e1 = -(Sin[tt1]*Sin[tt2])/2 - (Sin[tt2]*Sin[tt3])/2 e2 = (Cos[tt1]*Sin[tt2])/2 - (Cos[tt3]*Sin[tt2])/2 e3 = (Cos[tt1]*Cos[tt2]*Sin[tt3])/2 - (Cos[tt3]*Sin[tt1])/ 2 - (Cos[tt1]*Sin[tt3])/2 + (Cos[tt2]*Cos[tt3]*Sin[tt1])/2 e = {e1, e2, e3};
e1 = -(Sin[tt1]*Sin[tt2])/2 - (Sin[tt2]*Sin[tt3])/2
e2 = (Cos[tt1]*Sin[tt2])/2 - (Cos[tt3]*Sin[tt2])/2
e3 = (Cos[tt1]*Cos[tt2]*Sin[tt3])/2 - (Cos[tt3]*Sin[tt1])/2 -
(Cos[tt1]*Sin[tt3])/2 + (Cos[tt2]*Cos[tt3]*Sin[tt1])/2
e = {e1, e2, e3};
and the range of tt1tt1 to tt3tt3 are all from 0 to 2Pi2 Pi. In this case, since we have 3 parameters, the endpoints of e should form a region rather than a surface, and I'd like to plot this region, or its boundary (preferably both).
The only relevant command I found is the RegionPlot3DRegionPlot3D command, but seems that one handles the Cartesian coordinates with inequalities as constraints, which is not true in my case.
Could someone give me a hand? Thanks a lot!
Update###Update
Thanks a lot for your reply! But whenWhen I try this code:
GroebnerBasis[
{e1 == -(Sin[tt1]*Sin[tt2])/2 - (Sin[tt2]*Sin[tt3])/2,
e2 == (Cos[tt1]*Sin[tt2])/2 - (Cos[tt3]*Sin[tt2])/2,
e3 == (Cos[tt1]*Cos[tt2]*Sin[tt3])/2 - (Cos[tt3]*Sin[tt1])/2 -
(Cos[tt1]*Sin[tt3])/2 + (Cos[tt2]*Cos[tt3]*Sin[tt1])/2,
Cos[tt1]^2 + Sin[tt1]^2 == 1,
Cos[tt3]^2 + Sin[tt3]^2 == 1},
{e1, e2, e3},
{Cos[tt1], Sin[tt1], Cos[tt3], Sin[tt3]}]
e2 == (Cos[tt1]*Sin[tt2])/2 - (Cos[tt3]*Sin[tt2])/2, e3 == (Cos[tt1]*Cos[tt2]*Sin[tt3])/2 - (Cos[tt3]*Sin[tt1])/ 2 - (Cos[tt1]*Sin[tt3])/2 + (Cos[tt2]*Cos[tt3]*Sin[tt1])/2, Cos[tt1]^2 + Sin[tt1]^2 == 1, Cos[tt3]^2 + Sin[tt3]^2 == 1}, {e1, e2, e3}, {Cos[tt1], Sin[tt1], Cos[tt3], Sin[tt3]}]'
it gives me an error: General::ivar: "-(1/2)\ Cos[tt3]\ Sin[tt1]+1/2\ Cos[tt2]\ Cos[tt3]\ Sin[tt1]-1/2\ Cos[tt1]\ Sin[tt3]+1/2\ Cos[tt1]\ Cos[tt2]\ Sin[tt3] is not a valid variable.
General::ivar: -(1/2)Cos[tt3]Sin[tt1] + 1/2Cos[tt2]Cos[tt3]Sin[tt1] - 1/2 Cos[tt1]Sin[tt3] + 1/2Cos[tt1]Cos[tt2]Sin[tt3] is not a valid variable.
I thought that the GroebnerBasisGroebner basis is used for solving equations, but in my case, I'm not trying to solve e1=e2=e3=0e1 == e2 == e3 == 0, instead, I'm trying to plot the region that can be traced by the endpoints of ee. Besides, besidestt1, t1 t2 t3tt2 and tt3 are independent variables, so I doubt if we could eliminate some of them, when I'm trying to plot the boundary of the region?
Thanks again!
Update###Update II
toTo Andre: thanks a lot, I tried the code in the link, and it probably gives me the correct result. Now that I already have this region, I wonder if I can do some further operations, such as obtaining some intersection of this region about some plane, for example the plane x+y=0$x+y=0$? Moreover, can I get the volume of this region?
The above answer should have already solved my problem, but I'm kind of curious about the method using the Grobener basis, as well, because I can hardly understand the code in the above link (sorry I'm quite new to mathematicaMathematica). you
You guys are right, the code works after I clear the variables, sorry for the mistake.. Now the code gives me:
{e1^4 + 2 e1^2 e2^2 + e2^4 - 2 e1^4 Cos[tt2] - 4 e1^2 e2^2 Cos[tt2] -
2 e2^4 Cos[tt2] + e1^4 Cos[tt2]^2 + 2 e1^2 e2^2 Cos[tt2]^2 +
e2^4 Cos[tt2]^2 - e1^2 Sin[tt2]^2 - e2^2 Sin[tt2]^2 +
2 e1^2 Cos[tt2] Sin[tt2]^2 + 2 e2^2 Cos[tt2] Sin[tt2]^2 -
e1^2 Cos[tt2]^2 Sin[tt2]^2 - e2^2 Cos[tt2]^2 Sin[tt2]^2 +
e3^2 Sin[tt2]^4}
2 e2^4 Cos[tt2] + e1^4 Cos[tt2]^2 + 2 e1^2 e2^2 Cos[tt2]^2 + e2^4 Cos[tt2]^2 - e1^2 Sin[tt2]^2 - e2^2 Sin[tt2]^2 + 2 e1^2 Cos[tt2] Sin[tt2]^2 + 2 e2^2 Cos[tt2] Sin[tt2]^2 - e1^2 Cos[tt2]^2 Sin[tt2]^2 - e2^2 Cos[tt2]^2 Sin[tt2]^2 + e3^2 Sin[tt2]^4}
I wonder what I can do with this to plot the 3D region? Sorry if my question is silly..
best
