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Ronan

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14 years, 32 days
East Grinstead, United Kingdom
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These are questions asked by Ronan

Can gcd be mapped across a list?...

Ronan 1401 Maple
gcd map
December 24 2024
2 2

I knowI can use a loop but I would like something neater. igcd can be applied to a list. Is there a way to map or use @ to apply gcd to a list?

lst:= [4, 2, 8, 4];
lst1 := [a^3/10, -a^2/2, a, a^4/4]
                      lst := [4, 2, 8, 4]

                        [1   3    1  2     1  4]
                lst1 := [-- a , - - a , a, - a ]
                        [10       2        4   ]


gd:=igcd(lst);



                            gd := 2

gd1=(gcd@op)(lst1);

 

Determine if a point lies on a 3D line...

Ronan 1401 Maple 2024
geometry 3d
December 21 2024
2 3

It is not that this a terribly difficult to work out, but I feel I am probably missing something. I need to check if a 3D point lies on a 3D line. What is a good approach here. I started of with the idea all alpha's are equal. but there are exceptions. See P3 and P4

restart

NULL

l := `<,>`(3+2*alpha, 1+6*alpha, 4-5*alpha)

Vector[column](%id = 36893489809910741940)

 (1)

NULL

P := [9, 19, -11]

[9, 19, -11]

 (2)

seq(solve({l[i] = P[i]}, alpha), i = 1 .. 3)

{alpha = 3}, {alpha = 3}, {alpha = 3}

 (3)

l1 := `<,>`(3+2*alpha, 1+0*alpha, 4-5*alpha)

Vector[column](%id = 36893489809910721460)

 (4)

P1 := [9, 1, -11]

[9, 1, -11]

 (5)

seq(solve({l1[i] = P1[i]}, alpha), i = 1 .. 3)

{alpha = 3}, {alpha = alpha}, {alpha = 3}

 (6)

l2 := `<,>`(3+2*alpha, 1+0*alpha, 4-0*alpha)

Vector[column](%id = 36893489809910705556)

 (7)

P2 := [9, 1, 4]

[9, 1, 4]

 (8)

seq(solve({l2[i] = P2[i]}, alpha), i = 1 .. 3)

{alpha = 3}, {alpha = alpha}, {alpha = alpha}

 (9)

l3 := `<,>`(3+2*alpha, 0+0*alpha, 4-0*alpha)

Vector[column](%id = 36893489809963852012)

 (10)

P3 := [9, 0, 4]

[9, 0, 4]

 (11)

seq(solve({l3[i] = P3[i]}, alpha), i = 1 .. 3)

{alpha = 3}, {alpha = alpha}

 (12)

P4 := [9, 0, -2]

[9, 0, -2]

 (13)

seq(solve({l3[i] = P4[i]}, alpha), i = 1 .. 3)

{alpha = 3}, {alpha = alpha}

 (14)

 

Download 2024-12-21_Q_3D_point_lies_on_3D_line.mw

How is the simplification achieved...

Ronan 1401 Maple 2024
radical simplification
December 20 2024
0 2

2024-12-20_Q_simplification_Question.mw
Solve the general cubic. Apply values and simplify. 

Could someone show how Maple simplifies to the value of X=3? I tried doing it manually and I could not figure it out. 

Also is there a Help assistant to see the setps?

> 

restart
> 

 

> 

 

> 

X^3+a*X=b

X^3+X*a = b

 (1)
> 

 

> 

 

> 

sol:=solve(X^3+a*X=b,[X])

[[X = (1/6)*(108*b+12*(12*a^3+81*b^2)^(1/2))^(1/3)-2*a/(108*b+12*(12*a^3+81*b^2)^(1/2))^(1/3)], [X = -(1/12)*(108*b+12*(12*a^3+81*b^2)^(1/2))^(1/3)+a/(108*b+12*(12*a^3+81*b^2)^(1/2))^(1/3)+((1/2)*I)*3^(1/2)*((1/6)*(108*b+12*(12*a^3+81*b^2)^(1/2))^(1/3)+2*a/(108*b+12*(12*a^3+81*b^2)^(1/2))^(1/3))], [X = -(1/12)*(108*b+12*(12*a^3+81*b^2)^(1/2))^(1/3)+a/(108*b+12*(12*a^3+81*b^2)^(1/2))^(1/3)-((1/2)*I)*3^(1/2)*((1/6)*(108*b+12*(12*a^3+81*b^2)^(1/2))^(1/3)+2*a/(108*b+12*(12*a^3+81*b^2)^(1/2))^(1/3))]]

