Lecture 9 - 09/15/2014 - MATH 497C, Fall 2014

Home , Affine geometry, Conjugate diameters

Today we are talking about ellipses and some of their aﬃne properties. Elliptic geometry

Definition 1. An ellipse is the image under an affine transformation of a circle. In one of our recitation classes we proved the following lemma Lemma 1. Every affine transformation can be represented as a composition of a similarity transformation and a contraction with respect to some line. Note that any similarity transformation maps any circle to another circle. So, we can say that an ellipse is the image of a circle under a contraction with respect to some line. (See Figure 1)

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

To find out the properties of ellipses, we need just to consider those properties of circles which are affine in nature. Affine properties of ellipses 1. Every ellipse has a center of symmetry, which we call the center of an ellipse.

Proof. The center of a circle is its center of symmetry. Affine transformations preserve ratios of lengths of segments on the same line. So, under affine transformations the center of symmetry is mapped to the center of symmetry. Definition 2. A diameter of an ellipse is a chord which passes through the center.

Sometimes lines which contain diameters are also called diameters.

It follows from the deﬁnition of the center of symmetry that the center of an ellipse is the middle of any diameter.

1 2. The middle points of a family of parallel chords in an ellipse lie on the same line, which is the diameter of the ellipse. This diameter is called conjugate to any of considered chords.

Proof. It is known from the plane geometry that the middle points of a family of parallel chords in a circle lie on the same line, which is the diameter orthogonal to these chords. Aﬃne transformations map parallel lines to parallel lines, and a diameter to a diameter. Moreover, under aﬃne transformations the middle point of a segment is mapped to the middle point of the image of that segment. The desired result follows. (See Figure 2)

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

Consider a diameter AB of an ellipse which is conjugate to a diameter CD, i.e. it divides into halves all chords parallel to CD. If an aﬃne transformation maps this ellipse to a circle then it maps conjugate diameters AB and CD to mutually orthogonal diameters A0B0 and C0D0 of the circle. So, the diameter C0D0 divides into halves all chords parallel to A0B0. Then, it follows that the diameter CD of the ellipse divides into halves all chords parallel to AB. Therefore, if some diameter is conjugate to another diameter, then the second diameter is conjugate to the ﬁrst one.

3. A line which has a unique intersection point with an ellipse is called a tangent line. An ellipse has a unique tangent line at every point. Any tangent line is parallel to the diameter of the ellipse, which is conjugate to the diameter passing through the point of tangency. In particular, tangent lines to an ellipse at the end points of the same diameter are parallel to each other.

Proof. Every circle has a unique tangent line at every point. Any tangent line is orthogonal to the radius passing through the point of tangency. Under aﬃne transformations orthogonal lines of a circle are mapped to lines which have conjugate directions with respect to an ellipse. (See Figure 3)

2 Figure 3: (from ”Ideas and methods of aﬃne and projective geometries” I.M.Yaglom, V.G.Ashkinuze).

Theorem 1. Every ellipse which is not a circle has a unique pair of orthogonal conjugate diameters, which are called the axes of the ellipse. In particular, these diameters are the axes of symmetry of the ellipse. The end points of the axes of an ellipse are called the vertices of an ellipse.

Theorem 2 (Apollonius theorem). Parallelograms constructed on pairs of conjugate semi-diameters of an ellipse have the same area.

Proof. See Figure 4. Consider an affine transformation which maps the given ellipse to a circle. Under this affine transformation parallelograms OKML and OAEC constructed on conjugate semi-diameters OK, OL and OA, OC, respectively (Figure 4 (a)), are mapped to the equal squares O0K0M 0L0 and O0A0E0C0 (Figure 4 (b)). Although affine transformations do not preserve areas but they preserve the ratios of areas. Therefore, we have the statement of the theorem.

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

3 Remark 1. If we apply an affine transformation, then the area of the image of a figure is equal to the area of the original figure multiplied by the absolute value of the determinant of the affine transformation. Exercise 1. Let E be an ellipse given in Euclidean coordinates by an equation x2 y2 + = 1 a2 b2 where a, b are nonzero numbers. Use affine transformations to prove that the area of an ellipse E is equal to πab.

Now we consider the transformations which map a given ellipse to itself. Elliptic rotation All points of a circle are equivalent. It means that any point (any diameter) of a circle can be mapped to any other point of that circle by rotation centered at the center of the circle. Any rotation centered at the center of a circle maps the circle onto itself. We see that points of an ellipse are equivalent from the point of view of the affine geometry Theorem 3. Given an ellipse and two points M,N on it, there exists a unique affine orientation preserving transformation, which moves M to N and maps the ellipse to itself. In particular, it preserves areas of figures on the Euclidean plane. Proof. Let φ be an affine transformation which maps a given ellipse E to a circle Ω. Denote M 0 and N 0 the images of points M and N, respectively, under φ. Let ω be a rotation centered at the center of the circle Ω which maps M 0 to N 0. Define a transformation ω¯ := φ−1ωφ from the ellipse E to itself. For this transformation we have that ω¯(M) = φ−1ωφ(M) = φ−1ω(M 0) = φ−1(N 0) = N. The transformationω ¯ is an affine transformation as it is the composition of three affine maps. Moreover, this transformation preserves areas of figures. If under φ areas of figures on the Euclidean plane is multiplied by λ > 0, then under φ−1 areas of figures on the plane 1 are multiplied by λ . Rotation ω does not change areas.

Figure 5: (from ”Ideas and methods of aﬃne and projective geometries” I.M.Yaglom, V.G.Ashkinuze).

4 The uniqueness of an affine orientation preserving transformation of an ellipse to itself which maps a point M to a point N follows from the following: Assume there exists an affine orientation preserving transformationω ¯1 different fromω ¯ which maps the ellipse E to itself and the point M to the point N. Then, there exists an −1 affine orientation preserving transformation ω1 = φω¯1φ different from ω which maps the circle Ω to itself and the point M 0 to the point N 0. The only affine transformation which maps a circle to itself is an isometry. The only orientation preserving isometry which maps a circle to itself is a rotation centered at the center of the circle. But, there exists a unique rotation centered at the center of the circle Ω which maps the point M 0 to the point N 0.

Deﬁnition 3. An aﬃne orientation preserving transformation which maps a given ellipse to itself is called an elliptic rotation.