Tapering (mathematics) should not be confused with Tapering (signal processing).
In mathematics, physics, and theoretical computer graphics, tapering is a kind of shape deformation.[1] [2] Just as an affine transformation, such as scaling or shearing, is a first-order model of shape deformation, tapering is a higher order deformation just as twisting and bending. Tapering can be thought of as non-constant scaling by a given tapering function. The resultant deformations can be linear or nonlinear.
To create a nonlinear taper, instead of scaling in x and y for all z with constants as in:
q=\begin{bmatrix} a&0&0\\ 0&b&0\\ 0&0&1 \end{bmatrix}p,
let a and b be functions of z so that:
q=\begin{bmatrix} a(pz)&0&0\\ 0&b(pz)&0\\ 0&0&1 \end{bmatrix}p.
An example of a linear taper is
a(z)=\alpha0+\alpha1z
a(z)={\alpha}0+{\alpha}1z+
| 2 | |
| {\alpha} | |
| 2z |
As another example, if the parametric equation of a cube were given by ƒ(t) = (x(t), y(t), z(t)), a nonlinear taper could be applied so that the cube's volume slowly decreases (or tapers) as the function moves in the positive z direction. For the given cube, an example of a nonlinear taper along z would be if, for instance, the function T(z) = 1/(a + bt) were applied to the cube's equation such that ƒ(t) = (T(z)x(t), T(z)y(t), T(z)z(t)), for some real constants a and b.