 (2)
> 

vals:=[a=6,b=45]

[a = 6, b = 45]

 (3)
> 

Nans:=(map(eval,sol,vals))

[[X = (1/6)*(4860+12*166617^(1/2))^(1/3)-12/(4860+12*166617^(1/2))^(1/3)], [X = -(1/12)*(4860+12*166617^(1/2))^(1/3)+6/(4860+12*166617^(1/2))^(1/3)+((1/2)*I)*3^(1/2)*((1/6)*(4860+12*166617^(1/2))^(1/3)+12/(4860+12*166617^(1/2))^(1/3))], [X = -(1/12)*(4860+12*166617^(1/2))^(1/3)+6/(4860+12*166617^(1/2))^(1/3)-((1/2)*I)*3^(1/2)*((1/6)*(4860+12*166617^(1/2))^(1/3)+12/(4860+12*166617^(1/2))^(1/3))]]

 (4)
> 

simplify(Nans)

[[X = 3], [X = (1/4)*(I*3^(1/2)*(180+44*17^(1/2))^(2/3)+(8*I)*3^(1/2)-(180+44*17^(1/2))^(2/3)+8)/(180+44*17^(1/2))^(1/3)], [X = -3/2-((1/2)*I)*51^(1/2)]]

 (5)
 

 

Download 2024-12-20_Q_simplification_Question.mw

DataFrame display alpha as a Greek lette...

Ronan 1401 Maple 2024
typesetting greek dataframe
December 01 2024
1 3

How do I get alpha to display as the Greek letter in DataFrame Row column?

> 

restart
> 

#varp:=alpha
> 

QQFProj := proc(q12::algebraic, q23::algebraic,
                q34::algebraic, q14::algebraic,{varp:=:-alpha},
                  prnt::boolean:=true) #{columns:=[QQFproj,Q13proj,Q24proj]}
  description "Projective quadruple quad formula and intermediate 13 and 24 quads. Useful for cyclic quadrilaterals";
  local qqf,q13,q24, sub1,sub2,sub3, R,values,DF,lens;
  uses   DocumentTools;
  sub1:= (q12 + q23 + q34 + q14);
  sub2:=-4*(q12*q23*q34);
  sub3:=64*q12*q23;
  qqf:=(sub1+sub2)^2-sub3;
  q13:=(q12-q23)^2;
  q24:=varp*(q23-q34)^2;
  if prnt then
  
   values:=<qqf,q13,q24>;
   DF:=DataFrame(<values>, columns=[`"Values Equations"`],rows=[`#1  QQF`,`#2  Q13`,cat(`#3  Q24 (`,varp,`)`)]);
   lens := [4 +8* max(op(length~(RowLabels(DF)))),4+ min(max( 10*(length~(values))),1000)];#op(length~(ColumnLabels(DF)0)
   Tabulate(DF,width=add(lens),widthmode = pixels,weights = lens);
  return qqf,q13,q24
  end if;
  return qqf,q13,q24
end proc:
> 

 q12:=1/2:q23:=9/10:q34:=25/26:q41:=9/130:#Cyclic quadrilateral
q12:=sqrt(17+a)/2:q23:=(r^2+t^2)^2/10:q34:=((a+b+c)^4/26):q41:=sqrt(17+b)/130:
> 

Q:=QQFProj(q12,q23,q34,q41,true):

Download 2024-12-01_Q_Data_Table_alpha_as_Greek_Letter.mw

Experimental format for projective vecto...

Ronan 1401 Maple 2024
procedure vector type
November 20 2024
1 7

I am experimenting using the this format of  Vector( [Vector] ) to make projective vectors a different data type to Vectors. I don't want to use 1 x 3 or 3 x 1 matrices. The format holds some promise.
I would like to be able to copy the Maple format of Vector or Vector[column]    and Vector[row] for my varaition. 

ProjVectoC and ProjVectorR    so ProjVector or ProjVector[column]   and ProjVector[row]
A secondary question  is on type checking (see previous question How to setup special type check in a procedure? - MaplePrimes  ). Would it be possible to have the type check return ProjVector[column] or ProjVector[row]?
The attached worksheet contains a procedure for factor reducing the vectors to to a minimal format of <x,y,z>. Also   Cross product and Dot product procedures to suit.

I am open to any efficiency improvements.

