A more general statement would be to say it will exactly fit four constraints. Each constraint can be a point, angle , or curvature which is the reciprocal of the radius of an osculating circle. Angle and curvature constraints are most often added to the ends of a curve, and in such cases are called end conditions. Identical end conditions are frequently used to ensure a smooth transition between polynomial curves contained within a single spline. Higher-order constraints, such as "the change in the rate of curvature", could also be added.
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A more general statement would be to say it will exactly fit four constraints. Each constraint can be a point, angle , or curvature which is the reciprocal of the radius of an osculating circle. Angle and curvature constraints are most often added to the ends of a curve, and in such cases are called end conditions. Identical end conditions are frequently used to ensure a smooth transition between polynomial curves contained within a single spline.
Higher-order constraints, such as "the change in the rate of curvature", could also be added. This, for example, would be useful in highway cloverleaf design to understand the rate of change of the forces applied to a car see jerk , as it follows the cloverleaf, and to set reasonable speed limits, accordingly. 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. An exact fit to all constraints is not certain but might happen, for example, in the case of a first degree polynomial exactly fitting three collinear points. In general, however, some method is then needed to evaluate each approximation. The least squares method is one way to compare the deviations.
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. 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.
The effect of averaging out questionable data points in a sample, rather than distorting the curve to fit them exactly, may be desirable. If a curve runs through two points A and B, it would be expected that the curve would run somewhat near the midpoint of A and B, as well.
This may not happen with high-order polynomial curves; they may even have values that are very large in positive or negative magnitude. Low-order polynomials tend to be smooth and high order polynomial curves tend to be "lumpy". To define this more precisely, the maximum number of inflection points possible in a polynomial curve is n-2, where n is the order of the polynomial equation.
An inflection point is a location on the curve where it switches from a positive radius to negative. We can also say this is where it transitions from "holding water" to "shedding water". Note that it is only "possible" that high order polynomials will be lumpy; they could also be smooth, but there is no guarantee of this, unlike with low order polynomial curves. A fifteenth degree polynomial could have, at most, thirteen inflection points, but could also have twelve, eleven, or any number down to zero.
The degree of the polynomial curve being higher than needed for an exact fit is undesirable for all the reasons listed previously for high order polynomials, but also leads to a case where there are an infinite number of solutions. For example, a first degree polynomial a line constrained by only a single point, instead of the usual two, would give an infinite number of solutions.
This brings up the problem of how to compare and choose just one solution, which can be a problem for software and for humans, as well.
For this reason, it is usually best to choose as low a degree as possible for an exact match on all constraints, and perhaps an even lower degree, if an approximate fit is acceptable.
Relation between wheat yield and soil salinity  Fitting other functions to data points[ edit ] Other types of curves, such as trigonometric functions such as sine and cosine , may also be used, in certain cases. In spectroscopy, data may be fitted with Gaussian , Lorentzian , Voigt and related functions. In agriculture the inverted logistic sigmoid function S-curve is used to describe the relation between crop yield and growth factors.
The blue figure was made by a sigmoid regression of data measured in farm lands. It can be seen that initially, i. Algebraic fit versus geometric fit for curves[ edit ] For algebraic analysis of data, "fitting" usually means trying to find the curve that minimizes the vertical y-axis displacement of a point from the curve e.
However, for graphical and image applications geometric fitting seeks to provide the best visual fit; which usually means trying to minimize the orthogonal distance to the curve e.
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Пляжная мода: как не нужно одеваться на пляж