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I noticed that whenever reflecting a point (x,y) about the line y=x the x and y coordinates become swapped in order to give (y,x). However, I do not know why this is the case. Is there any way to geometrically/mathematically prove that this will always happen?

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  • Hint : Write $(x,y) = x e_1 + y e_2$ and notice that the linear transformation applied to $e_1$ and $e_2$ gives $e_2$ and $e_1$ respectively. – Amateur Apr 05 '14 at 21:32
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    Think about where the coordinate axes get mapped to. – David H Apr 05 '14 at 21:32

2 Answers2

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Because the segment joining $(x,y)$ to $(y,x)$ is in the direction of the vector $(x,y)-(y,x)=(x-y)\cdot(1,-1)$ hence it is orthogonal to the line $L=\{(u,v)\mid u=v\}$ whose direction is the vector $(1,1)$, and because its middle point is on $L$. These two properties (orthogonality+middle point) characterize the reflection about $L$.

Did
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  • In case anyone else did a double take parsing the notation: Vector(x,y) - Vector(y,x) = Scalar(x - y) * Vector(1, -1) – Joseph Garvin Oct 14 '17 at 16:31
  • @JosephGarvin What would be non-canonical here? – Did Oct 14 '17 at 18:58
  • nothing, except that I initially parsed the right hand side as a dot product because when I saw the word orthogonal I thought you were going to be showing that a dot product is 0. – Joseph Garvin Oct 14 '17 at 19:22
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Consider the points $A(x,x)$, $B(y,y)$, $M(x,y)$, and M' the reflexion of $M$.

$AM=BM=|x-y|$, and therefore by reflexion $AM'=BM'$. Thus $AMBM'$ is a parallelogram (we proved its a rhombus, and it's actually a square). From there coordinate calculations, be it by vector equalities or the fact that $[MM']$ and $[AB]$ share a middlepoint $\left(\frac{x+y}{2},\frac{x+y}{2}\right)$, gives you the result.

imj
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  • For M you used a comma to separate x and y, but for A and B you used semicolons. Is that a typo or are the semicolons some kind of notation? – Joseph Garvin Oct 14 '17 at 17:11
  • sorry for late answer. As the comma is the decimal point in french (eg 3.14 is written 3,14), we use semicolumns for coordinates (otherwise (3,1,4) would be ambiguous). This is just my french habits taking the helm here. I'll fix it. – imj Dec 06 '17 at 19:38