The axes thus chosen for the ellipse and hyperbola are called the principal axes.
In figs. 54, 55, 56 in order, conics of the three species, thus referred, are depicted.
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| Fig. 54 | Fig. 55 |
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| Fig. 56. |
The oblique straight lines in fig. 56 are the asymptotes x/a = ±y/b of the hyperbola, lines to which the curve tends with unlimited closeness as it goes to infinity. The hyperbola would have an equation of the form xy = c if referred to its asymptotes as axes, the coordinates being then oblique, unless a = b, in which case the hyperbola is called rectangular. An ellipse has two imaginary asymptotes. In particular a circle x2 + y2 = a2, a particular ellipse, has for asymptotes the imaginary lines x = ±y √−1. These run from the centre to the so-called circular points at infinity.
20. Tangents and Curvature.—Let (x′, y ′) and (x′ + h, y ′ + k) be two neighbouring points P, P′ on a curve. The equation of the line on which both lie is h(y − y ′) = k(x − x′). Now keep P fixed, and let P′ move towards coincidence with it along the curve. The connecting line will tend towards a limiting position, to which it can never attain as long as P and P′ are distinct. The line which occupies this limiting position is the tangent at P. Now if we subtract the equation of the curve, with (x′, y ′) for the coordinates in it, from the like equation in (x′ + h, y ′ + k), we obtain a relation in h and k, which will, as a rule, be of the form 0 = Ah + Bk + terms of higher degrees in h and k, where A, B and the other coefficients involve x′ and y ′. This gives k/h = −A/B + terms which tend to vanish as h and k do, so that −A : B is the limiting value tended to by k : h. Hence the equation of the tangent is B(y − y ′) + A(x − x′) = 0.
The normal at (x′, y ′) is the line through it at right angles to the tangent, and its equation is A(y − y ′) − B(x − x′) = 0.
In the case of the conic (a, b, c, f, g, h) (x, y, 1)2 = 0 we find that A/B = (ax′ + hy ′ + g)/(hx′ + by ′ + f).
We can obtain the coordinates of Q, the intersection of the normals QP, QP′ at (x′, y ′) and (x′ + h, y ′ + k), and then, using the limiting value of k : h, deduce those of its limiting position as P′ moves up to P. This is the centre of curvature of the curve at P (x′, y ′), and is so called because it is the centre of the circle of closest contact with the curve at that point. That it is so follows from the facts that the closest circle is the limit tended to by the circle which touches the curve at P and passes through P′, and that the arc from P to P′ of this circle lies between the circles of centre Q and radii QP, QP′, which circles tend, not to different limits as P′ moves up to P, but to one. The distance from P to the centre of curvature is the radius of curvature.
21. Differential Plane Geometry.—The language and notation of the differential calculus are very useful in the study of tangents and curvature. Denoting by (ξ, η) the current coordinates, we find, as above, that the tangent at a point (x, y) of a curve is η − y = (ξ − x)dy/dx, where dy/dx is found from the equation of the curve. If this be ƒ(x, y) = 0 the tangent is (ξ − x) (∂f/∂x) + (η − y) (∂f/∂y) = 0. If ρ and (α, β) are the radius and centre of curvature at (x, y), we find that q(α − x) = −p(1 + p2), q(β − y) = 1 + p2, q2ρ2 = (1 + p2)3, where p, q denote dy/dx, d2y/dx2 respectively. (See Infinitesimal Calculus.)
In any given case we can, at all events in theory, eliminate x, y between the above equations for α − x and β − y, and the equation of the curve. The resulting equation in (α, β) represents the locus of the centre of curvature. This is the evolute of the curve.
22. Polar Coordinates.—In plane geometry the distance of any point P from a fixed origin (or pole) O, and the inclination xOP of OP to a fixed line Ox, determine the point: r, the numerical measure of OP, the radius vector, and θ, the circular measure of xOP, the inclination, are called polar coordinates of P. The formulae x = r cos θ, y = r sin θ connect Cartesian and polar coordinates, and make transition from either system to the other easy. In polar coordinates the equations of a circle through O, and of a conic with O as focus, take the simple forms r = 2a cos (θ − α), r{1 − e cos (θ − α)} = l. The use of polar coordinates is very convenient in discussing curves which have properties of symmetry akin to that of a regular polygon, such curves for instance as r = a cos m θ, with m integral, and also the curves called spirals, which have equations giving r as functions of θ itself, and not merely of sin θ and cos θ. In the geometry of motion under central forces the advantage of working with polar coordinates is great.
