What is the classification of this degenerate quadratic surface?

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    Quadratic Surface
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Discussion Overview

The discussion focuses on classifying and describing a specific degenerate quadratic surface defined by the equation x² - 2xy + 2y² - 2yz + z² = 0. Participants explore the implications of rewriting the equation and the geometric interpretation of the resulting conditions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants propose rewriting the equation as (x² - 2xy + y²) + (y² - 2yz + z²) = 0 to gain insights.
  • Others note that this can be expressed as (x - y)² + (y - z)² = 0, leading to questions about its implications for the surface.
  • It is suggested that if the sum of two squares equals zero, both squares must be zero, leading to the conditions x = y and y = z.
  • Participants discuss the geometric interpretation, suggesting that the degenerate surface corresponds to the intersection of planes defined by x = y and y = z.
  • There is a query about whether the degenerate surface is simply a line, with some participants affirming this while clarifying that it is not along the y-axis.
  • One participant concludes that the line described by the equations passes through the points (0, 0, 0) and (1, 1, 1).

Areas of Agreement / Disagreement

Participants generally agree on the interpretation that the degenerate surface corresponds to a line defined by the conditions x = y and y = z. However, there is some uncertainty regarding the exact nature of this line and its position in three-dimensional space.

Contextual Notes

The discussion includes assumptions about the properties of squares and their implications for the quadratic surface, but these assumptions are not universally accepted or resolved among participants.

Jamie2
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The question is to classify/describe the following degenerate quadratic surface:

x2 - 2xy +2y2 - 2yz + z2 = 0
 
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Jamie said:
The question is to classify/describe the following degenerate quadratic surface:

x2 - 2xy +2y2 - 2yz + z2 = 0
Write it as (x^2- 2xy+ y^2)+ (y^2- 2yz+ z^2)= 0

Does that give you any ideas?
 
HallsofIvy said:
Write it as (x^2- 2xy+ y^2)+ (y^2- 2yz+ z^2)= 0

Does that give you any ideas?
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
 
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

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?
 
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?

That they are equal to each other?
Or that (x-y)2 = -(y-z)2
 
Jamie said:
That they are equal to each other?
Or that (x-y)2 = -(y-z)2

That they are both zero!
If either of them would be not zero, the sum would be positive, and therefore not equal to 0.
 
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.

right, I knew that too. But what does that mean for the equation's 3-dimensional surface?
 
Jamie said:
right, I knew that too. But what does that mean for the equation's 3-dimensional surface?

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?
 
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?

on the y axis? is the degenerate surface just a line?
 
  • #10
Jamie said:
on the y axis? is the degenerate surface just a line?

Indeed, the degenerate surface is just a line... but it is not the y axis...
Try to find a point that is on the line...
 
  • #11
Jamie said:
That they are equal to each other?
Or that (x-y)2 = -(y-z)2
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).
 

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