Difference between pages "Weight" and "Duality"

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#REDIRECT [[Bulk and weight]]
[[Image:Duality.svg|400px|thumb|right|'''Figure 1.''' The coordinates $$(p_x, p_y, p_z, p_w)$$ can be interpreted as the one-dimensional span of a single vector representing a homogeneous point or as the $$(n - 1)$$-dimensional span of all orthogonal vectors representing a homogeneous plane. Geometrically, these two interpretations are dual to each other, and their distances to the origin are reciprocals of each other.]]
The concept of duality can be understood geometrically in an ''n''-dimensional projective setting by considering both the subspace that an object occupies and the complementary subspace that the object concurrently does not occupy. The dimensionalities of these two components always sum to ''n'', and they represent the ''space'' and ''antispace'' associated with the object. The example shown in '''Figure 1''' demonstrates the duality between homogeneous points and planes in a four-dimensional projective space. The quadruplet of coordinates $$(p_x, p_y, p_z, p_w)$$ can be interpreted as a vector pointing from the origin toward a specific location on the projection subspace $$w = 1$$. This vector corresponds to the one-dimensional space of the point that it represents. The dual of a point materializes when we consider all of the directions of space that are orthogonal to the single direction $$(p_x, p_y, p_z, p_w)$$. As illustrated by the figure, these directions span an $$(n - 1)$$-dimensional subspace that intersects the projection subspace at a plane when $$n = 4$$. In this way, the coordinates $$(p_x, p_y, p_z, p_w)$$ can be interpreted as both a point and a plane, and they are duals of each other.
 
== See Also ==
 
* [[Complements]]

Revision as of 04:14, 17 June 2022

Figure 1. The coordinates $$(p_x, p_y, p_z, p_w)$$ can be interpreted as the one-dimensional span of a single vector representing a homogeneous point or as the $$(n - 1)$$-dimensional span of all orthogonal vectors representing a homogeneous plane. Geometrically, these two interpretations are dual to each other, and their distances to the origin are reciprocals of each other.

The concept of duality can be understood geometrically in an n-dimensional projective setting by considering both the subspace that an object occupies and the complementary subspace that the object concurrently does not occupy. The dimensionalities of these two components always sum to n, and they represent the space and antispace associated with the object. The example shown in Figure 1 demonstrates the duality between homogeneous points and planes in a four-dimensional projective space. The quadruplet of coordinates $$(p_x, p_y, p_z, p_w)$$ can be interpreted as a vector pointing from the origin toward a specific location on the projection subspace $$w = 1$$. This vector corresponds to the one-dimensional space of the point that it represents. The dual of a point materializes when we consider all of the directions of space that are orthogonal to the single direction $$(p_x, p_y, p_z, p_w)$$. As illustrated by the figure, these directions span an $$(n - 1)$$-dimensional subspace that intersects the projection subspace at a plane when $$n = 4$$. In this way, the coordinates $$(p_x, p_y, p_z, p_w)$$ can be interpreted as both a point and a plane, and they are duals of each other.

See Also