T ECHNISCHE U NIVERSIT ÄT B ERLIN
Institut für Mathematik
Geometry I - WS 2016/2017
Prof. Dr. Alexander I. Bobenko
Jan Techter
http://www3.math.tu-berlin.de/geometrie/Lehre/WS16/GeometryI/
Exercise Sheet 7
Exercise 1: Circumscribed quadrilateral.
(4 pts)
Consider four distinct points A, B, C, D on a non-degenerate non-empty conic in RP2 .
Let P = tA ∩ tB , Q = tB ∩ tC , R = tC ∩ tD , S = tD ∩ tA denote the intersection points
of the tangents. Show that the four lines P R, QS, AC, and BD pass through one point.
Exercise 2: Harmonic points and reflections.
(4 pts)
Let C be a non-degenerate non-empty conic in RP2 defined by the symmetric bilinear
form b. Let P be a point and ` its polar line. Show that for any point X ∈ C the points
Q = P X ∩ ` and Y = P X ∩ C satisfy
cr(P, X, Q, Y ) = −1.
This defines a map f : C → C with X 7→ Y . Show that this map is given by
Y = f ([x]) = [x − 2
b(x, p)
p],
b(p, p)
where X = [x] and P = [p]. Show that f is the restriction of a projective map from RP2
to RP2 .
Exercise 3: Paraboloid.
(4 pts)
3
Consider a rotationally symmetric paraboloid in R with focal point F . Show that rays
emitted from F are reflected to parallel rays.
Exercise 4: Confocal conics.
(4 pts)
2
Consider two points A and B in R and a circle K1 that does not contain the points.
Construct the four tangents from A and B to the two circle and call the intersection
points of these lines C, D, E, F . The quadrilateral CDEF is circumscribed around the
circle K1 . Construct two circles K2 and K3 tangent to the lines ACF , ADE, BEF and
ACF , BCD, BEF , respectively. Add two more tangents BG to K2 and AH to K3 .
Show that the points {C, E}, {F, G, H}, and {D, F } lie on three confocal conics with
focal points A and B (as shown on the picture).
Figure created with GeoGebra – you can use it to draw figures for your solution.
Due Thursday, December 15, before the lecture.