Can it so happen that "adding" two points of Elliptic Curve (over finite field) never "hits" any third point of that curve? For example Y^2 = X^3 - 6*X + 7 over GF19 looks like this, (unless there is mistake):
Taking two leftmost points P = (0, 11) and Q = (3, 15) it seems the tangent gradient is k = (15 - 11) * inv(3 - 0) where inv(3) seems to be 13, and thus k = 4 * 13 = 14. The "line" between these points is Y = 14 * X + 11, right?
Seemingly it never got to any other of the marked points except these two. Or perhaps I'm doing something wrong, or coefficients A, B, P should be chosen with some special rule to make points addition "more lucky"?