Linear Algebra: Homogeneous Coordinates and Matrix Transformations

Homogeneous Coordinates

• A 2D translation can be represented as a 3x3 matrix using homogeneous coordinates.
• Each 2x2 matrix from Chapter 2 corresponds to a 3x3 matrix with the same effect for homogeneous coordinates, with a 1 in the lower right-hand corner.

Advantages of Homogeneous Coordinates

• Allow translations to be represented as matrices.
• Enable the representation of points with very large coordinates.
• Make it possible to show points at infinity and transform 3D drawings into perspective drawings to create an illusion of depth.

Projectivities

• A projectivity is a transformation that preserves lines and collinearity.
• The intersection of the cross joins of all pairs of corresponding points determines a third line called the axis of projectivity or axis of homology.
• The axis of projectivity is unique and contains the intersection of the cross joins of all pairs of corresponding points.

Theorem 4.2.4

• A projectivity between two sets of points on two distinct lines determines a third line called the axis of projectivity.
• The axis of projectivity contains the intersection of the cross joins of all pairs of corresponding points.

Duality

• The duals of the theorems presented in this section are interesting in their own right.
• The proofs given depend on the concept of duality, even though the theorems could be proved directly.

Theorem 4.2.5

• A projectivity between the sets of lines on two points in a plane is determined by three concurrent lines and their images.

Theorem 4.2.6

• A projectivity between two sets of lines on two distinct points determines a third point, called the center of projectivity or center of homology, that lies on the joins of the cross intersections of corresponding lines.

Center of Projectivity

• The center of projectivity can be used to construct additional pairs of corresponding lines in a projectivity.

Homogeneous Coordinates and Projective Geometry

• The Cartesian coordinate system of ordinary analytic geometry is not sufficient for projective geometry, as it includes ideal points.
• A more general system of coordinates, called homogeneous coordinates, is needed to include ideal points.

Relationship Between Coordinate Systems

• The two systems of coordinates are related, but homogeneous coordinates are needed to distinguish among an infinite number of ideal points.

Defining Homogeneous Coordinates

• Homogeneous coordinates must be defined in a way that distinguishes among an infinite number of ideal points and does not require any restrictions in the way these points are handled.

Harmonic Conjugate

• The harmonic conjugate of a point with respect to two other points can be calculated using homogeneous coordinates.
• The truth of this assertion can be explained using the notation of Figure 7.37.

Cross Ratio

• The cross ratio of four collinear points is a number that can be calculated using homogeneous coordinates.
• The cross ratio is a generalization of the concept of harmonic set.