Active and passive transformation

Active and passive transformation

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Encyclopedia
In the physical sciences
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...

, an active transformation is one which actually changes the physical position of a system
System
System is a set of interacting or interdependent components forming an integrated whole....

, and makes sense even in the absence of a coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

 whereas a passive transformation is a change in the coordinate description of the physical system (change of basis
Change of basis
In linear algebra, change of basis refers to the conversion of vectors and linear transformations between matrix representations which have different bases.-Expression of a basis:...

). The distinction between active and passive transformation
Transformation (mathematics)
In mathematics, a transformation could be any function mapping a set X on to another set or on to itself. However, often the set X has some additional algebraic or geometric structure and the term "transformation" refers to a function from X to itself that preserves this structure.Examples include...

s is important. By default, by transformation, mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

s usually mean active transformations, while physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...

s could mean either.

Put differently, a passive transformation refers to observation of the same event from two different coordinate frames. On the other hand, the active transformation is a new mapping of all points from the same coordinate frame. If we describe successive displacements of a rigid body, the active transformation is useful. If we describe the displacements of individual arms of a robot, each with its own coordinate frame, the passive interpretation is useful to depict all the arm displacements from a common perspective.

In short, the active transform changes object's position while the observer moves in passive transform. When the screen scrolls up under "up" key press, the transform is active, and passive otherwise.

Example


As an example, in the vector space ℝ2, let {e1,e2} be a basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

, and consider the vector v = v1e1 + v2e2. A rotation through angle θ is given by the matrix:
which can be viewed either as an active transformation or a passive transformation, as described below.

Active transformation


As an active transformation, R rotates v . Thus a new vector v' is obtained. For a counterclockwise rotation of v with respect to the fixed coordinate system:

If one views {Re1,Re2} as a new basis, then the coordinates of the new vector v′ in the new basis are the same as those of v in the original basis. Note that active transformations make sense even as a linear transformation into a different vector space. It makes sense to write the new vector in the unprimed basis (as above) only when the transformation is from the space into itself.

Passive transformation


On the other hand, when one views R as a passive transformation, the vector v is left unchanged, while the basis vectors are rotated. In order for the vector to remain fixed, the coordinates in terms of the new basis must change. For a counterclockwise rotation of frames:
From this equation one sees that the new coordinates (i.e., coordinates with respect to the new basis) are given by
so that
Thus, in order for the vector to remain unchanged by the passive transformation, the coordinates of the vector must transform according to the inverse of the active transformation operator.