes for this question... Again any help is appreciated.

With the ##z## variable missing, isn't this a cylindrical surface? For any fixed ##(x,y)## satisfying the equation, what happens to ##(x,y,z)## as you vary ##z##?
 

FAQ: Help with Proving Ruled Surface Equation

1. What is a ruled surface equation?

A ruled surface is a type of surface that can be created by moving a straight line (called a generator) along a set of fixed curves (called directrices). The ruled surface equation is a mathematical representation of this type of surface. It describes the relationship between the coordinates of the points on the surface and the parameters that define the shape of the surface.

2. What is the purpose of proving a ruled surface equation?

The purpose of proving a ruled surface equation is to verify that the mathematical representation accurately describes the shape of the surface. This is important in order to use the equation for further analysis and calculations.

3. How do you prove a ruled surface equation?

To prove a ruled surface equation, you can use different methods such as parametric equations, vector equations, or implicit equations. These methods involve substituting the coordinates of points on the surface into the equation and checking if they satisfy the equation. If they do, then the equation is proven to accurately represent the surface.

4. What are some common challenges when proving a ruled surface equation?

One common challenge is finding the correct parameters to define the surface. This requires a good understanding of the geometry of the surface. Another challenge is dealing with complicated equations that involve multiple variables and parameters.

5. Are there any applications of ruled surface equations in real life?

Yes, ruled surface equations have various applications in fields such as engineering, computer graphics, and architecture. For example, they can be used to model the shape of objects such as airplanes and cars, or to create 3D graphics in video games and movies.

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