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 />
2. If det A = 0 (this is, b 03 = 0) the reduced equation of the affine quadric is<br />
and we distinguish various cases:<br />
a) If b 00 ≠ 0 we have<br />
0 = b 00 + λ 1 x 2 1 + λ 2 x 2 2<br />
1) If sign(λ 1 ) = sign(λ 2 ), this is J > 0, the reduced equation of the<br />
affine quadric is of the form<br />
0 = c + a 2 x 2 1 + b 2 x 2 2<br />
which is the equation of an elliptic imaginary cylinder if c > 0 or<br />
elliptic cylinder if c < 0.<br />
2) If sign(λ 1 ) ≠ sign(λ 2 ), this is J < 0, the reduced equation of the<br />
affine quadric is of the form<br />
0 = c + a 2 x 2 1 − b 2 x 2 2<br />
which is the equation of a hyperbolic cylinder.
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