Geometry II

Syllabus

• Perspective projection and ideal points; projective space, projective lines and planes.
• Cross-ratio: definition, invariance and permutation effects; harmonic ranges; Pappus-type results (overview).
• Coordinates in the projective plane: homogeneous coordinates; incidence; joins and meets via representative vectors; duality principle.
• Projective transformations induced by linear maps; fundamental theorem of projective geometry (statement); PGL and basic structure.
• Affine transformations and analytic descriptions; determining an affine map from a triangle/tetrahedron and its image.
• Isometries in ℝ^n: translations/reflections; orthogonal linear maps; canonical forms and classification in dimensions 1–3; orientation-preserving vs reversing.
• Quaternions and SO(3): representing rotations via quaternion conjugation; computational advantages (overview).
• Infinitesimal isometries (Killing fields): analytic form X(p)=Mp+v; planar classification; instantaneous center and pole curves (overview).
• Selected applications (high-level): gear geometry ideas; basic robot geometry vocabulary; Denavit–Hartenberg convention; direct/inverse kinematics; singular configurations.