How can we bend a line by an angle?

Question

We have a set of points on the $y$-axis as $(0,y_1), (0,y_2), (0,y_3)$. We want to bend the line containing our points by $cy$, where $c$ is the angle in degree.
$c$ is not the angle from $(0,0)$, but the angle in an imaginary circle on which the bent line lies.

In fact, we want to turn the line $(0,y_1)$ to a slice of a circle with an angle of $c . y_1$.
In a simple case, we want to bend the line of $(0,360/c)$ to a full circle, or the line of $(0,180/c)$ to a semi-circle, or $(0,1)$ to a $c$ slice of the circle.
For example, for $(0,9)$ we have the transformations of
$$c=10° to (4.5,4.5)$$
$$c=20° to (9,0)$$
I can find typical solutions by putting the circle at $(90/c,0)$, but I cannot systematically solve the problem to get a general formula to transform the points on the straight line to the points on the arc.

Ethan Bolker · Answer

Not a complete answer, but possibly helpful (if I understand the question correctly).
First, redraw the picture so that the circle is the unit circle in the plane and the line is the set of points $(1,y)$ along the vertical tangent to the circle at $(1,0)$.
Then the central angle of the line between the origin and $(1,y)$ is $arctan y$. You want to end up at the point on the circle with central angle $c arctan y$. That point has coordinates
$$
(cos(c arctan { y}), sin(c arctan y)) .
$$
The radius you drew suggests the answer above.
Note that this wrapping does not convert distances along the line to distances along the circle. If that is what you want, the map is in fact easier: the point $(1,y)$ maps to $(cos cy, sin cy)$.
These calculations are all in radians. Convert to degrees and move the coordinate system as you wish.
(Perhaps someone will edit this answer to include a picture.)

How can we bend a line by an angle?

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