How to find the coordinates of points on an arc knowing the coordinates of points on a chord?

Question

Mathematics

Good afternoon friends!
Help solve this problem. There are coordinates of the points of the segment AB through which the circle passes, that is, in fact, these are the coordinates of the chord. We also know the radius of the circle R. How to find the coordinates of intermediate points on the arc that divide the arc into 1/4, 1/2 and 3/4?
Thanks in advance!

Answer 1

Using the Pythagorean theorem, we find the distance from the center to the chord: d = sqrt(R² − [(x2−x1)² + (y2−y1)²]/4).
Find the midpoint of segment AB (let's call it (x3, y3)). We find the direction vector of the segment AB (x4,y4) = ((x2−x1)/|AB|, (y2−y1)/|AB|), and there are two variants of the center — (x0,y0) = (x3±d ) y4, y3∓d x4).
And then through atan2 we get the corners, arrange them and get as many points as we like through the corners.

Answer 2

You need to find the angle of the sector for these points (start-end):
x = r*cos(fi)
y = r*sin(fi)
divide it properly and find the coordinates of the points
, it is assumed that the center of the circle is in zero coordinates.

Answer 3

1. find the coordinates of the center (the center can be either on one side or on the other side of the segment)
   
2. find the angle from the x-axis to point A
   
3. find angle from x-axis to point B
   
4. find the angle from the x-axis to the desired point on the arc.
   The arc length is proportional to the angle. So 1/2 of the length is at 1/2 of the angle:угол_к_В + (угол_к_А - угол_к_В) / 2
   
5. find coordinates of a point based on center, angle and radius
   

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