Tapering (mathematics)

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:

${\displaystyle q={\begin{bmatrix}a&0&0\\0&b&0\\0&0&1\end{bmatrix}}p,}$

let a and b be functions of z so that:

${\displaystyle q={\begin{bmatrix}a(p_{z})&0&0\\0&b(p_{z})&0\\0&0&1\end{bmatrix}}p.}$

An example of a linear taper is ${\displaystyle a(z)=\alpha _{0}+\alpha _{1}z}$, and a quadratic taper ${\displaystyle a(z)={\alpha }_{0}+{\alpha }_{1}z+{\alpha }_{2}z^{2}}$.

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.