Question

How do you find the equation for a curve between two points?

Answer 1

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.

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 `y - x = 0`

or for that matter any circle through the origin given by

`f(x, y; a, b) = x^2+y^2+2ax+2by=0`

If the circle passes though (0, 1) and (1, 0), a = b = `-1/2`, and the circle is given by `x^2+y^2-x-y=0`..

If the the number of parameters is more than two, two points are not sufficient. For example, consider

`f(x,y;a, b, c)=y-a sin ( bx + c )=0`

This has three parameters. So, three points are required to determine this sine curve,