Conics

Conic sections are usually introduced as the first curves (as opposed to lines) and are given as the locus of points satisfying an equation of the form

ax² + 2bxy + cy² + 2dx + 2ey + f = 0

We have also studied that the behaviour of this equation is controlled by the following determinants

D = det$$\begin{pmatrix}a&b&d \\ b&c&e \\ d&e&f \end{pmatrix}$$; d = det$$\begin{pmatrix}a&b \\ b&c \end{pmatrix}$$

If D = 0 then the locus is a pair of lines which are parallel (and could coincide) if d = 0. We get curves if D $\neq$ 0 which are the hyperbola, parabola and ellipse (or circle) if d is negative, zero or positive (the circle corresponds to the latter when a = c). Since we have defined addition and multiplication in geometric terms it should not amaze us to see that the above definition can be made without reference to an algebraic equation.

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picture(6774,5871)(2239,-5965)

In order to understand this statement let us a fix a projective plane and work within it. The collection of all lines through a fixed point O are in natural 1-1 correspondence with the points on a line l not containing O; any point B on l determines a unique line b joining O and B and conversely, any line b through O meets l in exactly one point B. The locus of lines through a fixed point is called a pencil.

picture(6432,5758)(1036,-5639) (1291,-5101)(0,0)[lb]O (1246,-556)(0,0)[lb]l (2296,-1216)(0,0)[lb]B (2611,-46)(0,0)[lb]b (1036,-5581)(0,0)[lb]The pencil of lines through a point O is in 1-1 correspondence with the points of l

If l and m are two lines, neither containing a point O, then we have natural correspondence given above between the points of each line and the points of the pencil through O. This sets up a 1-1 correspondence between the two lines. Explicitly, for each point A of l let a be the line joining O and A and let B be the point of m that lies on a. This correspondence is called the perspectivity between the points of l and m with center O.

Similarly, if a pair of points A and B are such that neither lies on a line l, then there is a natural 1-1 correspondence between the pencils through A and B, since both the pencils have a 1-1 correspondence with the points of l. Explicitly, if p is a line containing A which meets l in P, we consider the line q joining B with P. This correspondence is called the perspectivity between the pencils through A and B with axis l.

picture(9414,6261)(154,-5650) (6316,-5596)(0,0)[lb]Axial perspectivity between pencils (5101,-3661)(0,0)[lb]A (6166,-4876)(0,0)[lb]l (9346,-3616)(0,0)[lb]B (8506,-1576)(0,0)[lb]q (6196,-1366)(0,0)[lb]p (7471,-46)(0,0)[lb]P (1501,-5581)(0,0)[lb]Central perspectivity between lines (2161,-4921)(0,0)[lb]O (211,-1186)(0,0)[lb]l (286,-2446)(0,0)[lb]m (1321,-766)(0,0)[lb]A (1816,-1576)(0,0)[lb]a (1711,-2536)(0,0)[lb]B

In both case a projective correspondence or projectivity is defined as a composition of perspectivities. Thus, the definition of a conic says, take a pair of points A and B and a projectivity $\pi$ between the pencils of lines through A and B. Let C be the locus of points of the form l $\cap$ $\pi$(l ) where l is a line through A and $\pi$(l ) the corresponding line through B, then C is a conic.

In order to understand projectivities better we note the following

Exercise 1   Given three points O, I and Z one a line l, and three points O', I', Z' on a line m. Each point A other than the given points on l corresponds to an element $\lambda$(A) of K - {0, 1} where K is the underlying field. Similarly, a point B of m other than the given points corresponds to $\mu$(B) in K - {0, 1}. A perspectivity (and more generally a projectivity) $\pi$ can be written as $\mu$($\pi$(A)) = (a + b$\lambda$(A))/(c + d$\lambda$(A)) where a, b, c and d are in K and ad - bc $\neq$ 0.

picture(6729,5335)(1519,-4649) (4756,-871)(0,0)[lb]I (7456,-316)(0,0)[lb]Z (2131,-3496)(0,0)[lb]O' (2506,-1186)(0,0)[lb]O (7111,-3931)(0,0)[lb]Z' (4261,-3631)(0,0)[lb]I' (3496,-2086)(0,0)[lb]I'' (7426,359)(0,0)[lb]A (3031,-661)(0,0)[lb]B (2011,-4591)(0,0)[lb]Central perspectivity from A and then B gives a general projectivity

The study of conics in projective geometry is a fascinating one but we leave it here with a pointer to the suggested readings on projective geometry. At the same time we note that it would be rather difficult to study more complicated curves such as the locus of points satisfying x³ + y³ = 1, using only the incidences between points and lines and not algebra; this is possible in principle since the addition and multiplication operations have been defined in terms of the incidence relations. From now on we will use all the familiar notions from algebra and deal with coordinate geometry.