Change position and orientation of 3D object

Question

I was trying to understand how 3D objects work in Mathematica, but I run into a problem I am not able to solve by myself. I have the following 3D plot:

f1=SphericalPlot3D[1 - Sin[t], {t, 0, Pi/2}, {f, 0, 2 Pi}]

What I would like is to change the position and orientation of this object, so that if $\vec a$ and $\vec b$ are any two vectors, I would get something like

Is this possible?

Maybe its even easier than this, becuase what I actually need is to place that figure into another (previously generated) plot, at a specific position specified by $\vec a,\vec b$. I thought I could use Epilog, together with some placing command or something like that, but Im unable to make it work. My intention is to do something like Plot3D[...,Epilog->Placed[f1,{a,b}]] (this is obvioulsy schematic, because the notation makes no sense), but I have no clue as to how to proceed.

Thank you in advance.

Answer 1

f1 =  First @ SphericalPlot3D[1 - Sin[t], {t, 0, Pi/2}, {f, 0, 2 Pi}];

a = {2, 2, 1};
b = {0, 2, 3};

Graphics3D[{
  Thick, Arrow[{{0, 0, 0}, a}], Arrow[{{0, 0, 0}, b}],

  GeometricTransformation[
   Translate[Scale[f1, {1, 1, 1} Norm[a - b], {0, 0, 0}], a],
   RotationTransform[{{0, 0, 1}, -a + b}, a]
   ]
  },
 Axes -> True, BoxRatios -> Automatic
 ]

Maybe more readable approach:

x = {0, 0, 1};
a = {1, 1, .5};
b = {0, 1, 2};
Graphics3D[{
  Red, Thick, Arrow[{{0, 0, 0}, a}],
  Blue, Arrow[{{0, 0, 0}, b}],
  Green, Arrow[{{0, 0, 0}, x}],
  {Opacity[.1], f1},
  GeometricTransformation[
   f1,
   {Norm[b - a] RotationMatrix[{x, b - a}], a}
   ]
  },
 Axes -> True, BoxRatios -> Automatic
 ]