Transformations taking straight lines to straight lines

In summary: As for proving this, one way is to show that any smooth transformation that takes straight lines to straight lines must be a combination of translations, rotations, reflections, and scalings, which are all affine transformations. This can be done using linear algebra and properties of matrices.
  • #1
Palindrom
263
0
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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  • #2
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
 
  • #3
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.
 
  • #4
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
 
  • #5
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.
 
  • #6
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
 

Related to Transformations taking straight lines to straight lines

1. What are transformations that take straight lines to straight lines?

Transformations that take straight lines to straight lines are geometric operations that map points on a straight line to points on another straight line, while preserving the straightness of the lines. These transformations include translation, rotation, reflection, dilation, and shear.

2. What is translation in terms of transforming straight lines?

Translation is a transformation that moves a straight line horizontally or vertically without changing its direction or shape. It involves shifting all points on the line by a fixed distance in a specified direction.

3. How does rotation transform a straight line?

Rotation is a transformation that turns a straight line around a fixed point called the center of rotation. This results in the straight line being rotated by a certain angle while maintaining its length and direction.

4. What is the difference between reflection and dilation in terms of straight line transformations?

Reflection is a transformation that flips a straight line over a line of reflection, resulting in a mirror image of the original line. Dilation, on the other hand, is a transformation that expands or shrinks a straight line by a certain scale factor, while keeping the same direction.

5. How does shear affect straight lines?

Shear is a transformation that skews a straight line by stretching or compressing one side of the line while keeping the other side fixed. This results in the line being transformed into a parallelogram shape.

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