Lecture 11 - 09/19/2014 - MATH 497C, Fall 2014

Today we are talking about hyperbolas and some of their aﬃne properties. Hyperbolic rotation A hyperbolic rotation is an aﬃne transformation that moves a given hyperbola on itself.

Theorem 1. Given a hyperbola and two points M,M 0 on it, there exists a unique aﬃne orientation preserving transformation, which moves M to M 0 and is a hyperbolic rotation.

Proof. Define a system of coordinates in the following way: the origin at the center of the hyperbola, and axes are the asymptotes of it. In this coordinate system the given hyperbola is defined by an equation xy = const = C > 0. If we consider an affine transformations which maps the hyperbola to itself, then the center of it should be preserved. Therefore, no translations are allowed. Let a a A = 11 12 . a21 a22 We show that there exists a matrix A such that the affine orientation preserving x transformation T ((x, y)) = A maps a point M to M 0. y First, we describe all affine transformations which map the given hyperbola to itself. Let (x, y) be a point on a hyperbola xy = C. Then, to guarantee that its image under a transformation T x a x + a y A = 11 12 y a21x + a22y lies on the same hyperbola, we need to have

(a11x + a12y)(a21x + a22y) = C.

As a result, we get an equality: C C (a x + a )(a x + a ) = C, 11 12 x 21 22 x

4 2 2 2 a11a21x + C(a11a22 + a12a21)x + C a12a22 = Cx . The equality above should be true for inﬁnitely many values of x, so it follows that

a11a21 = 0, i.e. a11 = 0 or a21 = 0,

a12a22 = 0, i.e. a12 = 0 or a22 = 0, and a12a21 + a11a22 = 1.

1 The two cases: 1) a11 = 0, a12 = 0 or 2) a21 = 0, a22 = 0, are impossible as they imply a12a21 + a11a22 = 0, which contradicts the third equality above. The case a11 = 0, a22 = 0 gives a solution 0 k A = 1 k 0 for some k 6= 0. But notice that det(A) = −1, so the map T reverses the orientation. The case a21 = 0, a12 = 0 gives a solution k 0 A = 1 0 k for some k 6= 0. Note if k > 0 then a transformation preserves the branches, and if k < 0 then it switches the branches.

C 0 C As a result, if we want to map a point M = (x1, ) to a point M = (x2, ), then the x1 x2 x2 following equality should be true kx1 = x2, i.e. k = . Therefore, there exists a unique x1 aﬃne orientation preserving transformation, which moves M to M 0, and it is a hyperbolic rotation. Now we discuss some interesting properties of a hyperbola which can be easily proved using aﬃne transformations.

Aﬃne properties of a hyperbola

1. Diameters of a hyperbola; conjugate diameters. Deﬁnition 1. The two hyperbolas are called conjugate if one is obtained from another by the rotation, by 90◦.

Any line, passing through the center of a hyperbola which does not coincide with its asymptotes, is called a diameter.

Figure 1: Diameters of a hyperbola(from ”Ideas and methods of aﬃne and projective geometries” I.M.Yaglom, V.G.Ashkinuze).

2 Any diameter of a hyperbola intersects either the hyperbola itself, or the conjugate hyperbola. Sometimes a segment of a diameter of a hyperbola with the end points being the points of intersection with the given hyperbola or its conjugate is also called a diameter (See Figure 1). This allows us to speak about lengths of diameters of a hyperbola. Theorem 2. The set of all middle points of all segments that are parallel to a given diameter lie on a line `, which is a diameter of the conjugate hyperbola. A line ` is called the conjugate diameter to the given diameter.

Proof. Denote a given diameter by MN. By the Theorem 1, there exists a hyperbolic rotation which maps a point M to a vertex A of the hyperbola. As the center O of the hyperbola is preserved under a hyperbolic rotation, we have that the diameter MN is mapped to a diameter AB (Aﬃne transformations map lines to lines). Moreover, aﬃne transformations map parallel lines to parallel lines. So, all parallel segments which are parallel to MN are mapped to segments parallel to a diameter AB which connects vertices of the hyperbola (See Figure 2, (a) becomes (b)).

Figure 2: (from ”Ideas and methods of aﬃne and projective geometries” I.M.Yaglom, V.G.Ashkinuze).

The axes of symmetry of a hyperbola divides all the segments parallel to AB into halves. Aﬃne transformations preserve ratios of lengths of segments on the same line. As a result, the preimage of the axes of symmetry divides all the segments parallel to MN into halves.

2. Tangent lines to a hyperbola. Any line which has only one common point with a hyperbola is called a tangent line to a hyperbola. Theorem 3. At any point M of a hyperbola there exists a unique tangent line; this tangent line is parallel to a diameter conjugate to the diameter MO.

3 Proof. As in the previous theorem, we use the fact from the Theorem 1 that we can map a point M to the vertex A of the hyperbola by hyperbolic map. For a point A there exists a tangent line m which is a line orthogonal to AO and passing through A. Notice that the line m is parallel to a diameter conjugate to the diameter AO. Affine transformations map parallel lines to parallel lines. Furthermore, if two figures have only one common point, then their images under affine transformations will have only one common point. Therefore, if we apply the inverse hyperbolic rotation, then we get the statement of the theorem for any point M.

3. Preservation of some particular areas.

Theorem 4. Let M,M 0 be points on a hyperbola, O be the center of a hyperbola. Tangent lines through points M,M 0 intersect asymptotes of a hyperbola at points P,Q and P 0,Q0, respectively. Then an area of a triangle P QO is equal to the area of a triangle P 0Q0O.

Proof. Exercise. See Figure 3.

Figure 3: Shaded triangles have equal areas(from ”Ideas and methods of aﬃne and projective geometries” I.M.Yaglom, V.G.Ashkinuze).

Theorem 5 (Apollonius theorem). All parallelograms constructed on conjugate semi-diameters of a hyperbola have the same area.

Proof. Exercise. See Figure 4.

4 Figure 4: Shaded parallelograms have equal areas(from ”Ideas and methods of aﬃne and projective geometries” I.M.Yaglom, V.G.Ashkinuze).