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I am having some issues on how to find the coordinate on a circle.

image

For example I have the below part of a circle where I know :

1) the chord d

2) the height h

Based on those 2 I can calculate the radius of the circle with this formula:

$$R=(d^2+4h^2)/(8h)$$

Now I want given any X coordinate to calculate the Y coordinate so that I get the point on the circles circumference

For example if c=d/2 what is the height h' ?

Any idea how may I do this, any idea ? Thanks in advance. Also I know that I have some questions closed but i wasn't able to answer them, so I'm sorry.

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  • $\begingroup$ Better use Google before mistagging your own question. $\endgroup$ – Rodrigo de Azevedo Oct 4 '19 at 23:48
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The equation of the circle in your setup is

$$\left(x-\frac d2\right)^2+(y+R-h)^2=R^2$$

So, given $x=c$, the corresponding $h’=y$ is

$$h’= \sqrt{ R^2-\left(c-\frac d2\right)^2}-(R-h)$$

where,

$$R=\frac{d^2+4h^2}{8h}$$

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Let consider the equation for the circle centered at the origin

$$X^2+Y^2=R^2 \implies Y=\sqrt{R^2-X^2}$$

and then consider a change in coordinates

  • $Y=y+(R-h)$
  • $X=x-\frac d2$
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