Matching point locations: Relative translation, preserving surrounding geometry

Question

How can I move the tip of the pyramid so that it sits at the top of Suzanne's ear (pictured below, the noted point is highlighted)?

1. The objects and their origins need to stay the same.

2. If all of the pictured geometry were part of one object, and #1 was not a requirement, how could this be achieved? (If this were a mesh level transformation instead of an object level transformation)

Answer 1

Multiple objects

This can be done with a combination of transform snapping and 3D cursor snapping.

1. Snap the cursor the the point vertex by selecting it in edit mode (↹ Tab) and pressing ⇧ ShiftS> Cursor to selected:

   

2. Set the pivot point to 3D cursor (.)

3. Set the snap element to vertex and the snap target to Center in 3D view > Header:

   

4. In object mode, grab the pointer object and move it over the target vertex while holding ⎈ Ctrl:

   

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Single mesh

If everything is in the same mesh, you don't even need to use the 3D cursor:

1. Select the geometry you want to move, making sure the vertex on the tip is active (last selected, appears white):

   

2. Set the snap target to Active:

   

3. Snap away by holding ⎈ Ctrl.

   

   Note that you can also enable snapping without holding ⎈ Ctrl by clicking the magnet icon in 3D view > header, then things will snap by default and holding ⎈ Ctrl will make them not snap.