Plane and solid analytic geometry; an elementary textbook . r-mal from the origin on a line, together with the angle 50 ANALYTIC GEOMETRY [Ch. IV, § 34 which this normal makes with the positive direction of , the line is completely determined. The perpen-dicular distance is represented by jt?, and the angle by a. Through 0 draw a linemaking an angle a withOX. If any distance Offis laid off on this lineeither in the positive di-rection (along the termi-nal line of the angle), orin the negative direction,and through H a lineAB, perpendicular to Off,is drawn, that line is com-pletely de
Plane and solid analytic geometry; an elementary textbook . r-mal from the origin on a line, together with the angle 50 ANALYTIC GEOMETRY [Ch. IV, § 34 which this normal makes with the positive direction of , the line is completely determined. The perpen-dicular distance is represented by jt?, and the angle by a. Through 0 draw a linemaking an angle a withOX. If any distance Offis laid off on this lineeither in the positive di-rection (along the termi-nal line of the angle), orin the negative direction,and through H a lineAB, perpendicular to Off,is drawn, that line is com-pletely determined. Itis convenient to restrict a to positive values from 0° to360°. In case we wish to speak of a complete set of parallellines without changing «, it will be necessary to allow p tobe either positive or negative, but every line in the planecan be determined by positive values of both a and p, andthis will always be understood unless otherwise have seen that the equation of the line AB in terms of its intercepts is - + %- = 1. a oline. Fig. 34. But for all positions of the V P and -=cos«, or a = cos a p f- = Sill «,0 or sin a Substituting these values of a and 5, the equation of AB becomes x cos a + y sin a = p. [15] Ch. IV, § 35] THE STRAIGHT LINE 51 This is called the normal form of the equation of astraight line. Let the student show that the equation of a straightline in oblique coordinates in terms of a and p is x cos a -f- y cos (o> — a) = p. Note. — The equations - = cos a and - = sin a are true for all cases,a b since if p is positive, a and cos a have the same sign, and also b and sin if p is negative, they have the opposite signs. PROBLEMS1. What is the equation of the straight line in which(a) a = 60°, and p = 5 ? (d) a = 225°, and p = 0 ?(6) a = 120°, and j> = 5 ? (e) a = 45°, u> = 60°, and p = 1 ?(c) «=330°, and p= -5 ? (/) a= -60°, <o=135°, and p = 6 ? 35. Reduction of the general equation to the normalform.
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