THE PLANE.
Cartesian Co-ordinates.—Any two straight lines, as XX' and YY', are assumed as lines of reference, or co-ordinate axes. The former is X-axis or axis of abscissa, the latter is Y-axis or axis of ordinates. The point 0 is the origin of distances; the (half) line OX, the origin of angles. Distances on or parallel to the X-axis are regarded positive (+) if meas ured to the right, negative (—) if to the left. Distances on or parallel to the Y-axis are re garded positive if measured upward, negative if downward. Angles are regarded positive or negative according as they are conceived to be generated by counter-clockwise or by clockwise rotation. (See TRIGONOMETRY ). Conceive drawn all lines parallel to the X-axis and all parallel to the Y-axis. Any pair of these lines, one of the former set and one of the latter, determine (intersect in) a point, and all points of the plane are thus determined. Conversely, any point determines (is the common point of) a pair of the lines, and all the pairs of the par allels are thus determined.. Obviously any line pair or its point determines two real numbers: the distances OF and FP in terms of any con venient unit. These, denoted respectively by x and y, are named respectively the abscissa and the ordinate, together the co-ordinates, of the point. Conversely, any pair of real numbers determine a point. It is thus seen that, by means of a pair (system) of axes, a one-to-one correspondence is established between the points of the plane and the assemblage of real number ,pairs. Any such point and its pair of coordinates are said to correspond; the point is said to depict or represent its pair of numbers geometrically, and the number pair is said to represent the point arithmetically or alge braically or analytically.
Transformation of Cartesian Co-ordinates. — It is plain that (the unit being the same) the co-ordinates of a point referred to one pair of axes will not coincide with its co-ordinates referred to a different pair. Formula . for expressing the old in terms of the new co ordinates are exceedingly useful. To find such
formula, consider first the case where the old and new origins coincide. Denote by a and IS the angles made with OX by OX' and OY' respectively. Let x and y be the old, and s' and y' the new, co-ordinates of any point P, and denote the angle XOY by w. The formula in question are readily seen to be: x sin la m= x' sin (6)—a)-f- y' sin (a—fl), y sin ga === x' sin a y' sin p. If the origins do not coincide and if h and k be the co-ordinates of the new origin 0' with reference to the old axes, the formulae of transformation are found by adding h sin and k sin id respectively to the right hand members of the foregoing .equations. Most commonly the axes are assumed to be rectangular. In that case, w 0- 90°, sin go - (90° + a), and the equations of trans formation become: x cos sin a + h. y ===x" sin a + cos a + k. The equations for effecting the inverse transformation are found by solving for x' and y' the equations of the direct transformation.
Polar Co-ordinates.— Though it is never necessary, it is often convenient to employ other than Cartesian co-ordinates to determine the position of a point. Of such other co ordinate systems, the most familiar is the polar. About any point 0 (as centre), called the pole, suppose drawn all possible concentric circles; also suppose drawn out from the pole all pos sible rays (half-lines). Any circle and any ray determine (intersect in) a point, and all points of the plane are thus determined; con versely, any point determines (is common to) a circle and a ray, and all pairs of such lines (circle and ray) are thus determined. A circle is given by its radius p. and a ray by its angle A made with a fixed ray, as OQ, called the initial line or polar axis. All the circles are obtained by letting p vary from A to co, and all rays by allowing 0 to vary from 0 to br If the division be exterior, i.e., if P be outside the segment, as at (P), one term of the ratio is negative, and the formula: are: x=(Mar-tfisXi) : (ffh—tris), y=(tnlyr-ttio):(tnr—tn2).