Active and passive transformation

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In physics and engineering, an active transformation , or alibi transformation , is a transformation that completely alters the physical position of a point, or a rigid body, which can be defined even in the absence coordinate system; while passive transformation , or alias transformation , is only a change in the coordinate system in which the object is described (change of coordinate map, or base change). By default, with transformations , mathematicians usually refer to active transformation, while physicists and engineers can mean as well.

In other words, the passive transformation refers to the same object description in two different coordinate systems. On the other hand, active transformation is the transformation of one or more objects in connection with the same coordinate system. For example, active transformation is useful to describe consecutive positions of stiff body. On the other hand, passive transformation may be useful in the analysis of human movement to observe the tibia movement relative to the femur, its movement relative to the coordinate system ( local ) that moves along with the femur, rather than the coordinate system ( global ) to the floor.

Video Active and passive transformation

Contoh

Sebagai contoh, di ruang vektor R ² , beri { e ₁ , e ₂ } menjadi dasar, dan pertimbangkan vektor v = v ¹ e ₁ v ² e ₂ . Rotasi vektor melalui sudut? diberikan oleh matriks rotasi:

${\ displaystyle R = {\ begin {pmatrix} \ cos \ theta & amp; - \ sin \ theta \\\ sin \ theta & amp; \ cos \ theta \ end {pmatrix }},}  ÂÂ$

which can be viewed as either active transformation or passive transformation (where the above matrix will be reversed), as described below.

Active transformation

Sebagai transformasi aktif, R memutar vektor awal v , dan vektor baru v ' diperoleh. Untuk rotasi berlawanan jarum jam v sehubungan dengan sistem koordinat tetap:

${\ displaystyle \ mathbf {v '} = R \ mathbf {v} = {\ begin {pmatrix} \ cos \ theta & amp; - \ sin \ theta \\\ sin \ theta & amp; \ cos \ theta \ end {pmatrix}} {\ begin {pmatrix} v ^ {1} \\ v ^ {2} \ end {pmatrix}}.}  ÂÂ$

If a person sees { R e ₁ R /sub>} as a new base, then the new vector coordinates v? on a new base equal to v on an initial base. Note that active transformation makes sense even as a linear transformation into a different vector space. It makes sense to write a new vector in an unpopular basis (as above) only when the transformation of space into itself.

Passive transformation

On the other hand, when a person sees R as a passive transformation, the initial vector v is left unchanged, while the basic coordinate and vector systems are played. For fixed vectors, the coordinates of the new base should change. For rotational rotation system anticlockwise:

${\ displaystyle \ mathbf {v} = v ^ {a} \ mathbf {e} _ {a} = v '^ {a} R \ mathbf { e} _ {a}.} Â Â$

Dari persamaan ini orang melihat bahwa koordinat baru (yaitu, koordinat sehubungan dengan dasar baru) diberikan oleh

${\ displaystyle v '^ {a} = (R ^ {- 1}) _ {b} ^ {a} v ^ {b}}  ÂÂ$

maka

${\ displaystyle \ mathbf {v} = v '^ {a} \ mathbf {e}' _ {a} = v ^ {b} (R ^ {- 1} ) _ {b} ^ {a} R_ {a} ^ {c} \ mathbf {e} _ c} = v ^ {b} \ delta _ {b} ^ {c} \ mathbf {e} _ {c } = v ^ {b} \ mathbf {e} _ {b}.}  ÂÂ$

Thus, in order for the vector to remain unchanged by a passive transformation, the vector coordinate must change according to the inverse of the active transformation operator.

Maps Active and passive transformation

See also

• Basic changes
• Covariance and vector contravant
• Axis rotation
• Translation axis

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References

• Dirk Struik (1953) Lectures on Analytical and Proactive Geometry , page 84, Addison-Wesley.

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External links

• UI ambiguity

Source of the article : Wikipedia