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39-542: [REDACTED] Look up fitting in Wiktionary, the free dictionary. Fitting can refer to: Curve fitting , the process of constructing a curve, or mathematical function, that has the best fit to a series of data points A dress fitting Piping and plumbing fitting , used in pipe systems to connect straight sections of pipe or tube, adapt to different sizes or shapes, and for other purposes Compression fitting ,

78-430: A diagram of a road or roof, or abstract . An application of the mathematical concept is found in the grade or gradient in geography and civil engineering . The steepness , incline, or grade of a line is the absolute value of its slope: greater absolute value indicates a steeper line. The line trend is defined as follows: Special directions are: If two points of a road have altitudes y 1 and y 2 ,

117-408: A 45° rising line has slope m = +1, and a 45° falling line has slope m = −1. Generalizing this, differential calculus defines the slope of a plane curve at a point as the slope of its tangent line at that point. When the curve is approximated by a series of points, the slope of the curve may be approximated by the slope of the secant line between two nearby points. When the curve is given as

156-401: 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

195-487: A curvature constraint. Many other combinations of constraints are possible for these and for higher order polynomial equations. If there are more than n  + 1 constraints ( n being the degree of the polynomial), the polynomial curve can still be run through those constraints. 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

234-404: A curve, then the slope given by the above definition, is the slope of a secant line to the curve. For a line, the secant between any two points is the line itself, but this is not the case for any other type of curve. For example, the slope of the secant intersecting y = x at (0,0) and (3,9) is 3. (The slope of the tangent at x = 3 ⁄ 2 is also 3 −  a consequence of

273-429: A degree as possible for an exact match on all constraints, and perhaps an even lower degree, if an approximate fit is acceptable. 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 biology, ecology, demography, epidemiology, and many other disciplines,

312-551: A fitting used to join two tubes or thin-walled pipes together Lightbulb socket or lamp fitting Persons with the surname Fitting [ edit ] Andrea Fitting , founder and CEO of Fitting Group Édouard Fitting (1898–1945), Swiss fencer Emma Fitting (1900–1986), Swiss fencer Frédéric Fitting (1902–1998), Swiss fencer Hans Fitting (1906–1938), German mathematician Willy Fitting (1925–2017), Swiss fencer See also [ edit ] Fit (disambiguation) Fitter (disambiguation) Fetting ,

351-625: A line in the plane containing the x and y axes is generally represented by the letter m , and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. This is described by the following equation: (The Greek letter delta , Δ, is commonly used in mathematics to mean "difference" or "change".) Given two points ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} and ( x 2 , y 2 ) {\displaystyle (x_{2},y_{2})} ,

390-412: 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

429-444: A separate function of arc length ; assuming that data points can be ordered, the chord distance may be used. Coope approaches the problem of trying to find the best visual fit of circle to a set of 2D data points. The method elegantly transforms the ordinarily non-linear problem into a linear problem that can be solved without using iterative numerical methods, and is hence much faster than previous techniques. The above technique

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468-562: A slope given as a percentage into an angle in degrees and vice versa are: and where angle is in degrees and the trigonometric functions operate in degrees. For example, a slope of 100 % or 1000 ‰ is an angle of 45°. A third way is to give one unit of rise in say 10, 20, 50 or 100 horizontal units, e.g. 1:10. 1:20, 1:50 or 1:100 (or "1 in 10", "1 in 20", etc.) 1:10 is steeper than 1:20. For example, steepness of 20% means 1:5 or an incline with angle 11.3°. Roads and railways have both longitudinal slopes and cross slopes. The concept of

507-399: A slope is central to differential calculus . For non-linear functions, the rate of change varies along the curve. The derivative of the function at a point is the slope of the line tangent to the curve at the point and is thus equal to the rate of change of the function at that point. If we let Δ x and Δ y be the distances (along the x and y axes, respectively) between two points on

546-459: A surname Fitling , a hamlet in the East Riding of Yorkshire, England Overfitting , production of an analysis that corresponds too closely or exactly to a data set Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Fitting . If an internal link led you here, you may wish to change the link to point directly to

