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 />
The generatrixs of the cone (lines of the cone) are the diameters tangent to<br />
the quadric.<br />
We call asymptotic plane to the polar planes of the points of the improper<br />
conic of Q (Q ∩ π ∞ = C) (if there exists any).<br />
Example 1. Let us consider the quadric Q ≡ x 2 1 + 3x 2 3 + 4x 1 x 2 + 2x 3 + 2 = 0.<br />
The matrix of Q is:<br />
⎛ ⎞<br />
2 0 0 1<br />
0 1 2 0<br />
A = ⎜ ⎟<br />
⎝ 0 2 0 0 ⎠<br />
1 0 0 3<br />
The determinant of A is det A = −20, quadric with proper center:<br />
⎛ ⎞<br />
2 0 0 1<br />
( )<br />
0 1 2 0<br />
z0 z 1 z 2 z 3 ⎜ ⎟<br />
⎝ 0 2 0 0 ⎠ = ρ ( 1 0 0 0 ) ,<br />
1 0 0 3
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