Do someone tell me how can I solve this exersise using a created procedure? Also I would like to know if using a procedure is useful and if in that form code lines will be reduce. If procedures are not useful. What is the best aproach?
Here the exercise and the code:
Before to begin to learn Maple, I used to do math with python. I usually built function and classes in order to reduce code lines and to generalize the problem, but I don't know if this aproach is useful with Maple. In fact, I have another question. Is it possible to write in a document some codeblocks using Maple and another using python? If answer is yes, how can I do it?
Thank you for helping me;
Greetings;
Lic. Marcus Vinicio Mora Salas
Chemist
Hi MaplePrimes team,
Is there a generalized way to fill an extremely CONCAVE closed 3d parametric plane curve in Maple with spacecurve command (NO implicitplot3d), preferably without doing any multi-decomposition or creating a particular skeleton, or algorithm/programming?
Here my example:
#Circle of reference
R0 := 1:
C0 := t-> < cos(t), sin(t), 0 >:
#Variable radius
R := t-> R0*(1 + 1/2*sin(t)):
#Variable phase
Dt := t-> Pi/2*sin(2*t):
#CONCAVE Curve
C := t-> R(t)*C0(t - Dt(t)):
'C(t)' = C(t);
#Range
t1, t2 := 0, 2*Pi:
#Plot
GC := spacecurve(C(t), t = t1..t2, color = "Red", thickness = 3
, linestyle = solid
, scaling = constrained, axes = frame, orientation = [50,40,0]):
GC;
We know how to do it easily for any convex 3d plane curve (circle, ellipse, regular polygon, etc ...).
We also know how to do it for 2d curve with the Maple’s plot option filled = true but obviously not in 3d.
Thank you for your time.
Best.
Guy.
restart:with(LinearAlgebra):with(plots):with(geometry):with(plottools):
On appelle alpha la moitié de l'angle de rotation de la roue menée par tour de roue menante.
alpha=Pi/n en radians?
soit Pi/8 pour 8 rainures..
On a alors les relations suivantes entre l'entaxe E, le rayon de la roue ùenante R1 et le rayon de la roue menée R2 :
R1=E.sin(alpha), R2=E*cos(alpha)
Intersection du cercle (O,R2) avec la droite tan(phi)x-r/cos(phi), on obtient les coordonnées de P3
sol:=allvalues(solve([tan(phi)*x-r/cos(phi)=y,y^2+x^2=R2^2],[x,y])):
Intersection de 2 cercles
sol1:=allvalues(solve([(x-E)^2+y^2=(R-a)^2,y^2+x^2=R2^2],[x,y])):
Coordonnées des points du pourtour de l'élément de croix
Oo:=point([0,0]):
phi:=Pi/8:R2:=5:r:=1/4:E:=R2/cos(phi):evalf(%):R:=R2*tan(phi):evalf(%):a:=0.5:
P1:=point([(R2/2-r)*cos(phi),(R2/2-r)*sin(phi)]):
P2:=point([(R2/2)*cos(phi)+r*sin(phi),(R2/2)*sin(phi)-r*cos(phi)]):
xP2:=(R2/2)*cos(phi)+r*sin(phi):yP2:=(R2/2)*sin(phi)-r*cos(phi):
xP1:=(R2/2-r)*cos(phi):yP1:=(R2/2-r)*sin(phi):
Equation paramétrique du segment OP1 (t varie de 0 à 1) ;
x1:=t*(0-xP1)+xP1:
y1:=t*(0-yP1)+yP1:
n1:=5:
dt:=1/(n1-1):#t varie entre 0 et 1
for i to n1 do
tau:=(i-1)*dt:
xx[i]:=evalf(subs(t=tau,x1)):
yy[i]:=evalf(subs(t=tau,y1)):
#print(i,xx[i],yy[i]);
od:
Equation paramétrique du quart de cercle P1P2 de la rainure (t varie de 0 à 1)
x2:=R2/2*cos(phi)+r*cos(t):#attention au sens de rotation du parcours de l'objet
y2:=R2/2*sin(phi)+r*sin(t):
n2:=6:
dt:=Pi/2/(n2-1):#arc de Pi/2
for i to n2 do
tau:=phi-Pi+(i-1)*dt:
xx[i]:=evalf(subs(t=tau,x2)):
yy[i]:=evalf(subs(t=tau,y2)):
od:
for i to n2 do Vector[row]([i,xx[i],yy[i]]) od:
droite:=plot((tan(phi)*x-r/cos(phi),x=0..3),linestyle=dot,color=blue):
sol[1]:
xP3:=evalf(subs(op(1,sol[1]),x)):yP3:=evalf(subs(op(1,sol[1]),y)):
xP2:yP2:
xP4:=evalf(subs(op(1,sol1[1]),x)):yP4:=evalf(subs(op(1,sol1[1]),y)):
xP5:=E-(R-a):yP5:=0:
x3:=t*(xP3-xP2)+xP2:
y3:=t*(yP3-yP2)+yP2:
n3:=10:
dt:=1/(n3-1):#t varie entre 0 et 1
for i to n3 do
tau:=(i-1)*dt:
xx[i+n2]:=evalf(subs(t=tau,x3)):
yy[i+n2]:=evalf(subs(t=tau,y3)):
od:
for i to n3 do Vector[row]([i,xx[i],yy[i]]) od:
x4:=xP5+R-a+(R-a)*cos(t):#attention au sens de rotation du parcours de l'objet
y4:=(R-a)*sin(t):
n4:=11:
eta:=arcsin(yP4/(R-a)):
dt:=(-eta)/(n2-1)/2:#arc de Pi/2
for i to n4 do
tau:=(Pi+eta)+(i-1)*dt:#recherche de tau ?
