Basic Examples (2) 

Convert a 2D curve into a spline function preserving the original shape with different accuracy:

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Parametrize 3D data:

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Scope (1) 

Parametrization of an arbitrary polygon with different scale factors:

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Options (7) 

AssignWeights

With option value "AssignWeights"→Automatic, original control points get weights equal to the sum of their Euclidean distances to the adjacent points, which is equivalent to "AssignWeights"→(Total[#2]&). Compare with a weighting function dependent on the interior angle at the node (argument #3):

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CurveClosed

With option value "CurveClosed"→True, a parametric function describing a closed contour is produced. Its geometric form is invariant to the rotation of the points:

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DuplicateNodes

With option value "DuplicateNodes"→n, original distinct control points will be duplicated n times. The default option value "DuplicateNodes"→False is equivalent to "DuplicateNodes"→1 (no point duplication). The option value "DuplicateNodes"→Automatic tries to preserve significant sharp features of the original shape and is equivalent to "DuplicateNodes"→(Max[Floor[Total[#2]/(2#4)],1]&). Compare them with a node duplication function dependent on the interior angle at the node (argument #3):

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FillGaps

With the default option value "FillGaps"→True, intermediate control points are added when the distance between successive nodes is at least twice the scale:

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Turn off introduction of intermediate control points by specifying "FillGaps"→False:

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The same happens when the scale factor is sufficiently large:

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Construct a parametric function from the intermediate points only (the original nodes will not be included):

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Applications (7) 

Look at some jagged contours:

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Parametrize the contours and plot them:

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With automatic node duplication, the sharp features are preserved:

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Provide a custom node duplication function in order to preserve only the features you are interested in:

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Automatic weights are trying to preserve only significant sharp features:

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Increasing the scale flattens out the features:

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Make some features in a particular region stronger by duplicating the corresponding control points:

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Properties and Relations (1) 

Compare the areas of an exact region, its discretized version obtained through Rasterize and a smoothed version of the latter obtained via CurveToBSplineFunction:

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Possible Issues (2) 

The default scaling factor of 1 may be insufficient to reproduce essential features of the shape:

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Decrease the scaling factor in order to reproduce the shape more closely:

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Without additional intermediate control points, significant sharp features may be lost:

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Neat Examples (2) 

Create a parametric function from a discretized implicit region:

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Compare the areas of the exact region, its discretized version obtained through BoundaryDiscretizeRegion and a smoothed version of the latter obtained via CurveToBSplineFunction:

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