I have a curve C and a point x in the curve. At the point x, I draw a circle B with radius r and centered at point x. That circle B will segment/intersect (with) the curve C as red sub-curve line. I would like to use English to describe the sub-curve. Could you see my describe and give me some correct and professional way for it? Thank in advance
This is my example image:
The sub-curve at point x, which intersects between the curve C and the circle B, is red color
Or
The arc of the point x which is segmented by the curve C and circle B, is red color
Which one is better to define the sub-curve? If it is possible, let suggest to me the other way
Red Curve is the intersection of disk, with center at $x$ with radius $r$, and curve C.
i-e $\text{Red curve} = (x-x_1)^2 + (y-y_1)^2 \le r^2 \cap C$.
where $ x =(x_1,x_2).$
So, instead of defining a circle define it as a 2-D region(disk).