Courses / Mathematics / Axiomatic Euclidean and non-Euclidean geometry Mathematics · unit version 1 Axiomatic Euclidean and non-Euclidean geometry A proof-based undergraduate sequence in incidence, order, congruence, continuity, Euclidean transformations and constructions, affine/projective foundations, and hyperbolic, spherical and elliptic models. Model calculations accompany written proofs; synthetic representation theorems and Gauss–Bonnet are explicitly identified dependencies, not conclusions certified by a diagram.
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Readiness and preparation Check your starting point Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.
What distinguishes a definition, an axiom and a theorem? Comparing geometries requires knowing which assumptions an argument uses.
Compare reasoning and find a refresher A definition specifies terminology, an axiom is an adopted starting assertion, and a theorem is deduced from assumptions. A diagram suggests a claim but cannot supply a missing axiom or prove every case.
Review geometric proof foundations
How can coordinates verify that a transformation preserves distance? Euclidean and hyperbolic models both test proposed isometries against their actual metrics.
Compare reasoning and find a refresher Apply the transformation to arbitrary points and compute their distance using the specified metric. Equality for every pair proves preservation; checking one segment or only the visual shape does not.
Review rigid motions
Why does a straightedge-and-compass construction lead to equations of degree at most two at each step? Constructibility obstructions connect geometric operations to field extensions.
Compare reasoning and find a refresher Line intersections solve linear equations; line-circle and circle-circle intersections reduce to quadratic equations after eliminating a variable. Starting from known coordinates, each step stays in a succession of extensions of degree at most two, with real intersection requirements checked.
Review geometric constructions
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Geometry: axioms, transformations and models
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Lesson details and objectives Separate geometric language from models and logical consequences.
Translate incidence statements into quantified claims. Use a model to refute an alleged consequence. Explain relative consistency without claiming absolute consistency. 6 practice tasks.
Lesson details and objectives Prove incidence properties in small algebraic models.
Construct the affine plane over the two-element field. Count projective points and lines over a finite field. Distinguish incidence axioms from ordered geometry. 6 practice tasks.
Lesson details and objectives Identify the order assumptions that diagrams usually conceal.
Verify betweenness in an ordered coordinate plane. Use Pasch to control triangle crossings. Separate segment order from incidence alone. 6 practice tasks.
Lesson details and objectives Separate equality of measurements from geometric transport.
State segment and angle transport requirements. Prove coordinate SAS with an orientation choice. Diagnose circular congruence arguments. 6 practice tasks.
Lesson details and objectives Expose the existence assumptions behind geometric constructions.
Separate ordered-field arithmetic from completeness. Show a rational plane misses a familiar circle intersection. Use a least-upper-bound argument to justify square roots. 6 practice tasks.
Lesson details and objectives Prove consequences without using a parallel axiom.
Trace an exterior-angle proof to neutral congruence and order. Use the theorem to construct nonintersecting lines. Identify the additional dependency in angle-sum bounds. 6 practice tasks.
Lesson details and objectives Compare existence, uniqueness and angle consequences of parallelism.
Separate a neutral existence proof from Playfair uniqueness. Prove the Euclidean coordinate parallel assertion. Explain why spherical geometry is not merely neutral geometry with zero parallels. 6 practice tasks.
Lesson details and objectives Check incidence, distance and order in R².
Verify the line and distance definitions algebraically. Use polarization to recover angles from distances. Distinguish model verification from categoricity. 6 practice tasks.
Lesson details and objectives Derive rigid motions from distances rather than assuming their form.
Prove an isometry of R² has affine orthogonal form. Use determinant to distinguish orientation. Compute fixed points from a linear system. 6 practice tasks.
Lesson details and objectives Build every plane isometry from elementary reflections.
Compute compositions of reflections with order explicit. Classify orientation-reversing isometries. Prove a bounded reflection factorization. 6 practice tasks.
Lesson details and objectives Determine what survives when rigid motions are enlarged.
Derive similarities as scaled isometries. Separate length, angle, area and affine invariants. Use an invariant to prove non-equivalence. 6 practice tasks.
Lesson details and objectives Translate ruler-and-compass steps into field operations.
Derive field operations from geometric constructions. Prove an intersection adds at most a square root. Construct a positive square root using a mean proportional. 6 practice tasks.
Lesson details and objectives Prove selected construction problems impossible under stated tools.
Use field degrees as necessary construction conditions. Prove cube duplication and one angle trisection impossible. Distinguish a necessary degree test from a full criterion. 6 practice tasks.
Lesson details and objectives Prove geometric statements without a metric.
Characterize affine combinations independent of origin. Show affine maps preserve ratios on a line. Use barycentric coordinates to prove concurrence. 6 practice tasks.
Lesson details and objectives Build the real projective plane from vector subspaces.
Use homogeneous coordinates without treating them as unique triples. Prove projective line intersection by linear algebra. Relate parallel affine lines to ideal points. 6 practice tasks.
Lesson details and objectives Find invariants of projective incidence and one-dimensional maps.
Prove point–line duality in homogeneous coordinates. Calculate cross-ratios with a fixed ordering convention. Show fractional linear maps preserve cross-ratio. 6 practice tasks.
Lesson details and objectives Use quadratic forms to explain projective conic transformations.
Represent conics by symmetric matrices. Derive tangent and polar equations. Separate affine shape from projective equivalence. 6 practice tasks.
Lesson details and objectives Derive distances and geodesics in a concrete non-Euclidean model.
Prove vertical hyperbolic distance from the length integral. Verify real Möbius metric invariance. Use a model to distinguish boundary and interior. 6 practice tasks.
Lesson details and objectives Transport hyperbolic geometry into a bounded picture.
Derive the Cayley transform and disk metric. Compute radial distance exactly. Explain why visual boundary proximity distorts length. 6 practice tasks.
Lesson details and objectives Compare straight geodesics with angle-preserving pictures.
Use the Lorentz form to define the hyperboloid model. Project geodesics to straight Klein chords. Calculate the disk–Klein coordinate conversion. 6 practice tasks.
Lesson details and objectives Relate fixed points to elliptic, parabolic and hyperbolic types.
Classify determinant-one real Möbius maps by trace squared. Find fixed points in the interior or ideal boundary. Distinguish translation length from displacement at an arbitrary point. 8 practice tasks.
Lesson details and objectives Use the Lorentz model to derive a non-Euclidean cosine law.
Derive the hyperbolic side cosine rule from tangent vectors. Compute right-triangle relations without Euclidean substitution. State precisely the dependency behind angle-defect area. 6 practice tasks.
Lesson details and objectives Derive spherical metric and trigonometry from unit vectors.
Compute short-arc distance using dot products. Derive the spherical cosine rule. Identify antipodal failures of unique joining arcs. 6 practice tasks.
Lesson details and objectives Compare the sphere with its projective metric quotient.
Define elliptic distance independently of representatives. Recover unique projective joins through antipodal identification. Explain why global betweenness still fails. 6 practice tasks.
Lesson details and objectives Separate exact curvature identities from small-scale approximations.
Apply the geodesic Gauss–Bonnet triangle identity with signs. Compare scale-normalized spherical and hyperbolic areas. Identify the assumptions behind a curvature inference. 6 practice tasks.
Lesson details and objectives Assemble model calculations into an explicit logical comparison.
Separate incidence, order, metric and topology in model comparisons. State the precise logical force of countermodels. Audit a proof portfolio for hidden Euclidean assumptions. 6 practice tasks.
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