Given the Y-axis values of a set (n > 3) of 2D-points that are known to be on a circle with an unknown center and radius, is it possible to find these points their corresponding X-axis values, if you know that the points are sampled at a constant, but unknown angle?

So there will be 2 circles: 1 to the left, 1 to the right of the Y-axis, that will return 2 sets of X-axis-values, that correspond to the given set of y-values.

It is known that:

• The y-values are strictly > 0
• These circles touch the Y-axis in 1 place: the first y-value (closest to the origin), where they also touch each other.
• These circles pass through the X-axis twice, but never through the origin (0,0)
• The center(s) of the circles are above the X-axis, at the same y-value, and they have the same radius, so they are mirror images.
• The first (closest to the origin) x-value is always 0

Or simply put: more than half of the circles lies slightly above the horizontal axis, touching the vertical axis in 1 point above the origin.

In the image below, -only- the green dots are initially known, all else must be calculated somehow. The points of which the green dots are the vertical coordinates, are known to be on a circle (the orange circles), and the lowest green dot is never on the origin (0,0): that is the only place where the orange circle touches the black, dotted, vertical Y-axis. The green dotted segments go from the (known) green dots to the (unknown) orange points on the (unknown) circle(s), and turn into orange segments to the corresponding (unknown) red dots (X-axis coordinates).

So what is required are the (X-axis) values of the red dots, if you only know the green dots. It is important to realize that the orange dots on the circle happen to be spread out over the circle at a constant, but unknown angle. So naturally, the arc length between green dots is also constant.

For example: given the following y-values:

yval<-c(0.10000,0.19877,0.29388,0.38298,0.46388,0.53459,0.59337,0.63877,0.66967,0.68531)

The corresponding x-values would be:

xval<-c(0.00000,-0.01564,-0.04654,-0.09194,-0.15072,-0.22143,-0.30233,-0.39143,-0.48654,-0.58531)

The radius $r$: $0.63726$

The circle center at: $(-0.6353,0.05002)$ For the left circle, and inverse these values for the dotted orange right circle.

Given these data, the coordinates of the first (lowest) orange point on the circle will be (0,0.10000) (not so clear in the image because it obviously coincides with the first green dot).

If the given y-values would not come from points that are sampled at a constant angle, a unique solution would not exist, but they are, and 1 solution for the left circle, and 1 for the right should be possible to find. All help much appreciated!

ps: the values are discretely sampled cumulative values of a sinusoid with a period of 40, divided by 10, starting at phase = $pi/2$, so $90°$: in R:

startpoint<-90
n<-10
phaseshift<-9 #so period = 40
cumsum(round(cos((startpoint+((0:(n-1))*phaseshift))*pi/180)/n,5))

pps: I used this simple method to estimate a circle from 3 known points (x,y): small bit of R code included per illustration: I'm sure there are computationally cheaper ways to do it:

circleFromThreePoints<-function(x1,x2,x3,y1,y2,y3)
{
  vara<-x1*(y2-y3)-y1*(x2-x3)+x2*y3-x3*y2;
  varb<-(x1*x1+y1*y1)*(y3-y2)+(x2*x2+y2*y2)*(y1-y3)+(x3*x3+y3*y3)*(y2-y1);
  varc<-(x1*x1+y1*y1)*(x2-x3)+(x2*x2+y2*y2)*(x3-x1)+(x3*x3+y3*y3)*(x1-x2);
  vard<-(x1*x1+y1*y1)*(x3*y2-x2*y3)+(x2*x2+y2*y2)*(x1*y3-x3*y1)+(x3*x3+y3*y3)*(x2*y1-x1*y2)
  varx<- -varb/(2*vara)
  vary<- -varc/(2*vara)
  varr<- (((varb*varb)+(varc*varc)-(4*vara*vard))/(4*vara*vara))^0.5
  # x, y , r: 
  # (x-x1)^2+(y-y1)^2 = r^2 
  # h,k,r for equation: (x-h)^2+(y-k)^2 = r^2
  # To plot: upp<-(((r^2)-((x-h)^2))^0.5)+k & dwn<--(((r^2)-((x-h)^2))^0.5)+k
  return(c(round(varx,5),round(vary,5),round(varr,5)))
}

This question is a more specific version (sampling at a constant angle) than my previously asked, related question