Transformations taking straight lines to straight lines

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SUMMARY

The discussion confirms that the only smooth transformations on R^n that map straight lines to straight lines are affine transformations. Affine transformations preserve parallelism, while projective transformations can also map straight lines to straight lines without this requirement. However, for a transformation to be a bijection on R^n and maintain this property, it must map parallel lines to parallel lines. Non-bijections may allow for projective transformations, which can include translations, rotations, and reflections.

PREREQUISITES
  • Understanding of affine transformations in R^n
  • Knowledge of projective transformations and their properties
  • Familiarity with bijections and their implications in transformations
  • Basic concepts of linear algebra, including translations and rotations
NEXT STEPS
  • Research the properties of affine transformations in R^n
  • Study projective transformations and their applications
  • Explore the concept of bijections in mathematical transformations
  • Learn about the role of reflections in geometric transformations
USEFUL FOR

Mathematicians, geometry enthusiasts, and students studying linear algebra or transformation theory will benefit from this discussion.

Palindrom
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On R^n, I'd say the only smooth transformations taking straight lines to straight lines are the affine transformations.

Would I be right saying that?:smile:

How would one go about proving that?
 
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Palindrom said:
On R^n, I'd say the only smooth transformations taking straight lines to straight lines are the affine transformations.

Would I be right saying that?:smile:

How would one go about proving that?
I believe so.

Best wishes

Pete
 
If I'm not mistaken, Affine Transformations require that parallelism be preserved... however, Projective Transformations also take straight lines to straight lines without requiring parallelism.
 
I'm not familiar with the terminology used here, but surely if a map takes a straight line to another straight line, it is made of a translation + a rotation. So something like
 
robphy said:
If I'm not mistaken, Affine Transformations require that parallelism be preserved... however, Projective Transformations also take straight lines to straight lines without requiring parallelism.

The problem here then is that on R^n (as opposed to R^n unioned with an (n-1)-sphere at infinity) the transformation wouldn't be onto. In other words, if a transformation is a bijection on R^n and maps lines to lines, it must needs to map parallel lines to parallel lines.

If we're talking about non-bijections as well, then the projective transformations might be allowable.
 
quasar987 said:
I'm not familiar with the terminology used here, but surely if a map takes a straight line to another straight line, it is made of a translation + a rotation. So something like

You can also do a reflection
 

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