AFFINE and PROJECTIVE GEOMETRY Quadrics

ETS Arquitectura. UPM Year 2010-2011. 1

1. Let us consider the quadric Q ≡ 2x0x1 +2x0x2 +2x0x3 −4x1x2 −6x1x3 + 4x2x3 = 0 in 3. Answer the following questions:

(a) Classify Q. (b) Polar plane of the point P (1, 0, −1, 2). (c) Pole of the plane π ≡ x − x = 0. 1 2 { x − x = 0 (d) Line conjugated with the line r ≡ 1 0 . x2 − x0 = 0 (e) Equation of the polarity associated with the quadric.

≡ 2 − 2 − − 2. Let us consider the quadric Q x1 x2 + 2x2x3 + x1x0 + x2x0 x3x0 2 5x0 = 0 in the projectivized aﬃne space 3 with respect to the coordinate system R = {O,B = {e1, e2, e3}} and in homogeneous coordinates. Answer the following questions:

(a) Classify Q. (b) Obtain the centers. { x − x + 5x = 0 (c) Planes tangent to the quadric, which contain the line r ≡ 1 2 0 . x3 = 0

≡ 2 2 2 − − 3. Let us consider the quadric Q 4x1 + 4x2 + 5x3 4x1x3 8x2x3 + − − 2 20x0x3 8x0x1 16x0x2 + 4x0 = 0. in the projectivized aﬃne space. Answer the following questions:

(a) Classify the quadric Q. (b) Singular points, in case there exists any. (c) Line of centers. (d) Classify the intersection conic of the quadric with the plane that contains the point and the line of centers.

≡ − 2 4. Classify the quadric Q x1x2 +x2x3 +x3x1 x0 = 0 of the projectivized aﬃne space 3.

x1 x2 x3 (a) Determine the diametral plane conjugated with the line 1 = 2 = 3 .

(b) Pole of the plane x1 − x2 + x0 = 0. (c) Study{ the intersection points of the quadric with the line x = x r ≡ 2 0 . x3 = −x0 ETS Arquitectura. UPM Mat II. Aﬃne and projective geometry. 2010-2011. 2

5. In the euclidean projectivized aﬃne space 3 with respect to the orthonor- mal coordinate system R = {O; e1, e2}, and in homogeneous coordinates, let us consider the quadrics

≡ 2 2 2 − − − 2 φ 25x1 + 16x2 + 22x3 + 4x1x2 20x1x3 16x2x3 72x0 = 0. ≡ 2 − 2 − − 2 − 2 γ x1 + 2x2x3 3x2 6x1x3 3x3 + 4x1x2 x0 = 0.

Answer the following questions:

(a) Classify them. (b) Main planes and axes. (c) Reduced forms.

6. In the projectivized aﬃne space A3 let us consider the quadric

≡ 2 2 − φ 3x1 + 4x2 12x1x3 + 6x0x1 + 12x0x3 = 0.

Answer the following questions:

(a) Determine the intersection points with the improper line of the plane x1 − x3 = 0. (b) Find the asymptotic planes of the quadric φ. (c) Classify φ. { x − x = 0 (d) Line conjugated with the line r ≡ 1 0 x1 + x2 = 0

7. In the euclidean projectivized aﬃne space let us consider the family of quadrics

2− 2 2 − − 2 λ(x1 x2+x3+2x2x3+x0x1+x0x2 x0x3 5x0) 2− 2 − − + µ(2x1 2x2+2x1x3 2x0x1+2x0x2 2x2x3) = 0. TF

Answer the following questions:

(a) Find all the values of λ and µ for which the obtained quadric is degenerate. (b) Find all the values of λ and µ for which the obtained quadric is a paraboloid. Specify in each case whether it is degenerate or not. (c) Are there values of λ and µ for which the obtained quadric is a sphere? ETS Arquitectura. UPM Mat II. Aﬃne and projective geometry. 2010-2011. 3

8. Let us consider the following quadrics of the euclidean projectivized aﬃne space:

≡ 2 2 − φ1 4x1 + x2 + 2x1x2 2x0x2 = 0, ≡ 2 2 2 − − − φ2 5x1 + 5x2 + 8x3 + 8x1x2 4x1x3 + 4x2x3 12x0x1 + 12x0x2 6x0x3 = 0,

φ3 ≡ 2x1x3 + 2x1x2 − 4x0x3 = 0, ≡ − 2 φ4 2x1x3 + 2x2x3 4x0x3 + 12x3 = 0, ≡ 2 − φ5 8x3 + 30x1x2 + 8x1x3 + 30x2x3 24x0x3 = 0.

Answer the following questions:

(a) Classify each one of them and obtain the reduced form. (b) If any of the former quadrics is of revolution, ﬁnd the equation of the circle which has as center the center of the quadric, as long as it is not a paraboloid. (c) Determine the tangent planes, for the quadrics which are not degenerate and contain the axis OX3. (d) Determine for the quadrics which are not paraboloids the diametral planes conjugated with the diameter that contains the improper point of the axis OX2. (e) If there exists any cylinder, determine its line of centers.

9. In the euclidean projectivized aﬃne space, determine the value of a ∈ R for which the quadric

2 2 2 − − − 2 x1 + x2 + x3 2x0x1 + 2x0x2 2ax1x2 + 2ax1x3 x0 = 0

is a sphere. Determine the radius of the circle situated in the diametral plane conjugated with the diameter which has the same improper point as the line

x x − 1 x + 1 1 = 2 = 3 . 1 0 −1