23. Trilinear and Areal Coordinates.—Consider a fixed triangle ABC, and regard its sides as produced without limit. Denote, as in trigonometry, by a, b, c the positive numbers of units of a chosen scale contained in the lengths BC, CA, AB, by A, B, C the angles, and by Δ the area, of the triangle. We might, as in § 6, take CA, CB as axes of x and y, inclined at an angle C. Any point P (x, y) in the plane is at perpendicular distances y sin C and x sin C from CA and CB. Call these β and α respectively. The signs of β and α are those of y and x, i.e. β is positive or negative according as P lies on the same side of CA as B does or the opposite, and similarly for α. An equation in (x, y) of any degree may, upon replacing in it x and y by α cosec C and β cosec C, be written as one of the same degree in (α, β). Now let γ be the perpendicular distance of P from the third side AB, taken as positive or negative as P is on the C side of AB or not. The geometry of the figure tells us that aα + bβ + cγ = 2Δ. By means of this relation in α, β, γ we can give an equation considered countless other forms, involving two or all of α, β, γ. In particular we may make it homogeneous in α, β, γ: to do this we have only to multiply the terms of every degree less than the highest present in the equation by a power of (aα + bβ + cγ)/2Δ just sufficient to raise them, in each case, to the highest degree.
We call (α, β, γ) trilinear coordinates, and an equation in them the trilinear equation of the locus represented. Trilinear equations are, as a rule, dealt with in their homogeneous forms. An advantage thus gained is that we need not mean by (α, β, γ) the actual measures of the perpendicular distances, but any properly signed numbers which have the same ratio two and two as these distances.
In place of α, β, γ it is lawful to use, as coordinates specifying the position of a point in the plane of a triangle of reference ABC, any given multiples of these. For instance, we may use x = aα/2Δ, y = bβ/2Δ, z = cγ/2Δ, the properly signed ratios of the triangular areas PBC, PCA, PAB to the triangular area ABC. These are called the areal coordinates of P. In areal coordinates the relation which enables us to make any equation homogeneous takes the simple form x + y + z = 1; and, as before, we need mean by x, y, z, in a homogeneous equation, only signed numbers in the right ratios.
Straight lines and conics are represented in trilinear and in areal, because in Cartesian, coordinates by equations of the first and second degrees respectively, and these degrees are preserved when the equations are made homogeneous. What must be said about points infinitely far off in order to make universal the statement, to which there is no exception as long as finite distances alone are considered, that every homogeneous equation of the first degree represents a straight line? Let the point of areal coordinates (x′, y ′, z′) move infinitely far off, and mean by x, y, z finite quantities in the ratios which x′, y ′, z′ tend to assume as they become infinite. The relation x′ + y ′ + z′ = 1 gives that the limiting state of things tended to is expressed by x + y + z = 0. This particular equation of the first degree is satisfied by no point at a finite distance; but we see the propriety of saying that it has to be taken as satisfied by all the points conceived of as actually at infinity. Accordingly the special property of these points is expressed by saying that they lie on a special straight line, of which the areal equation is x + y + z = 0. In trilinear coordinates this line at infinity has for equation aα + bβ + cγ = 0.
On the one special line at infinity parallel lines are treated as meeting. There are on it two special (imaginary) points, the circular points at infinity of § 19, through which all circles pass in the same sense. In fact if S = O be one circle, in areal coordinates, S + (x + y + z)(lx + my + nz) = 0 may, by proper choice of l, m, n, be made any other; since the added terms are once lx + my + nz, and have the generality of any expression like a′x + b′y + c′ in Cartesian coordinates. Now these two circles intersect in the two points where either meets x + y + z = 0 as well as in two points on the radical axis lx + my + nz = 0.
24. Let us consider the perpendicular distance of a point (α′, β′, γ′) from a line lα + mβ + nγ. We can take rectangular axes of Cartesian coordinates (for clearness as to equalities of angle it is best to choose an origin inside ABC), and refer to them, by putting expressions p − x cos θ − y sin θ, &c., for α &c.; we can then apply § 16 to get the perpendicular distance; and finally revert to the trilinear notation. The result is to find that the required distance is
where {l, m, n}2 = l2 + m2 + n2 − 2mn cos A − 2nl cos B − 2lm cos C.
In areal coordinates the perpendicular distance from (x′, y ′, z′)
