half-turns and central conics When we write the equation of an ellipse or hyperbola C in standard form x2/a2 ±y2/b2 = 1 we see that the transformation r((x,y)) = (-x,-y) maps C to itself. This transformation is a rotation about the origin through angle π, a half-turn. The existence of a half-turn mapping a conic to itself has many interesting consequences. We begin by looking at the half-turn about a general point. facts about half-turns A half-turn about the point C with position vector c is r(x) = 2c - x. The transformation r(x) = b - x is a half-turn about the point ½b. The composite of two half-turns is a translation. proofs of these facts The point C which is fixed by a half-turn r is the centre of r. From the definition of a half-turn as a rotation, it clearly belongs to E(2), the euclidean group. Of course, it also belongs to the larger groups S(2) and A(2). An element t of the affine group A(2) has the from t(x) = Ax + b, where A is a non-singular matrix and b a vector. We shall call this A the matrix of t. As an element of A(2), a half-turn has the form r(x) = (-I)x + b. In affine geometry, we do not have the concept of angle, so we take as the definition of a half-turn the property that the matrix is -I. The centre is ½b. It is clearly the only fixed point of r. The affine half-turn theorem If r is the half-turn with centre C, and t is any affine transformation, then torot-1 is the half-turn with centre t(C). Definitions (1) A plane conic is central if it is an ellipse or hyperbola. (2) The centre of a central conic is the centre of the half-turn in the symmetry group. (3) A diameter of a central conic is a line which passes through the centre. Note that the centre is well defined since a half-turn is a euclidean transformation, and we know that the euclidean symmetry group of such a conic contains only one. From the affine half-turn theorem, we can deduce The central conic theorem If C is a central conic, with centre C, and t is an affine transformation, then t(C) is central, with centre t(C). The centre of a conic is thus an affine concept. Suppose that t belongs to the affine symmetry group of a central conic C, with centre C. Then t(C) = C. By the theorem, t(C) has centre t(C), so t(C) = C. Thus, every affine symmetry of C fixes C. Note that the fixed point theorem does not apply, since the affine symmetry group of a conic is infinite, as we shall shortly see. The centre and diameters have many interesting properties. Some of the nicest will be obtained in a unifed way for all conics - including parabolas which do not appear to have a centre. This involves projective ideas. For the moment, we give just one example. The central property If C is a central conic with centre C, and A is any point on C, then AC meets C again at B, and C is the mid-point of AB.
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