(1) Euclidean Geometry. Axiomatization of Euclidean geometry. Perpendicularity and parallelism of lines and planes. Skew lines. Theorem of three perpendiculars. Dihedral and solid angles. Polyhedra and solids. Euler s Theorem. Generalized prism, pyramid, cone, cylinder. Sphere. Tetrahedra. Volume. Regular solids, Archimedean solids, antiprisms. Isometries, inversions.
(2) Hyperbolic Geometry. Euclid s 5th Postulate and its negations. Riemann sphere and its cross-ratio of four points. Mobius transformations. Axiomatization of hyperbolic geometry. Models of the hyperbolic plane, hyperbolic isometries. Parallelism, hyperparallelism, relativity and absoluteness of measurements. Areas.
(3) Projective Geometry. Intersection and projection. Perspectivities, homologies, points at infinity. Axiomatization of projective geometry, duality. Desargues Theorem, Pappus Theorem. Simple and cross-ratios, harmonic quadruples of points, harmonically conjugate points. Quadrilaterals and tetrahedrons. Circles in the projective plane, connectedness and orientability. Homogeneous coordinates. Conic sections. Apollonius problems.
(4) Descriptive Geometry. Traces, rabattement of planes. Representation with projections on two planes (Monge). Representation with projections on one plane with elevations. Axonometry. Representation of plane section with a solid. Representation of cylindrical helix. Graphical solution of design and metric problems: drawing a roof plan, determining visibility and drawing a tunnel on a topographic map, finding true lengths, angles and areas.
(5) Other Geometries. Elements of absolute and affine geometry, fractals.
- Teacher: Δημήτριος Κοντοκώστας
ECTS : 4
Language : el