xx[i+n2+n3]:=evalf(subs(t=-tau,x4)):
yy[i+n2+n3]:=evalf(subs(t=-tau,y4)):
od:
for i to n4 do Vector[row]([i,xx[i],yy[i]]) od:
n:=n2+n3+n4;
n := 27
for i to n do Vector[row]([i,xx[i],yy[i]]) od:
figure:=NULL:
for i from 0 to n do
xx[0]:=0:yy[0]:=0:
figure:=figure,[xx[i],yy[i]]:
#print(i,xx[i],yy[i]);
od:
polygonplot([figure],scaling=constrained,color=yellow,view=[-0.1..5,-0.1..3]):
for i to n do X[i]:=xx[i]: Y[i]:=yy[i] od:
d1:=plottools[disk]([xP1,yP1],0.05,color=blue):
d2:=plottools[disk]([xP2,yP2],0.05,color=red):
d3:=plottools[disk]([xP3,yP3],0.05,color=green):
d4:=plottools[disk]([xP4,yP4],0.05,color=green):
d5:=plottools[disk]([xP5,yP5],0.05,color=green):
fig:=pointplot([figure],scaling=constrained):
Po:=pointplot([[xP1,yP1],[xP2,yP2],[xP3,yP3]],color = blue, symbol = asterisk):
Cir:=plot([R2*cos(t),R2*sin(t),t=0..Pi/2],color=black):
Arc:=plot([E+(R-a)*cos(t),(R-a)*sin(t),t=3*Pi/4..Pi],linestyle=dot,color=blue):
textplot({[1, 2, "one point in 2-D"], [3, 2, "second point in 2-D"]}):
texte:=textplot([[xP1-0.2,yP1,"P1"],[xP2,yP2-0.3,"P2"],[xP3+0.2,yP3+0.2,"P3"],
[xP4+0.2,yP4+0.1,"P4"],[xP5-0.2,yP5+0.2,"P5"]]):
display({Arc,Cir,d1,d2,d3,d4,d5,Po,fig,droite,texte},scaling=constrained,view=[-1..7,-1..6]):
with(plottools):
printlevel:=3:
Miroir : symétrie par rapport à l'axes des x
for i from 0 to n/2 do
tt:=yy[i]:
yy[i]:=yy[n-i+1]:
yy[n-i+1]:=tt:
tt:=xx[i]:
xx[i]:=xx[n-i+1]:
xx[n-i+1]:=tt
od:
for i from 0 to n-1 do
xx[2*n-i]:=xx[i]:
yy[2*n-i]:=-yy[i]:
#print(i,xx[i],yy[i])
od:
Poly:=NULL:
for i from 0 to 2*n-1 do
xx[0]:=0:yy[0]:=0:
Poly:=Poly,[xx[i],yy[i]]:od:
polygonplot([Poly],color=yellow,scaling=constrained):
pointplot([Poly],color = blue, scaling=constrained,symbol = asterisk,view=[-1..5,-3..3]):
Rotation
unassign('xt','yt'):
#gc():
zt:=8:#8 rainures ou faisceaux
xt:=Vector(63,[]):
yt:=Vector(63,[]):
xt:=Vector((2*n-1),zt,[]):
yt:=Vector((2*n-1),zt,[]):
j:=0:
for k from 0 to zt-1 do
j:=0:
phi:=2*Pi*k/zt:
cs:=cos(phi):
sn:=sin(phi):
for kk from 1 to 2*n-1 do
j:=j+1:
xt[j][k]:=evalf(xx[kk]*cs-yy[kk]*sn):
yt[j][k]:=evalf(xx[kk]*sn+yy[kk]*cs):
od:
od:
N1:=j:
points:=seq(seq([xt[i][j], yt[i][j]], j=0..zt-1), i=1..2*n-1):
p_cross:= pointplot([points], scaling = constrained, color = black,linestyle=solid, filled=[yellow]):
polygonplot([points], color = yellow, scaling = constrained);
NULL;
display([p_cross]);#How to draw this cross with a line without points. Thank you.
Hello, I'm using Maple 2021.2 to approximate the solution to a differential equation using a pseudospectral expansion in the basis of Hermite polynomials. After setting up some equations, I get a non-linear system of equations that I try to solve using the Maple's function solve, but Maple suddenly crashes during the execution of this function.
My code is the following:
HN := (n, y) -> exp(-1/2*y^2)*simplify(HermiteH(n, y))/sqrt(sqrt(Pi)*2^n*n!):
u7 := a0*HN(0, y) + a1*HN(1, y) + a2*HN(2, y) + a3*HN(3, y) + a4*HN(4, y) + a5*HN(5, y) + a6*HN(6, y):
resid := diff(u7, y $ 2) + 4*y*diff(u7, y)*u7 + 2*u7^2:
r := evalf(allvalues(RootOf(HermiteH(7, x), x)));
equations := {evalf(eval(u7, y = 0) - 1)};
for root_r in r do
if root_r != 0 then
equations := {op(equations), evalf(eval(resid, y = root_r))};
end if:
end do:
solutions := solve(equations, {a0, a1, a2, a3, a4, a5, a6}):
Does anyone have any idea about what is happening? Is there any better way of solving a non-linear system of equations?
