A few scholars have questioned the passage, contending that the absence of Christian tampering or interpolation does not itself prove authenticity.

However, it is the interpolation method that in theory exactly reconstructs any given bandlimited x ( t ) with any bandlimit B < 1 /( 2T ); any other method that does so is formally equivalent to it.

However, highly localized lighting effects ( such as specular highlights, e. g. the glint of reflected light on the surface of an apple ) will not be rendered correctly, and if a highlight lies in the middle of a polygon, but does not spread to the polygon's vertex, it will not be apparent in a Gouraud rendering ; conversely, if a highlight occurs at the vertex of a polygon, it will be rendered correctly at this vertex ( as this is where the lighting model is applied ), but will be spread unnaturally across all neighboring polygons via the interpolation method.

Additionally they try to do linear interpolation along a line of pixels to simplify the set-up ( compared to 2d affine interpolation ) and thus again the overhead ( also affine texture-mapping does not fit into the low number of registers of the x86 CPU ; the 68000 or any RISC is much more suited ).

In general, the difference methods can be a good choice when one does not know how many points, what degree of interpolating polynomial, will be needed for the desired accuracy, and when one wants to look first at linear and other low-degree interpolation, successively judging accuracy by the difference in the results of two successive polynomial degrees.

Because this response is a smooth continuous function, sampling with ever-faster TRs does not help ; it just gives more points on the response curve obtainable by simple linear interpolation anyway.

Similar to interpolation in many respects, a key difference is that the surface does not, in general, pass through the central control points ; rather, it is " stretched " toward them as though each were an attractive force.

" The term does not appear consistently in the manuscripts and Menachem Meiri ( 1249 – c. 1310 ) in his commentary on the Talmud Beit HaBechirah regarded it as a late interpolation.

Many scanners use interpolation techniques to achieve a higher effective SPI rating, with some manufacturers offering nearly one million SPI, though the quality is primarily limited by the optical resolution ; interpolated SPI does not provide additional pixels in the scanned image.

Z-fighting which cannot be entirely eliminated in this manner is often resolved by the use of a stencil buffer, or by applying a post transformation screen space z-buffer offset to one polygon which does not affect the projected shape on screen, but does affect the z-buffer value to eliminate the overlap during pixel interpolation and comparison.

The present book describes the hedonist ' Yang ' chapter as ' so unlike the rest of that it must be from another hand … The thought is certainly very different, and it does show the signs of editing and interpolation by the Taoist author … But although close scrutiny generally reveals marked differences in style between the body of the book and passages borrowed from earlier sources, I could find none to distinguish the hedonist chapter from the rest.

In contrast to Lagrange interpolation and Hermite interpolation, a Birkhoff interpolation problem does not always have a unique solution.

The question now becomes: does a Birkhoff interpolation problem with a given incidence matrix have a unique solution for any choice of the interpolation points?

Two of the versions are very similar and represent one manuscript tradition ; they incorporate a lengthy excerpt from a certain Book of Zoroaster appended to the Apocryphon ( as chapters 15: 29 – 19: 8f ) A shorter version of the Apocryphon found at Nag Hammadi does not contain the interpolation and represents another manuscript tradition.

The circuitry built into the slider counts the bars as the slider moves and does a linear interpolation based on the magnitudes of the capacitors to find the precise position of the slider.

After this, Afroman mentions his name and birthplace ( East Palmdale ), says the tumbleweed he smokes is " bomb as hell " and then says he does not believe in Hitler ( this line is actually an interpolation of John Lennon's song " God ").

: The most obvious and perhaps the correct explanation of this passage is to declare it a late Christian interpolation, yet “ the man ” does not fit the medieval Christian concept of Jesus.

For example, in linear interpolation, a line that increases one ordinate ( y-value ) for every two abscissa ( x-value ) has a ratio ( also known as slope or rise-over-run ) of 1 / 2.

In the latter case, the method of interpolation affects the reliability of individual isolines and their portrayal of slope, pits and peaks.

Premultiplied alpha has some practical advantages over normal alpha blending because premultiplied alpha blending is associative and linear interpolation gives better results, although premultiplication can cause a loss of precision and, in extreme cases, a noticeable loss of quality.

and is equivalent to linear interpolation.

* The curve begins at P < sub > 0 </ sub > and ends at P < sub > n </ sub >; this is the so-called endpoint interpolation property.

While this method is traditionally attributed to a 1965 paper by J. W. Cooley and J. W. Tukey, Gauss developed it as a trigonometric interpolation method.

This interpolation is not unique: aliasing implies that one could add N to any of the complex-sinusoid frequencies ( e. g. changing to ) without changing the interpolation property, but giving different values in between the points.

First, it consists of sinusoids whose frequencies have the smallest possible magnitudes: the interpolation is bandlimited.

In contrast, the most obvious trigonometric interpolation polynomial is the one in which the frequencies range from 0 to ( instead of roughly to as above ), similar to the inverse DFT formula.

" Although the purported citation of Euclid by Archimedes has been judged to be an interpolation by later editors of his works, it is still believed that Euclid wrote his works before those of Archimedes.

When that interval is 0. 5 / T, the applicable reconstruction formula is the Whittaker – Shannon interpolation formula.

