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Bottom: evolution of the normalised sum of the squares of the errors. Polynomial curves fitting points generated with a sine function. A line will connect any two points, so a first degree polynomial equation is an exact fit through any two points with distinct x coordinates. This will exactly fit a simple curve to three points. This will exactly fit four points. A more general statement would be to say it will exactly fit four constraints. Angle and curvature constraints are most often added to the ends of a curve, and in such cases are called end conditions.
The first degree polynomial equation could also be an exact fit for a single point and an angle while the third degree polynomial equation could also be an exact fit for two points, an angle constraint, and a curvature constraint. Many other combinations of constraints are possible for these and for higher order polynomial equations. In general, however, some method is then needed to evaluate each approximation. There are several reasons given to get an approximate fit when it is possible to simply increase the degree of the polynomial equation and get an exact match. Even if an exact match exists, it does not necessarily follow that it can be readily discovered. Depending on the algorithm used there may be a divergent case, where the exact fit cannot be calculated, or it might take too much computer time to find the solution. This situation might require an approximate solution.