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The Grassmann Determinant Principle for the Plane
The Grassmann Principle for Space
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affine geometry affine transformation analytic geometry angle arbitrary axes axioms axis called coefficients complete components configurations conic consider coordinate system corresponding course cross ratio defined definite determinant directed line-segment discussion distance elementary elements equation Euclid Euclid's Elements euclidean geometry example expression fact field finite follows formulas four points free plane-magnitude free vector functions fundamental given Grassmann Hence homogeneous coordinates imaginary spherical circle infinitely distant infinity intersection invariant theory lectures Leipzig linear substitution manifolds mathematics means metric geometry Mobius motion multiplication non-euclidean geometry null-system obtain origin parallel parallel axiom parameters pencil plane at infinity point coordinates polar polygon position postulates principle projective geometry projective transformations quadratic form rays real points rectangular coordinate relation represent rotation scalar segment sense space sphere straight line surface syzygies tangent tensor tetrahedron theorem theory of invariants tion trans translation triangle unchanged values variables xi yi