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.

See also

• 3D projection

References

1. Shirley, Peter; Ashikhmin, Michael; Marschner, Steve (2009). Fundamentals of Computer Graphics (3rd ed.). CRC Press. p. 426. ISBN 9781568814698.
2. Barr, Alan H. (July 1984). "GLOBAL AND LOCAL DEFORMATIONS OF SOLID PRIMITIVES" (PDF). Computer Graphics. 18 (3): 21–30. Retrieved 4 May 2015.

External links

• , Computer Graphics Notes. University of Toronto. (See: Tapering).
• , 3D Transformations. Brown University. (See: Nonlinear deformations).
• , ScienceWorld article on Tapering in Image Synthesis.