In most cases, N is chosen equal to the length of non-zero portion of s. Increasing N, known as zero-padding or interpolation, results in more closely spaced samples of one cycle of S < sub > 1 / T </ sub >( ƒ ).

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.

A different problem which is closely related to interpolation is the approximation of a complicated function by a simple 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 accuracy.

There is also another very different kind of interpolation in mathematics, namely the " interpolation of operators ".

Points through which curve is spline ( mathematics ) | splined are red ; the blue curve connecting them is interpolation.

The general scholarly view is that while the Testimonium Flavianum is most likely not authentic in its entirety, it is broadly agreed upon that it originally consisted of an authentic nucleus with a reference to the execution of Jesus by Pilate which was then subject to Christian interpolation.

In between a process of interpolation is used, generally a linear one.

François Callet's seven-place table ( Paris, 1795 ), instead of stopping at 100, 000, gave the eight-place logarithms of the numbers between 100, 000 and 108, 000, in order to diminish the errors of interpolation, which were greatest in the early part of the table ; and this addition was generally included in seven-place tables.

After 6. cxd5 Nxg5 7. Nxg5 e6, White has 8. Qd2 exd5 9. Qe3 +, with attacking chances, or the more usual 8. Nf3 exd5 ( though the interpolation 8 ... h6 9. Nf3 exd5 is a significant alternative ), after which play generally proceeds on lines analogous to the Queen's Gambit Declined, Exchange Variation, with a queenside minority attack by White ( b2 – b4 – b5xc6 ), as Black aims for his traditional kingside play with ... f7 – f5 – f4 and, in this case, ... g6 – g5.

Resampling may be referred to as decimation or interpolation, accordingly as the sampling rate decreases or increases – as in sampling and reconstruction generally, the same criteria generally apply in both cases, and thus the same filter can be used.

: Aircraft seldom fly exactly through weather gridpoints or at the exact heights at which weather predictions are available, so some form of horizontal and vertical interpolation is generally needed.

To proceed from one stage to the next a sufficient number of feed states must be investigated to allow for interpolation ; ;

For 3D animation Bézier curves are often used to define 3D paths as well as 2D curves for keyframe interpolation.

where the coefficients X < sub > k </ sub > are given by the DFT of x < sub > n </ sub > above, satisfies the interpolation property for.

In Ovid's interpolation, when Hera learned of Argus ' death, she took his eyes and placed them in the plumage of the peacock, accounting for the eye pattern in its tail.

Plutarch identified this Amphidamas with the hero of the Lelantine War between Chalcis and Eretria and he concluded that the passage must be an interpolation into Hesiod's original work, assuming that the Lelantine War was too late for Hesiod.

( Köstenberger pages 104-105 ) Köstenberger concurs with John Meier that it is highly unlikely for the passage to be a Christian interpolation given that in New Testament texts James is referred to as the " brother of the Lord " rather than the " brother of Jesus ", and that a Christian interpolator would have provided a more detailed account at that point.

According to Evans, if the passage had been an interpolation after the emergence of conflicts between Jews and Christians, it would have had a more accusative tone, but in its current form reads as one would expect it to read for a passage composed by Josephus towards the end of the first century.

Y Gododdin is known only from a 13th-century manuscript, so it is impossible to determine whether this passage is original or a later interpolation, but John Koch's view that the passage dates from a 7th-century or earlier version is regarded as unproven ; 9th-or 10th-century dates are often proposed for it.

These same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations.

Perl, for example, with its ' here document ' can hold multiple 4GL programs, as well as multiple JavaScript programs, in part of its own perl code and use variable interpolation in the ' here document ' to support multi-language programming.

As an advocate for humanity he gains semi-divine status at Athens, where the episode in Theogony in which he is liberated is interpreted by Casanova as a post-Hesiodic interpolation.

A designer of a system that deals with sampling and reconstruction processes needs a thorough understanding of the signal to be sampled, in particular its frequency content, the sampling frequency, how the signal is reconstructed in terms of interpolation, and the requirement for the total reconstruction error, including aliasing, sampling, interpolation and other errors.

In the end the whole process can be implemented for discrete sound signals as interpolation between values with similar phase and similar time.

Whereas Gregory distributed grain to the Roman populace as invasion loomed, Sabinian sold it for high prices ( though this may be a later interpolation by Gregory's biographers ).

Shen Kuo was also known for hydraulic clockworks, as he invented a new overflow-tank clepsydra which had more efficient higher-order interpolation instead of linear interpolation in calibrating the measure of time.

In interpolation-sequential search, interpolation is used to find an item near the one being searched for, then linear search is used to find the exact item.

Using big-O notation, the performance of the interpolation algorithm on a data set of size N is O ( N ); however under the assumption of a uniform distribution of the data on the linear scale used for interpolation, the performance can be shown to be O ( log log N ).

Practical performance of interpolation search depends on whether the reduced number of probes is outweighed by the more complicated calculations needed for each probe.

It can be useful for locating a record in a large sorted file on disk, where each probe involves a disk seek and is much slower than the interpolation arithmetic.

When sort keys for a dataset are uniformly distributed numbers, linear interpolation is straightforward to implement and will find an index very near the sought value.

On the other hand, for a phone book sorted by name, the straightforward approach to interpolation search doesn't apply.