CHAPTER III: CONICS AND QUADRICS - OCW UPM
CHAPTER III: CONICS AND QUADRICS - OCW UPM
CHAPTER III: CONICS AND QUADRICS - OCW UPM
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AFFINE <strong>AND</strong> PROJECTIVE GEOMETRY, E. Rosado & S.L. Rueda<br />
If det A = d 0 λ 1 λ 2 λ 3 = 0 (this is, d 0 = 0 and rank(A) = 3) then they are<br />
degenerate quadrics with reduced equation:<br />
We can distinguish two cases:<br />
λ 1 x 2 1 + λ 2 x 2 2 + λ 3 x 2 3 = 0<br />
1. If sign(λ 1 ) = sign(λ 2 ) = sign(λ 3 ), the reduced equation of the affine<br />
quadric is<br />
0 = λ 1 x 2 1 + λ 2 x 2 2 + λ 3 x 2 3<br />
which is the equation of an imaginary cone.<br />
2. If sign(λ 1 ) = sign(λ 2 ) ≠ sign(λ 3 ), the reduced equation of the affine<br />
quadric is of the form<br />
which is the equation of an cone.<br />
0 = a 2 x 2 1 + b 2 x 2 2 − c 2 x 2 3
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