approximate non-linear transformation of affine combination
However, sometimes it is desirable to compute the image as if were a linear (or affine) transformation; the result is then hoped to be a good approximation to the true value. (See below for an application.)
Actually, it is possible to show that, provided that is twice continuously differentiable, the approximation is good to first order, despite the absence of any derivatives of in the formula. The domain and range of may be any normed vector spaces.
First, we write:
If , then , and so the error term in the Taylor expansion can be simplified to .
Substituting another Taylor expansion
into the first, we obtain:
Furthermore, it is not hard to see, by accounting the error from the Taylor expansions more carefully, that we have the bound:
where is the maximum, as ranges inside the convex hull formed by the points , of the quantity . Finally, the point over which we performed Taylor expansions can be replaced by any other point , and so correspondingly can be replaced by .
Application in computer graphics
The principle just derived is often applied in vector-based computer graphics when curved objects are drawn by cubic Bézier curves:
which are affine combinations of the control points . To compute and display a smooth transformation of such curves, it may be too much work to compute repeatedly for many parameter values . Provided is not too wavy, computing and displaying is vastly more efficient, and may result in little or no visually perceptible difference.
As a concrete example, consider bending a straight line segment into a circle. Mathematically, we are mapping the interval via . If the interval is split into sub-segments, each considered as a cubic Bézier curve with its interior control points both set at the midpoint of the line segment, then a circle can be approximated by transforming these control points. The following diagram shows the approximation for 24 segments (three Bézier curves per arc).
|Title||approximate non-linear transformation of affine combination|
|Date of creation||2013-03-22 16:50:35|
|Last modified on||2013-03-22 16:50:35|
|Last modified by||stevecheng (10074)|