1. "RE: Steiner Construction II"
In response to message #0
Probably because projective transformations do not preserve parallelism. The properties of a construction that depends on the existence of a parallelogram do not carry over by a projective transformation.
2. "RE: Steiner Construction II"
In response to message #1
>Probably because projective transformations do not preserve >parallelism. The properties of a construction that depends >on the existence of a parallelogram do not carry over by a >projective transformation.
Does this mean that the point at infinity of a pair of parallel lines is mapped into a point of intersection?
3. "RE: Steiner Construction II"
In response to message #2
>Does this mean that the point at infinity of a pair of >parallel lines is mapped into a point of intersection?
If the images intersect, yes.
In projective geometry all lines intersect. The points at infinity are singled out in transition from the affine to the projective. In the latter all points are the same. Projective transformations map points of intersection onto points of intersection. From the affine view point, it may happen that either image or preimage is a point at infinity.
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