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Suppose we have a circle stroke that is 100px in width. We now want to change it so that the center left side has a stroke with of 50px and the width should increase symmetrically such that when the top and bottom sides reach the right half of the circle the width of the top and bottom left half is 100px?

Also, this might be the same technique, but suppose the center point of the right side of the circle stroke has a width of 100px and we want this to decrease symmetrically and continuously until we reach the left side center point which has a 50px width.

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    Well you can do something like this with width tool. But it may be easier on you if you dont consider the stroke at all but instead 2 separate shapes made into a compoind object. – joojaa Jun 1 '18 at 8:01
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Set the Display Units in the Document Properties (on the Page tab) to pixels. And set up some vertical guides to help you measure the stroke widths.

The following method uses two circles to do this. It's easier and quicker than using a PowerStroke path effect for varying stroke widths - but that is another way you could do it.

  1. Draw the larger circle first. Hold down Ctrl as you click and drag to constrain the proportions of the circle. Use snapping so that everything lines up with the guides.

  2. Then draw the smaller circle.

  3. Using snapping, move them into the correct position

  4. Select both circles, then click Path > Difference (Shortcut: Ctrl+-).

  5. Set a fill and no stroke.

enter image description here

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Your text wants the right half of the circle to have constant stroke width and in left half the width increases upwards and downward.

There are infinite number of solutions where edge direction changes continuously. One possiblity is to match ellipse and circle:

enter image description here

  1. from the top : red circle, green ellipse, orange circle, blue rectangle

  2. orange circle and the rectangle are intersected to make a half circle

  3. the ellipse and the half circle are united

  4. the small circle is subtracted

This way you get continuous direction changing to the outer edge. A critical watcher can see that the curvature doesn't change continuously in the top and bottom joints. That can be helped by dragging the left side handles a little to left. The handles should be LR symmetric.

enter image description here

The cyan rectangle is drawn to measure the approximate lengths of the right side. It's done by eye only.

The orange copy is made to drag the handles as much. Have snap to paths, nodes and handles on and hold the Ctrl key as you drag.

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