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Jamie2
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The question is to classify/describe the following degenerate quadratic surface:
x2 - 2xy +2y2 - 2yz + z2 = 0
x2 - 2xy +2y2 - 2yz + z2 = 0
Write it as [tex](x^2- 2xy+ y^2)+ (y^2- 2yz+ z^2)= 0[/tex]Jamie said:The question is to classify/describe the following degenerate quadratic surface:
x2 - 2xy +2y2 - 2yz + z2 = 0
well that's the same as (x-y)2 + (y-z)2 = 0HallsofIvy said:Write it as [tex](x^2- 2xy+ y^2)+ (y^2- 2yz+ z^2)= 0[/tex]
Does that give you any ideas?
Jamie said:well that's the same as (x-y)2 + (y-z)2 = 0
but I don't know how to use that to help me describe the quadratic surface
I like Serena said:Hi Jamie! Welcome to MHB! :)
Did you know that a square is always at least zero?
Suppose the sum of 2 squares is equal to zero, what does that say about those squares?
Jamie said:That they are equal to each other?
Or that (x-y)2 = -(y-z)2
I like Serena said:That they are both zero!
If either of them would be not zero, the sum would be positive, and therefore not equal to 0.
Jamie said:right, I knew that too. But what does that mean for the equation's 3-dimensional surface?
I like Serena said:It means that $x=y$ and $y=z$.
Both are equations of planes.
The degenerated quadratic surface is where they intersect.
Where do they intersect?
Jamie said:on the y axis? is the degenerate surface just a line?
Both! The only way a sum of squares can be 0 is if each is 0. x- y= 0 and y- z= 0 which is the same as the z= y= x. That is the line through (0, 0, 0) and (1, 1, 1).Jamie said:That they are equal to each other?
Or that (x-y)2 = -(y-z)2
A degenerate quadratic surface is a type of mathematical surface that can be described by a quadratic equation in three variables. It is called "degenerate" because it has special characteristics that make it different from a regular quadratic surface.
A regular quadratic surface is a smooth, continuous surface that can be visualized in three-dimensional space. A degenerate quadratic surface, on the other hand, has special characteristics that make it difficult to visualize, such as self-intersections or missing points.
Some examples of degenerate quadratic surfaces include a cone, a cylinder, and a plane. These surfaces have special characteristics that make them different from regular quadratic surfaces, such as having a vertex or having infinite extent in one direction.
Studying degenerate quadratic surfaces is important in various fields, such as computer graphics, robotics, and computer-aided design. Understanding their properties and characteristics can help in creating more accurate and efficient mathematical models for these applications.
Some techniques for studying degenerate quadratic surfaces include using algebraic methods, such as finding the roots of the quadratic equation, and geometric methods, such as visualizing the surface in three-dimensional space. Computer simulations and numerical methods can also be used to study these surfaces.