. Plane and solid analytic geometry . and themotions which they generate are called shearing motions. Examjyle 1. Subject the curve(3) y = a^ + 2x TRANSFORMATIONS OF THE PLANE 353 to the shearHere x-. = x, 2/ = -2; r + y. (4) X = x, y = + 2/, and (3) becomesor 2x +y = x^ + 2 x\ (5) y = x^ Conversely, the curve (5) is carried by the shear (4) into thecurve (3). The shear (4) adds to the ordinate of a point(x, y) the amount 2x equal to the cor-responding ordinate of the line y = 2 , the ordinates of the curve(3) can be obtained by adding to the ordi-nates of the line 2/ ==2x t
. Plane and solid analytic geometry . and themotions which they generate are called shearing motions. Examjyle 1. Subject the curve(3) y = a^ + 2x TRANSFORMATIONS OF THE PLANE 353 to the shearHere x-. = x, 2/ = -2; r + y. (4) X = x, y = + 2/, and (3) becomesor 2x +y = x^ + 2 x\ (5) y = x^ Conversely, the curve (5) is carried by the shear (4) into thecurve (3). The shear (4) adds to the ordinate of a point(x, y) the amount 2x equal to the cor-responding ordinate of the line y = 2 , the ordinates of the curve(3) can be obtained by adding to the ordi-nates of the line 2/ ==2x the correspondingordinates of the curve (5), whose graph isknown. Thus the curve (3) can be easilyplotted. It is tangent to the line y — 2xat the origin. Examjjle 2. Construct the curve y = 4:0^ — X. This is done by plotting the line y = — xand the curve y = 4: x^, and then adding their ordinates al-gebraically for a new ordinate — that of the required process is equivalent to subjecting the curve y= 4:x^ tothe shear. Fig. 15
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