find coordinates of rotated sigmoidal curve

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Given the equation for sigmoidal function and the coordinates of point A (see fig.), how can I find the coordinates (blue points) for a rotated curve passing through a point A? Please see the attached figure.enter image description here

My current thoughts: In the case of two lines, one could just easily find an angle (θ) between two lines and get the new coordinates with the formula on figure 2. Maybe in the case of two sigmoidal curves, it is also possible to find an angle between them and then rotate each point individually by this angle.

enter image description here