A line.
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In classical or Euclidean plane geometry two points defines exactly one line. On a sphere two points can define infinitely many lines only one of which will represent the shortest distance between the points. On other curved surfaces, or in non-Euclidean geometries, the number of lines determined by two points can vary. Even in the Euclidean plane, two points determine infinitely many lines that are not straight!
A Line ;)
An open curve
The shape described is a plane, which is a two-dimensional surface that extends infinitely in both width and length. In geometry, a plane can be uniquely determined by any three non-collinear points on the plane. This is known as the "three-point" or "unique determination" property of a plane. The three points define the plane's orientation and position in three-dimensional space.
Asymptotes