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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra drawpak.spad}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package DRAWCX DrawComplex}
<<package DRAWCX DrawComplex>>=
)abbrev package DRAWCX DrawComplex
++ Description: \axiomType{DrawComplex} provides some facilities
++ for drawing complex functions.
C ==> Complex DoubleFloat
S ==> Segment DoubleFloat
PC ==> Record(rr:SF, th:SF)
INT ==> Integer
SF ==> DoubleFloat
NNI ==> NonNegativeInteger
VIEW3D ==> ThreeDimensionalViewport
ARRAY2 ==> TwoDimensionalArray
DrawComplex(): Exports == Implementation where
Exports == with
drawComplex: (C -> C,S,S,Boolean) -> VIEW3D
++ drawComplex(f,rRange,iRange,arrows?)
++ draws a complex function as a height field.
++ It uses the complex norm as the height and the complex argument as the color.
++ It will optionally draw arrows on the surface indicating the direction
++ of the complex value.\newline
++ Sample call:
++ \spad{f z == exp(1/z)}
++ \spad{drawComplex(f, 0.3..3, 0..2*%pi, false)}
++ Parameter descriptions:
++ f: the function to draw
++ rRange : the range of the real values
++ iRange : the range of imaginary values
++ arrows? : a flag indicating whether to draw the phase arrows for f
++ Call the functions \axiomFunFrom{setRealSteps}{DrawComplex} and
++ \axiomFunFrom{setImagSteps}{DrawComplex} to change the
++ number of steps used in each direction.
drawComplexVectorField: (C -> C,S,S) -> VIEW3D
++ drawComplexVectorField(f,rRange,iRange)
++ draws a complex vector field using arrows on the \spad{x--y} plane.
++ These vector fields should be viewed from the top by pressing the
++ "XY" translate button on the 3-d viewport control panel.\newline
++ Sample call:
++ \spad{f z == sin z}
++ \spad{drawComplexVectorField(f, -2..2, -2..2)}
++ Parameter descriptions:
++ f : the function to draw
++ rRange : the range of the real values
++ iRange : the range of the imaginary values
++ Call the functions \axiomFunFrom{setRealSteps}{DrawComplex} and
++ \axiomFunFrom{setImagSteps}{DrawComplex} to change the
++ number of steps used in each direction.
setRealSteps: INT -> INT
++ setRealSteps(i)
++ sets to i the number of steps to use in the real direction
++ when drawing complex functions. Returns i.
setImagSteps: INT -> INT
++ setImagSteps(i)
++ sets to i the number of steps to use in the imaginary direction
++ when drawing complex functions. Returns i.
setClipValue: SF-> SF
++ setClipValue(x)
++ sets to x the maximum value to plot when drawing complex functions. Returns x.
Implementation == add
-- relative size of the arrow head compared to the length of the arrow
arrowScale : SF := (0.125)::SF
arrowAngle: SF := pi()-pi()/(20::SF) -- angle of the arrow head
realSteps: INT := 11 -- the number of steps in the real direction
imagSteps: INT := 11 -- the number of steps in the imaginary direction
clipValue: SF := 10::SF -- the maximum length of a vector to draw
-- Add an arrow head to a line segment, which starts at 'p1', ends at 'p2',
-- has length 'len', and and angle 'arg'. We pass 'len' and 'arg' as
-- arguments since thet were already computed by the calling program
makeArrow(p1:Point SF, p2:Point SF, len: SF, arg:SF):List List Point SF ==
c1 := cos(arg + arrowAngle)
s1 := sin(arg + arrowAngle)
c2 := cos(arg - arrowAngle)
s2 := sin(arg - arrowAngle)
p3 := point [p2.1 + c1*arrowScale*len, p2.2 + s1*arrowScale*len,
p2.3, p2.4]
p4 := point [p2.1 + c2*arrowScale*len, p2.2 + s2*arrowScale*len,
p2.3, p2.4]
[[p1, p2, p3], [p2, p4]]
-- clip a value in the interval (-clip...clip)
clipFun(x:SF):SF ==
min(max(x, -clipValue), clipValue)
drawComplex(f, realRange, imagRange, arrows?) ==
import Point SF
delReal := (hi(realRange) - lo(realRange))/realSteps::SF
delImag := (hi(imagRange) - lo(imagRange))/imagSteps::SF
funTable: ARRAY2(PC) :=
new((realSteps::NNI)+1, (imagSteps::NNI)+1, [0,0]$PC)
real := lo(realRange)
for i in 1..realSteps+1 repeat
imag := lo(imagRange)
for j in 1..imagSteps+1 repeat
z := f complex(real, imag)
funTable(i,j) := [clipFun(sqrt norm z), argument(z)]$PC
imag := imag + delImag
real := real + delReal
llp := empty()$(List List Point SF)
real := lo(realRange)
for i in 1..realSteps+1 repeat
imag := lo(imagRange)
lp := empty()$(List Point SF)
for j in 1..imagSteps+1 repeat
p := point [real, imag, funTable(i,j).rr, funTable(i,j).th]
lp := cons(p, lp)
imag := imag + delImag
real := real + delReal
llp := cons(lp, llp)
space := mesh(llp)$(ThreeSpace SF)
if arrows? then
real := lo(realRange)
for i in 1..realSteps+1 repeat
imag := lo(imagRange)
for j in 1..imagSteps+1 repeat
arg := funTable(i,j).th
p1 := point [real,imag, funTable(i,j).rr, arg]
len := delReal*2.0::SF
p2 := point [p1.1 + len*cos(arg), p1.2 + len*sin(arg),
p1.3, p1.4]
arrow := makeArrow(p1, p2, len, arg)
for a in arrow repeat curve(space, a)$(ThreeSpace SF)
imag := imag + delImag
real := real + delReal
makeViewport3D(space, "Complex Function")$VIEW3D
drawComplexVectorField(f, realRange, imagRange): VIEW3D ==
import Point SF
-- compute the steps size of the grid
delReal := (hi(realRange) - lo(realRange))/realSteps::SF
delImag := (hi(imagRange) - lo(imagRange))/imagSteps::SF
-- create the space to hold the arrows
space := create3Space()$(ThreeSpace SF)
real := lo(realRange)
for i in 1..realSteps+1 repeat
imag := lo(imagRange)
for j in 1..imagSteps+1 repeat
-- compute the function
z := f complex(real, imag)
-- get the direction of the arrow
arg := argument z
-- get the length of the arrow
len := clipFun(sqrt norm z)
-- create point at the base of the arrow
p1 := point [real, imag, 0::SF, arg]
-- scale the arrow length so it isn't too long
scaleLen := delReal * len
-- create the point at the top of the arrow
p2 := point [p1.1 + scaleLen*cos(arg), p1.2 + scaleLen*sin(arg),
0::SF, arg]
-- make the pointer at the top of the arrow
arrow := makeArrow(p1, p2, scaleLen, arg)
-- add the line segments in the arrow to the space
for a in arrow repeat curve(space, a)$(ThreeSpace SF)
imag := imag + delImag
real := real + delReal
-- draw the vector feild
makeViewport3D(space, "Complex Vector Field")$VIEW3D
-- set the number of steps to use in the real direction
setRealSteps(n) ==
realSteps := n
-- set the number of steps to use in the imaginary direction
setImagSteps(n) ==
imagSteps := n
-- set the maximum value to plot
setClipValue clip ==
clipValue := clip
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>
<<package DRAWCX DrawComplex>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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