If P be the interior mid-point, nii=na, and x2):2, y= (y2+ y2):2. If In, P is the exterior mid-point and its'co-brdinates are both infinite unless the segment is parallel to an axis, in which case but one of the co ordinates is infinite.
or 360°. Obviously, to any pair of values (within the ranges of variation mentioned) of p and 0 there corresponds one point, and con versely. The pair of numbers (p, 8) determin ing or determined by a point P are called the polar co-ordinates of P. In particular, p is called the radius vector, and A the vectorial angle, of P.
Transformations from Cartesian to Polar Co-ordinates.— We present here only the simplest and most important case, viz., that wherein the Cartesian axes are rectangular, the origin coinciding with the pole, and the positive half of the X-axis with the polar axis. Let P be any point. It is dear, Fig. 3, that the tions of direct transformation are: p cos e; p sin O. Solving these for p and 0, the equa tions of the inverse transformation are found to be: V xi ± • Distance between Points.— Henceforth the axes will be assumed to he rectangular. Let P and P' be any points, of co-ordinates (x, y) and (x', y) respectively. From Fig. 4, by the Pythagorean theorem, di= whence the distance between any two points (x, y) and (x', y') is found to be d=\/ (y--y')'. Trans forming to polar co-ordinates (p. It) and (p', 8'), and reduc ing. there results d=--\,/ p't cos (0-0% in agreement with the Law of Cosines. (See Taicoxostrrint).
Division of Line-segment in Given Ratio. — Suppose P divides the segment P.P. in the ratio, mi:n22. By hypothesis, nu: nu; hence, from similar triangles, • these equations, solved for x and y, yield x mix2 mai) : (m2 + m2) and y (miy2 mo):(mi + m2).
Locus of Equation.— An equation, f(x, y) 0, between the variables x and y, defines a system or aggregate or assemblage of pairs of numbers, viz., the assemblage of pairs of values of x and y that satisfy the equation. To each of such pairs of (real) values, a system of axes and a unit of distance being chosen, there corresponds a point. The assemblage of all the points so determined constitute the (real) locus of the equation. In general, as x or y varies continuously, y or x will vary continuously, and accordingly the correspond ing point will trace a continuous path, some curve, the locus in question: Conversely, if a point move subject to some geometric condition, its path will be a curve such that the co ordinates of its points and of no other satisfy some equation. An .equation and its locus or curve are each said to represent the other, and, from the properties of either, corresponding properties of the other can be inferred. An equation defines its locus, a locus defines its equation. Any equation, f(x, is, of course, satisfied by countless pairs of values of which either (or both) is imaginary or complex. To such a pair no real or ((visible" point of the plane corresponds. Nevertheless, in order that the geometric and analytic lan guages shall be coextensive, it is customary to say that any pair of numbers of which at least one is complex represents an 'imaginary point* of the plane. Accordingly the locus (in gener
alized sense) of an equation is composed of a i real part and an imaginary part, the latter consisting of all imaginary points whose co ordinates satisfy the equation. The intersection of two loci or curves consists of the points (real and imaginary) whose co-ordinates satisfy the equations of both curves. The foregoing remarks respectingequations in Cartesian co ordinates apply equally to equations in polat co-ordinates.
The Straight Line and the Linear Egos tion.,—Let (1) be any line through the origin, and denote by 0 its angle with OX, and Its O. Obviously the x and y of any (every) point of (1) and of no other point are connected by the equation 3=mx, which there fore defines, and is called the equation of, line (1). To each line through 0 there corresponds one value of the :tope m, and conversely. Any line not through 0 is parallel to a line through 0. Hence (2), parallel to (1), represents any line not through 0. Clearly by adding b to any y of (1) the corresponding y of (2) is found, while corresponding x's are the same. Hence the equation of (2) is 3=mx-Fb. The quantity b is called the V-intercept of the line; if b is zero, (2) goes through 0, and conversely. The equation represents a line for every pair of real values of m and b; conversely, every line has a slope and a Y-intercept (positive or nega tive). Hence every equation of 1st degree in x and y represents a straight line, and con versely. Solving the equation y 0 of the X-axis and the equation of (2) as simultane b ous, the intersection is found to be (— ' ' 0) or (a, 0), whence the equation of (2) may be x y written— a +—.= called the intercept or sym b metric form, a being the X-intercept. The equation + b is called the slope form. Other forms are readily obtainable, of which one of the most important is the so-called standard or normal form, .r cos a + y sin a — P =0, readily deducible from the figure, where p is the length of the perpendicular (normal) from 0 to (2), a is the angle indicated, and FP is any point of (2). The general equation Ax -I- By 4- C 0 can be reduced to any of the foregoing forms. To reduce it to the normal form, it suffices to multiply by the normalizing factor, 1 :\/A' B', yielding Ax By C • V Ai- where cos a= A : V sin a B C: that sign before the radical be ing chosen which renders the constant (or absolute) term negative. A line is determined by yet other pairs of data, as by its slope m and a given point (x,, y,) or by two points (x2, y,), (x2, y,), and its equation assumes cor responding forms, which may be called respect ively the point-slope form and the two-point form. The former is y = ,n(x — xi). The necessary and sufficient condition that this line shall pass through (x,, y2) is y2 — yi— m x2— x2). Combining the two equations by division, there results the two-point form, (y:(y,Yi)(X— Xi) equa tion of the line fixed by the points (x,, y,), (xl, Distance from Point to The equa tion of any line (3), parallel to (2), Fig. 6, is x cos a + y sin a — 0, where p'= p + d.