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Topic: Mapping from straight line to straight line
Replies: 11   Last Post: Nov 3, 2013 4:35 PM

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Timothy Murphy

Posts: 657
Registered: 12/18/07
Re: Mapping from straight line to straight line
Posted: Nov 3, 2013 8:47 AM
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Virgil wrote:

>> The argument can be put more simply as follows.
>> It is easy to construct the mid-point C of two points A,B
>> using only straight lines.
>> (Eg take any point P not on the line AB,
>> and take any line l parallel to AB not going through P or A.
>> Suppose the line cuts AP,BP at E,F.
>> Let EB,FA meet in X. Then PX cuts AB in C.
>> This is the standard construction of 4 points with cross-ratio -1,
>> the 4th point in this case being the point
>> where AB meets the line at infinity.)
>> By the same argument, given A,B we can construct the point D
>> such that B is the mid-point of AD, etc.
>> It follows that a bijective map sending straight lines into straight
>> lines must preserve mid-points.
>> It is evident that this will give a dense set of points on the line AB
>> which must be mapped into corresponding points on any line A'B'.

> That would only be enough to make it an affine transformation if it
> were not also required that origin map to origin.

The original problem was given in two forms.
The second added the condition that the origin was fixed.

Obviously if this condition is not added
one can only conclude that the map is affine.

Timothy Murphy
e-mail: gayleard /at/
School of Mathematics, Trinity College, Dublin 2, Ireland

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