Affine equivalence

Two sets $A,B\in \mathbb {R} ^{n}$ ${\ displaystyle A, B \ in \ mathbb {R} ^ {n}}$ ${\ displaystyle A, B \ in \ mathbb {R} ^ {n}}$ are called affine equivalent if there is an affine transformation $f:\mathbb {R} ^{n}\to \mathbb {R} ^{n}$ ${\ displaystyle f: \ mathbb {R} ^ {n} \ to \ mathbb {R} ^ {n}}$ $f: {\ mathbb {R}} ^ {{n}} \ to {\ mathbb {R}} ^ {{n}}$ translating $A$ ${\ displaystyle A}$ $A$ at $B$ ${\ displaystyle B}$ $B$ , i.e. $f(A)=B$ ${\ displaystyle f (A) = B}$ $f (A) = B$ .

Affine equivalence is an equivalence relation on the set of all subsets ${\mathcal {P}}(\mathbb {R} ^{n})$ ${\ displaystyle {\ mathcal {P}} (\ mathbb {R} ^ {n})}$ ${\ displaystyle {\ mathcal {P}} (\ mathbb {R} ^ {n})}$ many $\mathbb {R} ^{n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ mathbb {R}} ^ {{n}}$ and in particular on any subset $X\subset {\mathcal {P}}(\mathbb {R} ^{n})$ ${\ displaystyle X \ subset {\ mathcal {P}} (\ mathbb {R} ^ {n})}$ ${\ displaystyle X \ subset {\ mathcal {P}} (\ mathbb {R} ^ {n})}$ .

For example, if $X\subset {\mathcal {P}}(\mathbb {R} ^{2})$ ${\ displaystyle X \ subset {\ mathcal {P}} (\ mathbb {R} ^ {2})}$ ${\ displaystyle X \ subset {\ mathcal {P}} (\ mathbb {R} ^ {2})}$ - is the set of all irreducible conics on the plane, then affine equivalence splits it into four equivalence classes , the representatives of which are four standard conics:

$x^{2}+y^{2}\,=1$ ${\ displaystyle x ^ {2} + y ^ {2} \, = 1}$ ${\ displaystyle x ^ {2} + y ^ {2} \, = 1}$ - real unit circle;
$x^{2}-y^{2}\,=1$ ${\ displaystyle x ^ {2} -y ^ {2} \, = 1}$ ${\ displaystyle x ^ {2} -y ^ {2} \, = 1}$ - equal-sided hyperbole;
$y=x^{2}$ ${\ displaystyle y = x ^ {2}}$ $y = x ^ 2$ - standard parabola;
$x^{2}+y^{2}\,=-1$ ${\ displaystyle x ^ {2} + y ^ {2} \, = - 1}$ ${\ displaystyle x ^ {2} + y ^ {2} \, = - 1}$ - imaginary circle.

In other words, affine equivalence gives an affine classification of conics on a plane: each irreducible conic on a plane is affinely equivalent to only one of the listed standard conics.

Affine equivalence

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