I'm given that the distance between two points in the Beltrami-Klein model is d ( X Y

minwaardekn

minwaardekn

Answered question

2022-06-16

I'm given that the distance between two points in the Beltrami-Klein model is
d ( X Y ) = 1 2 l n ( X Q ¯ Y P ¯ X P ¯ Y Q ¯ )
where P and Q are ideal points lying on the boundary of the unit disc, and X Q ¯ denotes the standard Euclidean distance between a point X inside the unit disc and an ideal point Q.
Given the ideal points P = ( 0 , 1 ) , Q = ( 0 , 1 ) , nad points A = ( 0 , 0 ) and B = ( 0 , 1 2 ) . I am asked to find the midpoint M = ( 0 , m ) between A and B.

Answer & Explanation

Haggar72

Haggar72

Beginner2022-06-17Added 25 answers

Step 1
Assume that M is the midpoint of XY. Then d ( M , X ) = d ( M , Y ) has to hold, so:
X Q M P X P M Q = M Q Y P M P Y Q
has to hold, and we must have:
( M P M Q ) 2 = X P Y P X Q Y Q
so it is quite easy to construct the midpoint of XY with straightedge and compass, but in general it is not the "euclidean" midpoint X + Y 2 .

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