InterpolationOrder

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`InterpolationOrder`

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InterpolationOrder

is an option for Interpolation, as well as ListLinePlot, ListPlot3D, ListContourPlot, and related functions, that specifies what order of interpolation to use.

Details

InterpolationOrder->n specifies that polynomials of degree n should be fitted between data points.
For multidimensional data, the polynomials are taken to be of degree n in each variable.
InterpolationOrder->None specifies that data points in plots should be joined without interpolation.
InterpolationOrder->0 yields a collection of flat regions, with steps at each data point.
InterpolationOrder->1 joins data points with straight lines in 2D, and with piecewise polygonal surface elements in 3D.
Higher interpolation orders generally lead to increasingly smooth curves or surfaces.
In functions such as NDSolve, InterpolationOrder->All specifies that the interpolation order should be chosen to be the same as the order of the underlying solution method.
InterpolationOrder can also be used in functions like Manipulate, to specify the smoothness of animations between control points such as bookmarks.

Examples

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Basic Examples (3)Summary of the most common use cases

Use different interpolation orders for curves:

In[1]:=1

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https://wolfram.com/xid/0puyxkh7ci-glqey3

In[2]:=2

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https://wolfram.com/xid/0puyxkh7ci-iffwcm

Out[2]=2

Use different interpolation orders for surfaces:

In[1]:=1

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https://wolfram.com/xid/0puyxkh7ci-2a9302

In[2]:=2

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https://wolfram.com/xid/0puyxkh7ci-n96cc1

Out[2]=2

Use different interpolation orders when constructing an InterpolatingFunction:

In[1]:=1

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https://wolfram.com/xid/0puyxkh7ci-c7hbzp

In[2]:=2

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https://wolfram.com/xid/0puyxkh7ci-ey1m5o

Out[2]=2

Scope (4)Survey of the scope of standard use cases

Use piecewise quintic interpolation to approximate the sine function:

In[1]:=1

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https://wolfram.com/xid/0puyxkh7ci-h3vy0n

Out[1]=1

Show the approximation error:

In[2]:=2

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https://wolfram.com/xid/0puyxkh7ci-b9ti1o

Out[2]=2

Show the smoothing effect of higher interpolation order in plotting:

In[1]:=1

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https://wolfram.com/xid/0puyxkh7ci-e54c3o

In[2]:=2

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https://wolfram.com/xid/0puyxkh7ci-crwdko

Out[2]=2

Show the smoothing effect of higher interpolation order for GCD data:

In[1]:=1

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https://wolfram.com/xid/0puyxkh7ci-jxrwi5

In[2]:=2

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https://wolfram.com/xid/0puyxkh7ci-bij57t

Out[2]=2

Get a solution that uses interpolation of the same order as the method from NDSolve:

In[1]:=1

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https://wolfram.com/xid/0puyxkh7ci-6149w

Out[1]=1

This is more time consuming than the default interpolation order used:

In[2]:=2

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https://wolfram.com/xid/0puyxkh7ci-tmf5n

Out[2]=2

It is much better in between steps:

In[3]:=3

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https://wolfram.com/xid/0puyxkh7ci-k48sku

Out[3]=3

Possible Issues (1)Common pitfalls and unexpected behavior

Very high-order interpolation can lead to large errors:

In[1]:=1

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https://wolfram.com/xid/0puyxkh7ci-c0b2fg

In[2]:=2

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https://wolfram.com/xid/0puyxkh7ci-n82dao

Interpolate with order 20:

In[3]:=3

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https://wolfram.com/xid/0puyxkh7ci-gqo2u3

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0puyxkh7ci-bn5jic

Out[4]=4

Piecewise interpolation with lower order makes a much better approximation:

In[5]:=5

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https://wolfram.com/xid/0puyxkh7ci-cbqvpu

Out[5]=5

Show the approximation error for different interpolation orders:

In[6]:=6

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https://wolfram.com/xid/0puyxkh7ci-du2lpy

Out[6]=6

Neat Examples (1)Surprising or curious use cases

Zero-order interpolation, with Voronoi cells having the constant value:

In[1]:=1

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https://wolfram.com/xid/0puyxkh7ci-5x05f

Out[1]=1

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