Offset a curve so that it passes through my point

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I have a spline curve with 4 control points and I have a point in space (Let's call it M) that I want my new curve to go through that point.

These are the steps that I am doing to offset my curve.

  • I get the closest point in the curve from my point M and I get the delta vector from it.

  • I then offset all my control points by that delta and construct my new curve.

In most situations my point M doesn't go through that curve.

Is there anything I am missing? Any pointers would be helpful.

Thank you.