Practical geometry & graphics; a text-book for students in technical and trade schools, evening classes, and for engineers, artisans, draughtsmen, architects, builders, surveyors, &c - Softcover
Synopsis
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1912 Excerpt: ...16 is the circle, Fig. 316, of radius /UTor 4. If the centre of the circle is at a point whose co-ordinates are h, k (Fig. 317), and if the co-ordinates of any point A on the circle are x y, then AC = AE-CE = AE-BD = y-. Hence 2 =--6, / =--3 2/ =-4, / =-2, and c =-3. = JF+P-= 4 + 3 = 16 = 4... The co-ordinates of the centre are--g,--f, or 3, 2, and the radius is 4. Examples of this kind may also be solved by making the left-hand side of the equation into the sum of two squares of the form (x--h)2 and (y--k)2, thus a;2_.y2_ga;_4y_3_:o mav ke written x2-G x) + (y2-ly) = 3. Make the expressions in brackets into complete squares. (x2-6 x + 9) + (y2-4y + 4) = 3 + 9 + 4 (x-3)2 + (y-22) = 16 (9 + 4 is added to the right side to balance that added to left.) By comparison with the equation (x--A)2 + (y--Jc)2 = a2, it is readily seen that the co-ordinates of the centre are 3, 2, and the radius 4. This method is equivalent to shifting the axes of reference so that the origin is at the centre of the circle. All ordinates then become (y--k), and abscissae (x--h), for the axis OY moves a distance Jc upwards, and OX moves a distance h to the right. 358. Polar Co-ordinates. Fig. 318. The polar coordinates of any point P are (1) its distance OP from a fixed point 0, called the pole or origin; (2) the angle 0 (XOP in Fig. 318) ivhich OP makes with a fixed line OX (called the initial line) pass ing through the origin. X The distance OP is called the radius vector, and is usually denoted by r. The angle 6 is always measured counter-clockwise. Hence the polar co-ordinates of a point are often called the "r$ co-ordinates." Any equation of a curve in rectangular co-ordinates may be readily transformed into one in polar co-ordinates, if ordinate, PQ is a maximum, and the...
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