585-410: Is (−2,4). The derivative of this function is d y ⁄ d x = 2 x . So the slope of the line tangent to y at (−2,4) is 2 ⋅ (−2) = −4 . The equation of this tangent line is: y − 4 = (−4)( x − (−2)) or y = −4 x − 4 . An extension of the idea of angle follows from the difference of slopes. Consider the shear mapping Then ( 1 , 0 ) {\displaystyle (1,0)}

624-466: Is an exact fit through any two points with distinct x coordinates. If the order of the equation is increased to a second degree polynomial, the following results: This will exactly fit a simple curve to three points. If the order of the equation is increased to a third degree polynomial, the following is obtained: This will exactly fit four points. A more general statement would be to say it will exactly fit four constraints . Each constraint can be

663-548: Is extended to general ellipses by adding a non-linear step, resulting in a method that is fast, yet finds visually pleasing ellipses of arbitrary orientation and displacement. Note that while this discussion was in terms of 2D curves, much of this logic also extends to 3D surfaces, each patch of which is defined by a net of curves in two parametric directions, typically called u and v . A surface may be composed of one or more surface patches in each direction. Many statistical packages such as R and numerical software such as

702-654: Is mapped to ( 1 , v ) {\displaystyle (1,v)} . The slope of ( 1 , 0 ) {\displaystyle (1,0)} is zero and the slope of ( 1 , v ) {\displaystyle (1,v)} is v {\displaystyle v} . The shear mapping added a slope of v {\displaystyle v} . For two points on { ( 1 , y ) : y ∈ R } {\displaystyle \{(1,y):y\in \mathbb {R} \}} with slopes m {\displaystyle m} and n {\displaystyle n} ,

741-472: Is the standard deviation of the y-values and s x {\displaystyle s_{x}} is the standard deviation of the x-values. This may also be written as a ratio of covariances : There are two common ways to describe the steepness of a road or railroad . One is by the angle between 0° and 90° (in degrees), and the other is by the slope in a percentage. See also steep grade railway and rack railway . The formulae for converting

780-461: Is the exact slope of the tangent. If y is dependent on x , then it is sufficient to take the limit where only Δ x approaches zero. Therefore, the slope of the tangent is the limit of Δ y /Δ x as Δ x approaches zero, or d y /d x . We call this limit the derivative . The value of the derivative at a specific point on the function provides us with the slope of the tangent at that precise location. For example, let y = x . A point on this function

819-451: 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.: 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

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858-480: Is −1. So these two lines are perpendicular. In statistics , the gradient of the least-squares regression best-fitting line for a given sample of data may be written as: This quantity m is called as the regression slope for the line y = m x + c {\displaystyle y=mx+c} . The quantity r {\displaystyle r} is Pearson's correlation coefficient , s y {\displaystyle s_{y}}

897-421: The x -axis is Consider the two lines: y = −3 x + 1 and y = −3 x − 2 . Both lines have slope m = −3 . They are not the same line. So they are parallel lines. Consider the two lines y = −3 x + 1 and y = ⁠ x / 3 ⁠ − 2 . The slope of the first line is m 1 = −3 . The slope of the second line is m 2 = ⁠ 1 / 3 ⁠ . The product of these two slopes

936-530: The gnuplot , GNU Scientific Library , Igor Pro , MLAB , Maple , MATLAB , TK Solver 6.0, Scilab , Mathematica , GNU Octave , and SciPy include commands for doing curve fitting in a variety of scenarios. There are also programs specifically written to do curve fitting; they can be found in the lists of statistical and numerical-analysis programs as well as in Category:Regression and curve fitting software . Slope In mathematics ,

975-470: The growth of a population , the spread of infectious disease, etc. can be fitted using the logistic function . 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.e. at low soil salinity, the crop yield reduces slowly at increasing soil salinity, while thereafter

