CS 15-491 A: COMPUTATIONAL PROJECTIVE GEOMETRY

Fall 1998


Topics Outlined by Chapters (subject to revision):

CHAPTER 1. A REVIEW OF SOME BASIC ALGEBRA.

Complex numbers and Argand diagrams; roots of an n-th order polynomial equation and the roots of unity; determinants, the line joining two points, and areas of polygons; points in space, the intersection of two lines, and projections between intersecting planes; vector spaces and polynomial algebra.

CHAPTER 2: HOMOGENEOUS COORDINATES.

Homogeneous coordinates for points and lines; projective representations of points and lines; dual spaces and bilinear paring; linear forms and the dual paring; exterior products of forms; meets and joins: computations and geometric nterpretation; points on a line and concurrent lines; Desargues' Theorem and computing the intersection of lines determined by points; Theorems of Menelaus, Ceva and Pappus; the uniqueness of the 4th harmonic and grids of squares.

CHAPTER 3: PROJECTIVE TRANSFORMATIONS.

Matrices for transformations and point transformations; transformation by substitution and composition of transformations; line transformations; transforming quadrangles and quadrilaterals, and mapping triangles, quadrangles and quadrilaterals; the notion of crossratio and positioning points on a line; preservation of crossratios by transformation; perspective correspondences and compositions of perspectives; classifying transformations by fixed points and fixed lines.

CHAPTER 4: POLYNOMIAL DUALITY.

Curves, envelopes, and intersections; conic sections and quadratic equations; intersecting lines with curves and curves with curves; polydifferentiation and the ring of differential operators; evaluation through polydifferentiation; duality at higher degree and the dual vector spaces of polynomials; linear combinations of powers of linear forms; contact, multiplicity, and tangency; polydifferential graded algebra for general polynomials; degenerate curves; the Hessian, and testing conics for degeneracy; the tangents at a point on a curve; intersecting lines with a curve from an (n-1)-tuple point.

CHAPTER 5: CONICS.

Poles and polars with respect to a conic; degenerate and non-degenerate conics and the relation to the Hessian; preservation of crossratios and the crossratio of four points on a conic; meets of conjugate lines, the envelope of a conic; the conic determined by five points; Pascal's Theorem; projectivities on a conic; conics determined by points and tangents; conics through four points; harmonic properties of conjugate points; involutions on a conic; the eight tangents at the intersection of two conics; pencils of conics.

CHAPTER 6: CUBICS AND SOME HIGHER-DEGREE CURVES.

Special points on cubics; independence of points; nodes and cusps; the invariance of the Hessian determinant; preserving polydifferentiation by projective transformations; inflection points of cubics; cubics intersecting at eight points; normal form for cuspidal cubics, for nodal cubics, and classifying non-singular cubics; the configuration of inflection points of a non-singular cubic; graphing and parameterizing curves in general, and the parameterization of the non-singular cubic; the maximum number of double points on a curve; cataloging degenerate cubics; parametrization of a quartic curve; a few examples of other curves; a rational curve with positive defficiency; finding the tangential equation of a curve; analyzing rational functions.