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I am trying to come up with a formula to calculate the y co-ordinate of the point on the circle in the attached picture (i.e. delta y) based on the circumference of the circle and the distance x. This formula needs to be applicable for any point along the circumference of the circle at any distance x along the chord and for any given radius R. Is this possible?

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I think this is not possible.

enter image description here

According to the stated, I draw the red circle whose radius is R as given. The dotted portion of it formed the locus of all the possible positions of the centers of the said circle. The light green and the dark green circles (with radii both equal to R; and centers at P and Q respectively) are two examples.

After drawing the x= 30 line, we can clearly see that the corresponding ⊿y s’ generated are of different lengths.

This leads to the conclusion that “⊿y cannot be uniquely determined unless the center of the circle is clearly specified.”.

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