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I've seen the lattice used to move objects into place by deforming the lattice but I want to do something different. I want to move an object to create the deformations of a lattice and apply that deformation to other objects.

I have 7 objects (green) and 7 target objects (red). I want to move each object to its corresponding target. I also have what I call free objects (blue cube) which I would like to be effected based from the deformation of the lattice.

Can I set a lattice so that when the green objects are moved to the red objects position the deformation from the lattice will move the blue cube?

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You can use the Hook Modifier to achieve this effect. You can use any objects as your hooks, including the green spheres you used in your question, but unless you also want the spheres themselves to be rendered, it’s natural to just use empties:

To easily set up the Hook modifiers, you can select both your hook object (in your example, one of the green spheres) and the lattice, and in Edit Mode, select all the lattice’s vertices, and press CtrlHHook to Selected Object. Do this for each of your hook objects, then set the falloff radius you want in each modifier’s options.

Then all you have to do is add a Lattice modifier to your blue cube to apply the lattice’s deformation, and now moving the hooks will move the cube:

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