Tapering (mathematics)

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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, there also exist higher-order deformations such as tapering, 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(p_z) & 0 & 0 \\
0 & b(p_z) & 0 \\
0 & 0 & 1 \\
\end{bmatrix} p

An example of a linear taper is: a(z) = {\alpha}_0 + {\alpha}_1z.

And a quadratic taper: a(z) = {\alpha}_0 + {\alpha}_1z + {\alpha}_2z^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[edit]

External links[edit]

  • [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.