In projective geometry, a homothetic transformation is a similarity transformation ( i . e ., fixes a given elliptic involution ) that leaves the line at infinity pointwise invariant.
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The whole family of circles can be considered as " conics passing through two given points on the line at infinity " at the cost of requiring complex coordinates.
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The fundamental property that singles out all projective geometries is the " elliptic " group of transformations can move any line to the " line at infinity " ).
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This is true of the line at infinity itself; it meets itself at its two endpoints ( which are therefore not actually endpoints at all ) and so it is actually cyclical.
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On the other hand, starting with the real projective plane, a Euclidean plane is obtained by distinguishing some line as the line at infinity and removing it and all its points.
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In the real projective plane, since parallel lines meet at a point on the line at infinity, the parallel line case of the Euclidean plane can be viewed as intersecting lines.
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Since an ellipse does not intersect the line at infinity, it properly belongs to the affine plane determined by removing the line at infinity and all of its points from the projective plane.
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Since an ellipse does not intersect the line at infinity, it properly belongs to the affine plane determined by removing the line at infinity and all of its points from the projective plane.
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Assuming a given direction for the axis of the parabola implicitly provides two of these points, because the direction of the axis determines where the parabola is tangent to the line at infinity.
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In geometry and topology, the "'line at infinity "'is a projective line that is added to the real ( affine ) incidence properties of the resulting projective plane.
