Let $v_{0}=(1,0,0)$. Then how to plot in 3D the geometric object $$\left\{ \left(v_{0}\cdot v_{1},v_{0}\cdot v_{2},v_{1}\cdot v_{2}\right):v_{1},v_{2}\in\mathbb{R}^{3} \text{are unit vectors}\right\} $$ in $\mathbb{R}^{3}$ in Mathemtica? Thanks.

  • $\begingroup$ As written, the set seems to consist of a single point. You should maybe check your expression once more. $\endgroup$ – Henrik Schumacher Mar 5 at 7:33
  • $\begingroup$ @HenrikSchumacher $v_1$ and $v_2$ vary, so it should not be a single point. Best regards. $\endgroup$ – Adele Mar 5 at 7:46
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    $\begingroup$ No, I do not know what it really means because you have not told me, yet. I am not kidding here. We are discussing here about a matter that is like me saying: "Uh, there is only half a bottle of milk in the fridge; so maybe you should consider to buy a new one." and you are like "Thanks, but no. The weather could change." That just does not make sense. I won't continue to discuss this with you. I am off here. $\endgroup$ – Henrik Schumacher Mar 5 at 8:10
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    $\begingroup$ Not really satisfactory: convex hull of 10000 randomly chosen points from the set ConvexHullMesh[With[{v0={1,0,0}},Map[{v0.#[[1]],v0.#[[2]],#[[1]].#[[2]]}&,RandomPoint[Sphere[],{10000,2}]]]] $\endgroup$ – მამუკა ჯიბლაძე Mar 5 at 8:47
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    $\begingroup$ More accurate version is ContourPlot3D[c^2+x1^2-2 c x1 x2+x2^2==1,{x1,-1,1},{x2,-1,1},{c,-1,1}] but it requires some math preprocessing $\endgroup$ – მამუკა ჯიბლაძე Mar 5 at 12:20

Could define it parametrically.

pr = ParametricRegion[{{x1, y1, x1*y1 + x2*y2 + x3*y3}, 
    x1^2 + x2^2 + x3^2 == 1 && y1^2 + y2^2 + y3^2 == 1},
{x1, x2, x3, y1, y2, y3}];

We can "see" it using Region.


enter image description here

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