How do you find the equation for a curve between two points?
In general, the equation of a curve in 2-d is of the form f(x, y; a, b, c, d, ..) = 0, where a, b, c, d, ... are parameters that take specific values for a particular curve. See explanation.
Two points are enough for determining the parameters m and c of the straight line
f(x, y; m, c) = y-mx-c=0,
If this passes through (0, 0) and (1, 1) then c = 0 and m = 1. The line is
or for that matter any circle through the origin given by
If the circle passes though (0, 1) and (1, 0), a = b =
If the the number of parameters is more than two, two points are not sufficient. For example, consider
This has three parameters. So, three points are required to determine this sine curve,