mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

, and theoretical

computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...

, tapering is a kind of

shape A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A pl ...

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 Scaling may refer to: Science and technology Mathematics and physics * Scaling (geometry), a linear transformation that enlarges or diminishes objects * Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...

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
a & 0 & 0 \\
0 & b & 0 \\
0 & 0 & 1
\end p,

let ''a'' and ''b'' be functions of ''z'' so that: :

q= \begin
a(p_z) & 0 & 0 \\
0 & b(p_z) & 0 \\
0 & 0 & 1
\end p.

An example of a linear taper is

a(z) = \alpha_0 + \alpha_1z

, and a quadratic taper

a(z) = _0 + _1z + _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''.

References

{{reflist

External links

Computer Graphics Notes. University of Toronto. (See: Tapering).

3D Transformations. Brown University. (See: Nonlinear deformations).

ScienceWorld article on Tapering in Image Synthesis. Linear algebra Functions and mappings

See also

References

External links