Kitonum

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11 years, 263 days

MaplePrimes Activity


These are replies submitted by Kitonum

The polynomial is more convenient to specify as a procedure:

P := x -> a0*x^3+a1*x^2+a2*x+a3:

You have 4 unknown coefficients and 4 conditions. Make a system of equations to find them.

@Rouben Rostamian   Of course, it’s good for everyone to know that this is called the shoelace formula  https://en.wikipedia.org/wiki/Shoelace_formula 

restart:
Fib:= proc(n)
option remember;
    if n=1 then return 1
    elif n=2 then return 2
    else Fib(n-1)+Fib(n-2)
    fi;
end proc:

Fib(30);

 

@goli Do you want several arrows on one curve?

A:=plot(x^2, x=0..3, color=red, thickness=3):
x0:=1.5:
B:=seq(plots:-arrow([x0,x0^2], 0.7*[1,2*x0], color=blue, head_width=0.15, head_length=0.4, shape=arrow), x0=[0.5,1.5,2.2]):
plots:-display(A, B);

          

 

@goli 

A:=plot(x^2, x=0..3, color=red, thickness=3):
x0:=1.5:
B:=plots:-arrow([x0,x0^2], [1,2*x0], color=blue, head_width=0.15, head_length=0.4, shape=arrow):
plots:-display(A, B);

                

 

@baharm31  I do not think that's possible.
Try

dsolve(odeA, {x(t),q(t)});


Maple returns  NULL .


Edit. If you want to investigate how solutions change depending on parameter changes, then you can use  Explore  command.

JAMET wrote:
"Let Q be a quadrilateral which has a inscribed circle and a circumscribed circle. Show that the centers of these 2 circles and the point of intersection of the diagonals of the quadrilateral are aligned."   

The proof of this statement for the general case can be read here  http://mathhelpplanet.com/viewtopic.php?f=28&t=68666   ( chebo's post, in Russian).

@Vrighty  It seems you did not understand me. When you are not using indexes, you should use eval command with your approach. With indexes you can not use eval:

restart;
F(y):=A+B+C;
A:=10*y;
B:=y²-4;
C:=5-y;
F(y):=eval(F(y));

F[y]:=A+B+C;
A:=10*y;
B:=y²-4;
C:=5-y;
F[y]:=F[y]; 

                             

 

@Thomas Dean  I do not know how to do this programmatically in the general case. You see that, as Thomas Richard pointed out, Maple 2020 successfully handled this example. Maybe one of the Maple developers will answer your questions.

@Carl Love  Thank you for your detailed analysis (vote up). 

Since this set of points on the plane is described by the polynomial equation  P(x)=0 ,
where  P(x)=x^8+4*x^6*y^2+6*x^4*y^4+4*x^2*y^6+y^8-12*x^7-36*x^5*y^2-36*x^3*y^4-12*x*y^6+252*x^6+324*x^4*y^2-108*x^2*y^4-180*y^6-5184*x^3*y^2-5184*x*y^4+7776*x^4+23328*y^4+116640*x^3+209952*x*y^2+839808*x^2+2519424*x+3779136-1944^2 ,
it would be great if we could split this polynomial  P(x)  into the product of 3 polynomials corresponding to each connected curve.

 

@Carl Love  It seems to me a manifestation of some disrespect for the participants of this forum who do not have Maple 2019 (I think this applies to most of them), the writing a code that obviously does not work in Maple <2019. 

Here is almost the same code, but it also works for Maple <2019. Since Fibonacci numbers are growing very fast, the code works great for extremely large ranges (I took 10^10000 instead of 10000):

restart;
a,b:= 0,1:  
while b<10^10000 do
s:=a+b; a,b:= b,s;  
if issqr(b) then print(b) fi;
od:

                                     1
                                    144

  

Where is the system itself? Copy it here in text form (not a picture).

@goli  The  explicit  option allows you to get the roots of an equation in explicit form, of course, if possible. See acer's answer  for  TT8 .

@goli  In the first example

RootOf(_Z^2*l^2+3*_Z^4-3)

, we have a simple biquadratic equation, the roots of which are easily expressed in terms of the coefficients of this equation. The equation in your example (I named it  A ) is much more complicated and probably just there is no formula expressing the roots of this equation in terms of its coefficients. As you can see, allvalues command does not help. Only when specifying the parameter value  do we get the solution explicitly.