1014-406: The mean value theorem .) By moving the two points closer together so that Δ y and Δ x decrease, the secant line more closely approximates a tangent line to the curve, and as such the slope of the secant approaches that of the tangent. Using differential calculus , we can determine the limit , or the value that Δ y /Δ x approaches as Δ y and Δ x get closer to zero ; it follows that this limit

1053-406: The slope or gradient of a line is a number that describes the direction of the line on a plane . Often denoted by the letter m , slope is calculated as the ratio of the vertical change to the horizontal change ("rise over run") between two distinct points on the line, giving the same number for any choice of points. The line may be physical – as set by a road surveyor , pictorial as in

1092-471: The above equation generates the formula: The formula fails for a vertical line, parallel to the y {\displaystyle y} axis (see Division by zero ), where the slope can be taken as infinite , so the slope of a vertical line is considered undefined. Suppose a line runs through two points: P  = (1, 2) and Q  = (13, 8). By dividing the difference in y {\displaystyle y} -coordinates by

1131-406: The change in x {\displaystyle x} from one to the other is x 2 − x 1 {\displaystyle x_{2}-x_{1}} ( run ), while the change in y {\displaystyle y} is y 2 − y 1 {\displaystyle y_{2}-y_{1}} ( rise ). Substituting both quantities into

1170-399: The curve that minimizes the vertical ( y -axis) displacement of a point from the curve (e.g., ordinary least squares ). 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.g., total least squares ), or to otherwise include both axes of displacement of a point from

1209-414: The curve. Geometric fits are not popular because they usually require non-linear and/or iterative calculations, although they have the advantage of a more aesthetic and geometrically accurate result. Most commonly, one fits a function of the form y = f ( x ) . The first degree polynomial equation is a line with slope a . A line will connect any two points, so a first degree polynomial equation

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1248-440: The decrease progresses faster. If a function of the form y = f ( x ) {\displaystyle y=f(x)} cannot be postulated, one can still try to fit a plane curve . Other types of curves, such as conic sections (circular, elliptical, parabolic, and hyperbolic arcs) or trigonometric functions (such as sine and cosine), may also be used, in certain cases. For example, trajectories of objects under

1287-482: The difference in x {\displaystyle x} -coordinates, one can obtain the slope of the line: As another example, consider a line which runs through the points (4, 15) and (3, 21). Then, the slope of the line is For example, consider a line running through points (2,8) and (3,20). This line has a slope, m , of One can then write the line's equation, in point-slope form: or: The angle θ between −90° and 90° that this line makes with

1326-578: The graph of an algebraic expression , calculus gives formulas for the slope at each point. Slope is thus one of the central ideas of calculus and its applications to design. There seems to be no clear answer as to why the letter m is used for slope, but it first appears in English in O'Brien (1844) who introduced the equation of a line as " y = mx + b " , and it can also be found in Todhunter (1888) who wrote " y = mx + c ". The slope of

1365-439: The image has slope increased by v {\displaystyle v} , but the difference n − m {\displaystyle n-m} of slopes is the same before and after the shear. This invariance of slope differences makes slope an angular invariant measure , on a par with circular angle (invariant under rotation) and hyperbolic angle, with invariance group of squeeze mappings . The concept of

1404-489: The influence of gravity follow a parabolic path, when air resistance is ignored. Hence, matching trajectory data points to a parabolic curve would make sense. Tides follow sinusoidal patterns, hence tidal data points should be matched to a sine wave, or the sum of two sine waves of different periods, if the effects of the Moon and Sun are both considered. For a parametric curve , it is effective to fit each of its coordinates as

1443-464: The intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Fitting&oldid=1133842450 " Categories : Disambiguation pages Disambiguation pages with surname-holder lists Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Curve fitting For linear-algebraic analysis of data, "fitting" usually means trying to find

1482-477: 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

1521-458: The rise is the difference ( y 2 − y 1 ) = Δ y . Neglecting the Earth's curvature , if the two points have horizontal distance x 1 and x 2 from a fixed point, the run is ( x 2 − x 1 ) = Δ x . The slope between the two points is the difference ratio : Through trigonometry , the slope m of a line is related to its angle of inclination θ by the tangent function Thus,

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