One benefit of using complex numbers is that you may forget about trigonometry.
load'plot' circle=._1^n=.(%~i:)60 ellipse=.(circle*-.a)+(+circle)*a=.0.8 hyperbola=.-:((+%)j.(-%))^n plot circle,ellipse,:hyperbola >________________________________ > Fra: km <[email protected]> >Til: "[email protected]" <[email protected]> >Sendt: 0:40 tirsdag den 17. september 2013 >Emne: Re: [Jprogramming] Plotting complex lists > > >Summary of results. The strategy of hyperbola below (plotting a complex >table) is not well known. Henry Rich found it and reported it. > >Bo Jacoby gave the best way to change the sign of the real part of a complex >number. >Simply do [: + - . > > > >NB. Complex Analytic Geometry > > >NB. How to calculate complex number lists and tables for >NB. plotting lines, circles, ellipses, and hyperbolas. How to >NB. modify these tables to achieve translations, rotations, >NB. and reflections. Begin with preliminaries: > > >steps =: {.@] + -~/@] * [ %~ [: i. >:@[ > >NB. n steps a,b produces n+1 equally spaced values from a to b > >to =: 512 steps , NB. Usage a to b for 512 steps from a to b > >sin =: 1&o. > >cos =: 2&o. > >sinh =: 5&o. > >cosh =: 6&o. > >arcsinh =: _5&o. > > >NB. Now, results > > >line =: 2 : 'm + (n-m)*]' > >NB. A line B [ t is point "t of the way from A to B". Command >NB. >NB. plot 0 line 1j1 [ _1 to 2 >NB. >NB. shows the line segment from _1j_1 to 2j2 > >NB. You are plotting a list of 513 complex numbers. > > >parabola =: 1 : '] j. (1 % 4 * m) * *:' > >NB. p parabola x produces point x j. y on parabola >NB. (*: x) = 4*p*y . Command >NB. >NB. plot 1r4 parabola _2 to 2 >NB. >NB. plots parabola y = *: x for x from _2 to 2 > >NB. You are plotting a list of 513 complex numbers. > > >ellipse =: 2 : '((m * cos) j. n * sin) 0 to 2p1' > >NB. Suggested by Henry Rich > >NB. Command >NB. >NB. plot a ellipse b >NB. >NB. plots the ellipse 1 = (*: x % a) + *: y % b . > >NB. If a = b you get the circle (*: x) + (*: y) = *: a > > >hyperbola =: 2 : '[: (,: +) (n * sinh) j. m * cosh' > >NB. Suggested by Henry Rich > >toh =: [: to/ [: arcsinh %~ > >NB. b toh c,d is (arcsinh c%b) to (arcsinh d%b) > >NB. Command >NB. >NB. plot a hyperbola b [ b toh c,d >NB. >NB. plots y^2/a^2 - x^2/b^2 = 1 for x from c to d. > >NB. Remember the pattern b [ b toh c,d > >NB. You are plotting rows of a 2 by 513 table to get the two >NB. branches of the hyperbola. > > >NB. Rotations, translations, and reflections > >NB. Multiply a complex number list or table by (^&j. theta) >NB. to rotate all of its points by theta radians. The center >NB. of rotation is the origin 0 = 0j0 . > >NB. Add 5j3 to a complex list or table to move all of its points >NB. the distance and direction of 5j3 from 0j0. > >NB. Use (+ list) or (+ table) (monadic + is conjugate) to >NB. reflect all the points of the list or table across the >NB. line through 0j0 and 1j0 -- the x-axis. Afterwards >NB. multiply by (^&j. theta) to achieve a reflection across >NB. the line through 0j0 and (^&j. theta). > >NB. Multiply a positive number p times a list or table to >NB. achieve an expansion from 0 or compression toward 0 >NB. according as p > 1 or p < 1 . > >NB. If you want to combine several operations do the >NB. reflection first and the translation last. > >NB. Example >NB. >NB. plot (^&j. theta) * p parabola _2 to 3 >NB. >NB. plots a parabola rotated by theta radians, with 0j0 >NB. the center of rotation. If theta is _1r2p1 (that is >NB. - pi%2 radians) you have converted a (*: x) = 4 * p * y >NB. parabola into a (*: y) = 4 * p * x parabola. > > >--Kip Murray > >Sent from my iPad > >---------------------------------------------------------------------- >For information about J forums see http://www.jsoftware.com/forums.htm > > > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
