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    Therefore the cross-ratio is not intrinsic to projective ... — Carmelics
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    Challenges→Cross-ratio can be defined intrinsically in projective geometry using quadruples of collinear points

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

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    1 reason for
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    Reasons For

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    Reason for
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    • 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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    Reasons Against

    1 perspective
    Reason against
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    • 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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    Related

    Cross-ratio can be defined intrinsically in projective geometry using quadruples...Cross-ratio can be defined purely combinatorially via harmonic conjugates using ...Cross-ratio definitions require distance ratios or harmonic division, which pres...Cross-ratio is projectively invariant under all projective transformations—the d...
    +3 moreShow less
    Historical development shows cross-ratio entered projective geometry via analyti...If Euclidean structure were essential, cross-ratio would vary under projective m...Projective geometry's primitive notions are only incidence and collinearity; cro...

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    2 (1 for, 1 against)
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