Computing degrees of freedom for planar linkages

Use R ::= Q[x,y,z,w,t,r];


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A three linkages chain :

[Maple Metafile]


I:=Ideal(x^2+y^2-1, (z-x)^2+(w-y)^2-1, (t-z)^2+(r-w)^2-1)

Dim(R/I)


3

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Last point linked to the origin :

I:=Ideal(x^2+y^2-1, (z-x)^2+(w-y)^2-1, (t-z)^2+(r-w)^2-1,t,r)


Dim(R/I)


1

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A Cuadrilateral:



[Maple Metafile]

Use R ::= Q[x,y,z,w,t,r,a,b];

I:=Ideal((z-x)^2+(w-y)^2-1, (t-z)^2+(r-w)^2-1,(t-a)^2+(r-b)^2-1, (a-x)^2+(b-y)^2-1)

Dim(R/I)


4

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With two diagonal bars :


I:=Ideal((z-x)^2+(w-y)^2-1, (t-z)^2+(r-w)^2-1,(t-a)^2+(r-b)^2-1, (a-x)^2+(b-y)^2-1,
(t-x)^2+(r-y)^2-2, (a-z)^2+(b-w)^2-2)


Dim(R/I)


3