Developer: RhinoScript
Summary: Demonstrates how to mark the center points of closed planar curves with a point object.

I have a pattern of circles from which I need to extract their center points. Is there a command or script that I can use to do this?

Yes. See the following RhinoScript code.

 Sub MarkCenterPoints()
   Dim curves, crv, pt, arr
   curves = Rhino.GetObjects("Select closed planar curves", 4, ,True)
   If IsArray(curves) Then
     Rhino.EnableRedraw False
     For Each crv In curves
       pt = vbNull
       If Rhino.IsCircle(crv) Then
         pt = Rhino.CircleCenterPoint(crv)
       Else
         arr = Rhino.CurveAreaCentroid(crv)
         If IsArray(arr) Then pt = arr(0)
       End If
       If IsArray(pt) Then
         Rhino.SelectObject Rhino.AddPoint(pt)
       End If
     Next        
     Rhino.EnableRedraw True
   End If
 End Sub

Here is a button macro version of the above script:

 ! _-Runscript (
 curves = Rhino.GetObjects("Select closed planar curves", 4, ,True)
 If IsArray(curves) Then
 Rhino.EnableRedraw False
 For Each crv In curves
 pt = vbNull
 If Rhino.IsCircle(crv) Then
 pt = Rhino.CircleCenterPoint(crv)
 Else
 arr = Rhino.CurveAreaCentroid(crv)
 If IsArray(arr) Then pt = arr(0)
 End If
 If IsArray(pt) Then
 Rhino.SelectObject Rhino.AddPoint(pt)
 End If
 Next        
 Rhino.EnableRedraw True
 End If
 )