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    Home/Original/inverse
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    Inverse View

    It is not the case that Therefore the cross-ratio is not intrinsic to projective geometry but depends on importing Euclidean or arithmetic structure through the back door.

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Cross-ratio is projectively invariant under all projective transformations—the defining criterion for intrinsic geometric properties.
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    • 2.Cross-ratio can be defined purely combinatorially via harmonic conjugates using only incidence relations, without importing metrics.
      ?

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    • 3.If Euclidean structure were essential, cross-ratio would vary under projective maps; its constancy proves independence from metric geometry.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Cross-ratio definitions require distance ratios or harmonic division, which presuppose metric concepts absent from pure projective axioms.
      ?

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    • 2.Projective geometry's primitive notions are only incidence and collinearity; cross-ratio emerges only after adding coordinate systems.
      ?

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    • 3.Historical development shows cross-ratio entered projective geometry via analytic geometry and perspective theory, not synthetic projection.
      ?

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