How can I describe the intersection between a circle and a curve?

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 cut/intersection (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

The sub-curve at point x, which intersects between the curve C and the circle B, is red color

• This is likely to get a better answer at the math site than the English site: math.stackexchange.com Nov 25, 2015 at 16:09
• I think that my english is not so good to explain it. I am looking for a good expression from English man Nov 25, 2015 at 16:11
• @MoonLee You are asking for a very particular technical term which people in that technical field will know better than the generalist ELU. You should ask over at math.SE Nov 25, 2015 at 18:38
• You apparently want your "sub curve" to "fit" your curve C using some (as yet undefined) metric for "goodness of fit". And you want this "fitting" to occur for a distance of r from the chosen point on your curve C. You need to define the equation that your "sub curve" must follow, and you need to define the equation for the metric of "goodness of fit". Nov 25, 2015 at 23:49
• I'm voting to close this question as off-topic because it really belongs in a math forum. Nov 25, 2015 at 23:51

The red "sub-curve" is an arc in mathematical parlance; possibly, segment might be acceptable in some fields. By analogy, line segment is to line as arc is to curve. In school geometry, the term arc is sometimes limited to arcs of a circle, but this is not the case elsewhere; Wolfram Alpha, for example, has a parabolic arclength calculator. In topology, it might represent a path.

But I'm afraid I haven't done this kind of mathematics since a previous lifetime, and couldn't tell you if there is a more specific technical term for an arc demarcated by the double intersection of two curves. Our sister site, Mathematics.SE may be better-equipped to assist.

• So, I think I will use as sentence "The arc which is segmented by the curve C and circle B, is red color". Is it right? Nov 25, 2015 at 16:47
• @MoonLee I do not know what the proper technical terminology would be. I might as a layman refer to it as the arc of curve C between its intersections with circle B. Nov 25, 2015 at 16:57
• After read more detail, I think that arc is good to describe the sub-curve. If I want to said about area inside and outside of circle B which segmented by the arc of curve, can I said as "the area inside and outside the arc "? Nov 25, 2015 at 17:48

Assuming that you're in the Cartesian plane, I would say "the subset of C lying in the interior of circle B" if you want the curve to be open and "the closure of the subset of C lying in the interior of circle B" for a closed set, but the mathematics you use to describe the set are going to depend on your application, context, and field.

There are a lot of ways to do it.

(BTW, the circle and intersection points should be assigned names for reference purposes).

The simplest is just to say

• the continuous segment of `C` consisting of all points `p` lying inside the circle `Q`.

That would be an open set, excluding the intersection points `y` and `z`.
If you wanted to include `y` and `z` in a closed set, you could say

• the continuous segment of `C` that starts at `y` and ends at `z`.

If you're assuming that `C` is a continuous curve in the Euclidean plane, then one can also speak of the Euclidean distance `δ(p,x)` between some point `p` in `C` and the point `x`.

• for the open set, the segment of `C` consisting of all points `p` such that `δ(p,x) < r`
• for the closed set, the segment of `C` consisting of all points `p` such that `δ(p,x) ≤ r`

Consider the curve segment inscribed in the circle.

"Inscribed" is the mathematical term used for a square whose summits are on a circle or a circle tangent to all square sides.

Try: The arc of the curve C within the circle B.

"Arc" is standard parlance in geometry:

https://en.wikipedia.org/wiki/Arc_%28geometry%29

And "within" is the natural choice for the relation of the arc with the circle:

http://dictionary.reference.com/browse/within?s=t