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I am given 2 end points of the chord $AB$ as well as the apothem, the distance from the center point of the circle to the chord. I can easily find the radius circle and midpoint of the chord I'm just unsure where to go from there. My end goal is to get the coordinate pair for the center of the circle.

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    $\begingroup$ Technically, if all you have are two points $A$, $B$ and the length $a$ of the apothem, if $a\ne0$ there are exactly two circles (the centers of which are distinct) such that $AB$ is a chord and the distance of the center from $AB$ is $a$. Namely, a point $C\notin AB$ and its symmetrical $C'$ w.r.t. $AB$. $\endgroup$ – user228113 Mar 4 '16 at 3:52
  • $\begingroup$ Isn't the center just the perpendicular "connector" of the chord of length of the apothem (which can go in two different directions as @G.Sassatelli stated). This is much easier to figure out if you use "vectors"--specifically write a line parameterized--as $x(t) = x_0 + m_xt$ and $y(t) = y_0 + m_yt$. $\endgroup$ – Jared Mar 4 '16 at 3:54
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The center of the line segment is $C =(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}) $.

Since the line to the center is perpendicular to the line segment, which has direction $(x_2-x_1, y_2-y_1) = (dx, dy) $, its direction is $D = (-dy, dx)$ (or $(dy, -dx)$).

Therefore, go from $C$ in that direction by the specified length. If that length is $L$, the center is at $C + L\frac{D}{|D|} $.

(Added later)

Note that, depending on which version of $D$ is used, you get two possible centers, one on each side of the line.

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