Convex Hull of Points on a Straight Line

Let L be a straight line on the cartesian plane. Pick a finite set of
points S on that line. What is the convex hull of S?

From my understanding of hull points, the convex hull is the two points
farthest away from each other in S. Is this true?

I ask because there is a version of Graham's algorithm in my discrete
mathematics book that does not consider the case when there are two
points whose segments with the first hull point have the same angle
with respect to the horizontal. I guess this version of the algorithm
is flawed in that respect.

.

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