right angle triangles within a minor circle segment

[Mulhacen]

Consider a circle with centre at point O and radius r. Consider a minor segment of the circle with a chord between points A and B.

I know the half-length of AB which is x0. I know the sagitta or height of the segment y0. From this it is straightforward to calculate the angle subtended at circle centre theta_0, and also the radius r.

Now given a general distance x along the chord from A at point D, how do I calculate the orthogonal distance y from the chord at point D to the circle perimeter.

Please see attached diagram.

Many thanks

Tim

 

[Dr.Peterson]

Write the equation of the circle with center [MATH](0, y_0 - r)[/MATH]. That will give you y as a function of x.

 

[Mulhacen]

Write the equation of the circle with center [MATH](0, y_0 - r)[/MATH]. That will give you y as a function of x.

Many thanks! I knew there would be a simple elegant solution. I found a separate one which was messy.

The reason behind the problem is to build a curved garden path.