Definition of vertex and edge.
So I'm trying to name things in a graphics library. Reading Wiki it says that a vertex is created by the meeting point of lines, line segments and rays. Can the meeting of curves create a vertex, or is there another name for this? Similarly can edges be curved or is there another name for a curved edge?
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$\begingroup$If two curves of any kind (including lines, segments, rays) meet locally in a single point, that point is in general called an "intersection point". (By "locally" I mean within some small distance about the intersection point, there is no other intersection - though there may be many other intersections further away.)
"Vertex" is generally used for a point where a curve turns a sharp corner. That is, there are two tangent rays to the curve at the point, one on each side, which make an angle. When two curves cross, there are $4$ such angles:
If you are discussing one of those angles, then effectively, you are creating a new curve from stitching together the section of one of the curves on one side of the angle with the section of the other curve on the other side of the angle. This new curve has the point of intersection as a vertex. So in this sense, you can consider any point of intersection to be a vertex. But generally that terminology is not used unless you are discussing some figure that involves an angle at the point.
That said, there are no Mathematical Terminology Police, ready to come breaking down your door if you dare to "misuse" a term. You have the freedom to bend the terminology to meet your needs. I do not think it would be wrong for you to refer to all points of intersection as "vertices" if it is convenient for your application.
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