> 

restart
> 

interface(rtablesize=50)

[10, 10]

 (1)
> 

with(LinearAlgebra):
> 

 

> 

FactReduce:=overload([
     proc(v::{list,Vector})
          option overload;
          description " removes linear factor from",
                      " a list, vector, matrix or expression";
          uses LinearAlgebra;
          local i, num,tgdc,dnm, V1;
          num:=`ifelse`(type(v,Vector),numelems(v),nops(v));
          dnm:=frontend(lcm, [seq(denom(v[i]),i=1..num)]);
          V1:=radnormal(v*~dnm);
          tgdc:=V1[1];

          for i from 2 to num do
               tgdc:=frontend(gcd, [tgdc, V1[i]]);
          end do;

          return  simplify(V1/~tgdc);
     end proc,

     proc(M::{Matrix})
          option overload;
          uses LinearAlgebra;
          local i, num,r,c, tgdc,dnm, V1, Ml;
          r,c:=Dimension(M);
          num:=r*c;
          V1:=convert(M,list);
          dnm:=frontend(lcm, [seq(denom(V1[i]),i=1..num)]);
          Ml:=radnormal(dnm*~M);
          V1:=convert(Ml,list);#print((dnm,V1));
          tgdc:=V1[1];#print("xx")

          for i from 2 to num do
               tgdc:=frontend(gcd, [tgdc, V1[i]])
          end do;

          return  simplify(Ml/~tgdc);   
     end proc,

     proc(l::{`+`,`*`,`=`, `symbol`,procedure},  {vars::list:=[:-x,:-y]})
          option overload;
          uses LinearAlgebra;
          local i, num,f1,f1a,lv,lr, tgdc,dnm, V1,Vs;
          f1 := `if`(l::procedure, l(vars[]), l);
               f1a:=`if`(f1::`=`,lhs(f1)-rhs(f1),f1)  ; # Remequal(f1);
          lr:=primpart(f1a,vars);
          return lr
end proc

]):
> 

ProjVectorC := proc(a, b, c)
local cfs, vectr;
description " A Projective Column (Line) Vector in Reduced format";
cfs := FactReduce([a, b, c]);
vectr := <[<cfs>]>;
end proc:
> 

 

> 

ProjVectorR := proc(a, b, c)
local cfs, vectr;
description " A Projective Row (Point) Vector";
cfs := sign(c)*FactReduce([a, b, c]);
vectr := <[<cfs>^%T]>^%T;
end proc:
> 

 

> 

`&otimes;` := proc(A, B)
local cp;
description "Cross Product of Projective Vectors in Reduced format";
cp :=sign(c)* FactReduce(LinearAlgebra:-`&x`(A[1], B[1]))^%T;
cp := ifelse(cp[3] <> 0, <[sign(cp[3]) *~ cp]>, cp); #makes sure format is [x,y,z] and not [x,y-z]
end proc:
> 

 

> 

`&odot;` := proc(A, B)
description "Dot Product of Projective Vectors";
(A[1]) . (B[1]);
end proc:
> 

 

V := ProjVectorR(2, 4, -6); W := ProjVectorR(11, 7, 5); S := ProjVectorC(6, -18, 24)

Vector[column](%id = 36893490982610361748)

 (2)

whattype(V); `~`[whattype](V)

Vector[row](%id = 36893490982610825812)

 (3)

whattype(S); `~`[whattype](S)

Vector[column](%id = 36893490982626471436)

 (4)

`~`[whattype](V[1])

Vector[row](%id = 36893490982558545668)

 (5)
> 

V[1] . V[1]

14

 (6)
> 

`&odot;`(V, V)

14

 (7)
> 

R := `&otimes;`(W, V)

Vector[column](%id = 36893490982630825980)

 (8)
> 

R := `&otimes;`(V, W)

Vector[column](%id = 36893490982630903548)

 (9)

whattype(R)

Vector[column]

 (10)

`~`[whattype](R)

Vector[column](%id = 36893490982598861396)

 (11)

`~`[whattype](R[1])

Vector[column](%id = 36893490982598866092)

 (12)
> 

`&otimes;`(R, S)

Vector[column](%id = 36893490982624872076)

 (13)
> 

`&odot;`(R, S)

-85

 (14)
> 

`&odot;`(W, R)

0

 (15)
> 

`&odot;`(R, `<,>`([`<,>`(x, y, 1)]))

15-31*x+38*y

 (16)
 

 

Download 2024-11-21_Q_Projective_Vector_Format.mw

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