What is "interpolation"

Interpolation \In*ter`po*la"tion\, n. [L. interpolatio an alteration made here and there: cf. F. interpolation.]

1. The act of introducing or inserting anything, especially that which is spurious or foreign.

2. That which is introduced or inserted, especially something foreign or spurious.

   Bentley wrote a letter . . . . upon the scriptural glosses in our present copies of Hesychius, which he considered interpolations from a later hand.
   --De Quincey.

3. (Math.) The method or operation of finding from a few given terms of a series, as of numbers or observations, other intermediate terms in conformity with the law of the series.

1610s, from French interpolation (early 17c.), or directly from Latin interpolationem (nominative interpolatio), noun of action from past participle stem of interpolare (see interpolate).

n. 1 (context music English) An abrupt change in elements, with continuation of the first idea. 2 (context mathematics science English) the process of estimating the value of a function at a point from its values at nearby points. 3 (context computing English) The process of including and processing externally-fetched data in a document or program; see interpolate. 4 That which is introduced or inserted, especially something foreign or spurious.

1. n. a message (spoken or written) that is introduced or inserted; "with the help of his friend's interpolations his story was eventually told"; "with many insertions in the margins" [syn: insertion]

2. (mathematics) calculation of the value of a function between the values already known

3. the action of interjecting or interposing an action or remark that interrupts [syn: interjection, interposition, interpellation]

In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.

In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate (i.e. estimate) the value of that function for an intermediate value of the independent variable. This may be achieved by curve fitting or regression analysis.

A different problem which is closely related to interpolation is the approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complex to evaluate efficiently. A few known data points from the original function can be used to create an interpolation based on a simpler function. Of course, when a simple function is used to estimate data points from the original, interpolation errors are usually present; however, depending on the problem domain and the interpolation method used, the gain in simplicity may be of greater value than the resultant loss in precision.

In the examples below if we consider x as a topological space and the function f forms a different kind of Banach spaces then the problem is treated as "interpolation of operators". The classical results about interpolation of operators are the Riesz–Thorin theorem and the Marcinkiewicz theorem. There are also many other subsequent results.

Interpolation (also known as replayed), especially 20th-century music and later, is an abrupt change of musical elements, with (almost immediate) resumption of the main theme or idea. Pieces that are cited as featuring interpolation, among other techniques, are Music for Brass Quintet by Gunther Schuller and Threnody to the Victims of Hiroshima by Krzysztof Penderecki, both 1960–61.

An interpolation, in relation to literature and especially ancient manuscripts, is an entry or passage in a text that was not written by the original author. As there are often several generations of copies between an extant copy of an ancient text and the original, each handwritten by different scribes, there is a natural tendency for extraneous material to be inserted into such documents over time.

Interpolations originally may be inserted as an authentic explanatory note (for example, ), but may also be included for fraudulent purposes. The forged passages and works attributed to the Pseudo-Isidore are an example of the latter. Similarly, the letters of Ignatius of Antioch were interpolated by Apollinarian heretics, three centuries after the originals were written. Charters and legal texts are also subject to forgery of this kind. In the 13th Century a medieval romance, the Prose Tristan, inserted another prose romance, the Vulgate Queste del Saint Graal, in its entirety in order to reinterpret the Quest for the Holy Grail through the optics of the Tristan story.

However, most interpolations result from the errors and inaccuracies which tend to arise during hand-copying, especially over long periods of time. For example, if a scribe made an error when copying a text and omitted some lines, he would have tended to include the omitted material in the margin. However, margin notes made by readers are present in almost all manuscripts. Therefore, a different scribe seeking to produce a copy of the manuscript perhaps many years later could find it very difficult to determine whether a margin note was an omission made by the previous scribe (which should be included in the text) or simply a note made by a reader (which should be ignored or kept in the margin).

Conscientious scribes tended to copy everything which appeared in a manuscript, but in all cases scribes needed to exercise personal judgement. Explanatory notes would tend to find their way into the body of a text as a natural result of this subjective process.

Modern scholars have developed techniques for recognizing interpolation, which are often apparent to modern observers, but would have been less so for medieval copyists.

The Comma Johanneum, for example, is commonly regarded as interpolation. The specific problem of Christian transmission of Jewish texts outside the Jewish and Christian canons is often described as Christian interpolation.

In the context of computer animation, interpolation is inbetweening, or filling in frames between the key frames. It typically calculates the in between frames through use of (usually) piecewise polynomial interpolation to draw images semi-automatically.

For all applications of this type, a set of "key points" is defined by the graphic artist. These are values that are rather widely separated in space or time, and represent the desired result, but only in very coarse steps. The computed interpolation process is then used to insert many new values in between these key points to give a "smoother" result.

In its simplest form, this is the drawing of two-dimensional curves. The key points, placed by the artist, are used by the computer algorithm to form a smooth curve either through, or near these points. For a typical example of 2-D interpolation through key points see cardinal spline. For examples which go near key points see nonuniform rational B-spline, or Bézier curve. This is extended to the forming of three-dimensional curves, shapes and complex, dynamic artistic patterns such as used in laser light shows.

The process can be extended to motions. The path of an object can be interpolated by providing some key locations, then calculating many in between locations for a smooth motion. In addition to position, the speed or velocity, as well as accelerations along a path, can be calculated to mimic real-life motion dynamics. Where the subjects are too large or complex to move, the camera position and orientation can be moved by this process. This last is commonly called motion control.

Going further, orientations (rotations) of objects and parts of objects can be interpolated as well as parts of complete characters. This process mimics that used in early cartoon films. Master animators would draw key frames of the film, then, junior animators would draw the in-between frames. This is called inbetweening or tweening and the overall process is called " key frame animation". To make these motions appear realistic, interpolation algorithms have been sought which follow, or approximate real life motion dynamics. This applies to things such as the motion of arms and legs from frame to frame, or the motion of all parts of a face, given the motion of the important, key points of the face. Defining the motion of key strands of hair, spread around an animal, can be made into full fur. Using custom algorithms, motions with unique, unnatural and entertaining visual characteristics can be formed. The color of an object can be defined by key color-locations or frames allowing the computation of smooth color gradients around an object or varying in time. Algorithms such as the Kochanek–Bartels spline provide additional adjustment parameters which allow customizing the in-between behavior to suit a wide variety of situations. The article on tweening currently demonstrates several of these.

Another important area of this subject is the computational burden of these algorithms. Algorithms with faster execution times are sought to produce more of these results in less time in order to complete these projects quicker. As the resolution increases to produce animated feature films, the amount of processing can increase greatly.

Interpolation is a method of constructing new data points within the range of a discrete set of known data points in the mathematical field of numerical analysis.

Interpolation may also refer to: