Tapering (mathematics)
Encyclopedia
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:
,
Let a and b be functions of z so that:
An example of a linear taper is:
.
And a quadratic taper:
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, and theoretical computer graphics
Computer graphics
Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....
, 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
Scaling
Scaling may refer to:* Scaling , a linear transformation that enlarges or diminishes objects* Reduced scales of semiconductor device fabrication processes...
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:
,Let a and b be functions of z so that:
An example of a linear taper is:
.And a quadratic taper:
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.
External links
- http://www.utsc.utoronto.ca/~jansen/cscd18/teaching09/notes/LN05_3dObjects.pdf, Computer Graphics Notes. University of Toronto. (See: Tapering).
- http://www.cs.brown.edu/~ls/teaching08/LS06_Meshes_and_3DTransforms.pdf, 3D Transformations. Brown University. (See: Nonlinear deformations).
- http://scienceworld.wolfram.com/physics/Tapering.html, ScienceWorld article on Tapering in Image Synthesis.