In mathematics, physics, and theoretical computer graphics, tapering is a kind of shape deformation. 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 = [ a 0 0 0 b 0 0 0 1 ] p , {\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:
q = [ a ( p z ) 0 0 0 b ( p z ) 0 0 0 1 ] p . {\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 a ( z ) = α 0 + α 1 z {\displaystyle a(z)=\alpha _{0}+\alpha _{1}z} , and a quadratic taper a ( z ) = α 0 + α 1 z + α 2 z 2 {\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.
See also 3D projection
References
External links [1], Computer Graphics Notes. University of Toronto. (See: Tapering). [2], 3D Transformations. Brown University. (See: Nonlinear deformations). [3], ScienceWorld article on Tapering in Image Synthesis.
