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Details
- No. of pages:
- 256
- Language:
- English
- Copyright:
- © Academic Press 1981
- Published:
- 28th May 1981
- Imprint:
- Academic Press
- eBook ISBN:
- 9780080956626
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Sequences of Transformations - Expii
The identity is the transformation that does nothing. It maps every element in the domain to itself. Unsurprisingly, it is often denoted I:x↦x.
The composition of two transformations is the transformation equivalent to doing one transformation first, and then the other. Keep in mind that composition is generally order-sensitive.
For example, the composition f∘g is the composition formed by applying g first and then f. Notice that the order is reversed. To remember this, one may remember f∘g as f(g(x)) and notice that the name of the function applied first shows up farthest to the right.
The identity function is not so interesting by itself, but it has many important properties related to composition.
The identity function, unlike most functions, is not order sensitive: I∘f=f∘I=f.
The inverse of a function, f−1, is the function g such that I=g∘f. The inverse function is also not order-sensitive, so f−1∘f=f∘f−1=I. Note that the domain of f is the range of f−1, and that the range of f is the domain of f−1.
Here are two other important properties of function composition, both of which are not difficult to prove: (f∘g)∘h=f∘(g∘h)(associativity) (f∘g)−1=g−1∘f−1
Sequence Transformations
Keywords
Counting Maxima algorithms boundary element method convergence decidability diagrams form iteration logarithm maximum selection theorem transformation turbulence
Authors and affiliations
- 1.Laboratoire d’Informatique Fondamentale de LilleUniversité des Sciences et Techniques de Lille Flandres ArtoisVilleneuve d’Ascq CedexFrance
Bibliographic information
- Book TitleSequence Transformations
- AuthorsJean-Paul Delahaye
- Series TitleSpringer Series in Computational Mathematics
- DOIhttps://doi.org/10.1007/978-3-642-61347-0
- Copyright InformationSpringer-Verlag Berlin Heidelberg1988
- Publisher NameSpringer, Berlin, Heidelberg
- eBook PackagesSpringer Book Archive
- Hardcover ISBN978-3-540-15283-5
- Softcover ISBN978-3-642-64802-1
- eBook ISBN978-3-642-61347-0
- Series ISSN0179-3632
- Edition Number1
- Number of PagesXXI, 252
- Number of Illustrations0 b/w illustrations, 0 illustrations in colour
- TopicsNumerical Analysis
- Buy this book on publisher's site
Sequence of Transformations
Composite Transformation Theorems
There is a connection between the three transformations: reflections, translations and rotations.
Parallel Lines Theorem:A composition of reflections across two parallel lines is a translation. When a figure is reflected in two parallel lines, the final image is a translation in the direction perpendicular to the parallel lines and twice the distance between them.
The following diagram shows that a composition of reflections across two parallel lines is a translation. Scroll down the page for more examples and solutions of reflections across parallel lines.
The following diagram shows that a composition of reflections across two intersecting lines is a rotation. Scroll down the page for more examples and solutions of reflections across intersecting lines.
Looking at composition of transformations - combining transformations in a series. Specifically looking at glide reflections: translation followed by reflection. Composition of Transformations (2)
Looking at composition of transformations - combining transformations in a series. Specifically looking at glide reflections, reflection followed by rotation. Composition of Transformations (3)
Composition Theorem:The composition of two or more isometries is an isometry.
Reflections in Parallel lines Theorem: If lines k and m are parallel then the reflection in line k followed by a reflection in line m is the same as a translation. Composition of Transformations (4)
Reflection in Intersecting lines Theorem: If lines k and m intersect at point, P, then a reflection in k followed by a reflection in m is the same as a rotation about the point, P.
Transformational Geometry (Translations, Rotations, Reflections)Defining transformations to match polygonsApply Composition of Transformations
Glide Reflection: Transformation followed by reflection
Translation followed by rotation
Composition of Transformations
Reflections in Parallel Lines Theorem
Reflections in Intersecting Lines Theorem
Composite Transformations in Geometric Figures
This video discusses composite transformations and work through some examples of how to do a composite transformation. Reflecting in Parallel Lines
Reflect the blue triangle in the blue-green line to form the green image. Reflect the green triangle in the green-red line to form the red image. The red triangle, being reflected twice, is oriented like the blue triangle.
Ignoring the green triangle, the result of the two reflections is to translate the blue triangle to the red triangle through twice the distance between the two mirror lines.
Reflecting in Parallel Lines from the Wolfram Demonstrations Project by George Beck
Reflecting in Intersecting Lines
Reflect the blue triangle in the blue-green line to form the green image. Reflect the green triangle in the green-red line to form the red image. The red triangle, being reflected twice, is oriented like the blue triangle.
Ignoring the green triangle, the result of the two reflections is to rotate the blue triangle to the red triangle about the intersection of the two mirror lines through twice the angle between them.
Reflecting in Intersecting Lines from the Wolfram Demonstrations Project by George Beck
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Of transformations sequence
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