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Hello. I'm a beginner. In the model shown, when I select all of the top vertices and then scale them to the center, the center of scaling is not the center of the 3d-model.

  1. How do I move the center of scaling?
  2. How do I move the manipulator widget where I want it after selecting?
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I assume your mesh is perfectly centered at the 3d space, you could first use ShiftS --> Cursor to center to make sure the 3d cursor is at the center of the 3d space.

Then (with the mouse pointer pointing to the 3d window) you can make the center of scaling (pivot point = little white circle) to be exactly the same as the 3d cursor by:

  • Pressing . on the keyboard

or

  • Select it from the menu enter image description here

By icon of coordinates I think you mean the red, green, blue arrows, sticking out of the pivot point

To show the pivot point after moving 3d cursor: enter image description here

And what pivot point at active element does: enter image description here

And pivot point at median point (origin is the orange dot): enter image description here

(Maybe it is a little bit too detailed) :)

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Here this should be the answer to the OP's question. Please note that you may have to check your model, it does looks to me that you have overlapping polygons. Make sure you don't have overlaps by running "remove double" from your "W" Key drop down menu.

enter image description here

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If the point where you want to use as the centre does not coincide with a vertex there are two ways:

1 Extend a vertex to the point you want, then move the cursor to that point, (shift s) then move centre to cursor (in object mode on left side panel) (then delete the vertex)

or

Use side panel 'n' to set the position of the cursor typing in co-ordinates or by using the slider bar. Holding Shift while using a slider to change values will slow the rate of change so more precise movements can be achieved.

enter image description here

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What you call the "center of scaling" is one of three objects called "Manipulator Widgets". The one shown in your screenshot is the default, called "Translate manipulator widget". There are two others, one for rotation and one for scaling. The location of the manipulator widget is always co-incident with the pivot point, which in your screenshot, appears to be set to "bounding box". If you want the Manipulator widget to be at the 3D cursor, change the pivot point to "3D cursor", using the method The Beachdancer illustrates. There are other options which have their own utility.

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