Ellipsometrian

Ellipsometrian is a theoretical discipline that studies ellipses and their transformations within various geometric frameworks. It emphasizes the analytic and algebraic description of elliptical curves, their invariants under affine and projective maps, and the relationships between ellipse parameters and physical or graphical representations.

In some textbooks and speculative writings, the term ellipsometrian is used to designate a focused study of ellipse properties beyond standard conic sections, sometimes as a bridge between pure geometry and applied fields such as computer graphics and optics.

Key concepts include ellipse geometry: eccentricity e, semi-major axis a, semi-minor axis b, foci, parametric form x = a cos t, y = b sin t; polar form r(θ); Steiner inellipse; conjugate diameters; invariants under affine transformations; ellipse fitting; envelope of tangent lines.

Methods: use of analytic geometry, linear algebra, and computational methods; ellipse fitting via least squares; conic sections classification; use of projective geometry to study ellipse images under transformations.

Applications include graphics rendering, CAD, optical lens design, path planning in robotics, and pattern recognition.

See also: ellipse, conic sections, affine geometry, projective geometry, analytic geometry.