 

restart;
A:=RootOf(6*_Z^3+(27+3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*sqrt(9-3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2))*_Z^2+(3*sqrt(9-3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2)*l^4*RootOf(_Z^2*l^2+3*_Z^4-3)^2-9*sqrt(9-3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2)*l^2+90*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2-18*l^4+6*l^6*RootOf(_Z^2*l^2+3*_Z^4-3)^2-81+45*RootOf(_Z^2*l^2+3*_Z^4-3)^2*sqrt(9-3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2))*_Z-324-3*l^8+l^10*RootOf(_Z^2*l^2+3*_Z^4-3)^2+108*RootOf(_Z^2*l^2+3*_Z^4-3)^2*sqrt(9-3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2)-3*sqrt(9-3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2)*l^6+sqrt(9-3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2)*l^8*RootOf(_Z^2*l^2+3*_Z^4-3)^2-63*sqrt(9-3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2)*l^2+30*sqrt(9-3*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2)*l^4*RootOf(_Z^2*l^2+3*_Z^4-3)^2+45*l^6*RootOf(_Z^2*l^2+3*_Z^4-3)^2+351*RootOf(_Z^2*l^2+3*_Z^4-3)^2*l^2-108*l^4, index = 1):
Sol:=allvalues(A);

RootOf(72*_Z^3+(-3*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2+3*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)+324)*_Z^2+(-12*l^8+12*(l^4+36)^(1/2)*l^6-3*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6+3*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-396*l^4+180*(l^4+36)^(1/2)*l^2-99*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2+45*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)-972)*_Z-2*l^12+2*(l^4+36)^(1/2)*l^10-(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^10+(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^8-126*l^8+90*(l^4+36)^(1/2)*l^6-48*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6+30*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-1998*l^4+702*(l^4+36)^(1/2)*l^2-486*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2+108*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)-3888, index = 1), RootOf(72*_Z^3+(-3*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2+3*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)+324)*_Z^2+(-12*l^8+12*(l^4+36)^(1/2)*l^6-3*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6+3*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-396*l^4+180*(l^4+36)^(1/2)*l^2-99*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2+45*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)-972)*_Z-2*l^12+2*(l^4+36)^(1/2)*l^10-(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^10+(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^8-126*l^8+90*(l^4+36)^(1/2)*l^6-48*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6+30*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-1998*l^4+702*(l^4+36)^(1/2)*l^2-486*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2+108*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)-3888, index = 1), RootOf(72*_Z^3+(-3*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2-3*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)+324)*_Z^2+(-12*l^8-12*(l^4+36)^(1/2)*l^6-3*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6-3*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-396*l^4-180*(l^4+36)^(1/2)*l^2-99*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2-45*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)-972)*_Z-2*l^12-2*(l^4+36)^(1/2)*l^10-(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^10-(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^8-126*l^8-90*(l^4+36)^(1/2)*l^6-48*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6-30*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-1998*l^4-702*(l^4+36)^(1/2)*l^2-486*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2-108*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)-3888, index = 1), RootOf(72*_Z^3+(-3*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2-3*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)+324)*_Z^2+(-12*l^8-12*(l^4+36)^(1/2)*l^6-3*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6-3*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-396*l^4-180*(l^4+36)^(1/2)*l^2-99*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2-45*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)-972)*_Z-2*l^12-2*(l^4+36)^(1/2)*l^10-(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^10-(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^8-126*l^8-90*(l^4+36)^(1/2)*l^6-48*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6-30*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-1998*l^4-702*(l^4+36)^(1/2)*l^2-486*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2-108*(l^4+36)^(1/2)*(36+2*l^4+2*(l^4+36)^(1/2)*l^2)^(1/2)-3888, index = 1)

(1)

nops([%]);
allvalues(Sol[1]);

4

 

RootOf(72*_Z^3+(-3*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2+3*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)+324)*_Z^2+(-12*l^8+12*(l^4+36)^(1/2)*l^6-3*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6+3*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-396*l^4+180*(l^4+36)^(1/2)*l^2-99*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2+45*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)-972)*_Z-2*l^12+2*(l^4+36)^(1/2)*l^10-(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^10+(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^8-126*l^8+90*(l^4+36)^(1/2)*l^6-48*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^6+30*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^4-1998*l^4+702*(l^4+36)^(1/2)*l^2-486*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)*l^2+108*(l^4+36)^(1/2)*(36+2*l^4-2*(l^4+36)^(1/2)*l^2)^(1/2)-3888, index = 1)

(2)

 


 

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