3.1 Projective space and homogeneous coordinates

Projective space and homogeneous coordinates are key concepts in algebraic geometry. They extend Euclidean space by adding points at infinity, allowing parallel lines to intersect. This unified framework simplifies geometric problems and provides a powerful tool for studying algebraic varieties.

Homogeneous coordinates represent points in projective space using tuples defined up to scalar multiples. This system elegantly handles points at infinity and enables the study of projective transformations, which preserve incidence relations between points and lines in projective space.

Projective space

Extension of Euclidean space

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• Projective space extends Euclidean space by including points at infinity
• Allows for the representation of parallel lines intersecting at infinity
• In projective space, two parallel lines always intersect at a unique point called a point at infinity or an ideal point (line at infinity)
• Projective space has the property that any two distinct points determine a unique line, and any two distinct lines intersect at a unique point, possibly at infinity (duality)

Projective plane and transformations

• The projective plane, denoted as $RP^2$, is a two-dimensional projective space
• Can be visualized as a Euclidean plane with additional points at infinity
• Projective transformations, such as projections and perspectivities, preserve the incidence relations between points and lines in projective space
• Examples of projective transformations include homographies and collineations
• Projective geometry studies the properties of figures that are invariant under projective transformations

Homogeneous coordinates

Representing points in projective space

• Homogeneous coordinates are a coordinate system used to represent points in projective space
• A point is represented by a tuple of numbers $(x_0, x_1, ..., x_n)$ defined up to a non-zero scalar multiple
• In the projective plane ($RP^2$), a point is represented by a triple $(x_0, x_1, x_2)$, where at least one coordinate is non-zero
• Two sets of homogeneous coordinates $(x_0, x_1, ..., x_n)$ and $(\lambda x_0, \lambda x_1, ..., \lambda x_n)$ represent the same point in projective space for any non-zero scalar $\lambda$

Lines and points at infinity

• Points at infinity in projective space are represented by homogeneous coordinates where the last coordinate $(x_n)$ is equal to zero
• Lines in the projective plane are represented by linear equations in homogeneous coordinates, such as $ax_0 + bx_1 + cx_2 = 0$
• The line at infinity in the projective plane is given by the equation $x_2 = 0$
• The intersection of a line with the line at infinity determines its point at infinity, representing its direction

Homogeneous vs affine coordinates

Conversion between coordinate systems

• Affine coordinates are the standard Cartesian coordinates used in Euclidean space, while homogeneous coordinates are used in projective space
• To convert from homogeneous coordinates $(x_0, x_1, ..., x_n)$ to affine coordinates $(y_1, y_2, ..., y_n)$, divide each coordinate by the last non-zero coordinate (usually $x_n$): $y_i = x_i / x_n$ for $i = 1, 2, ..., n-1$
• To convert from affine coordinates $(y_1, y_2, ..., y_n)$ to homogeneous coordinates $(x_0, x_1, ..., x_n)$, append a 1 as the last coordinate: $(x_0, x_1, ..., x_n) = (y_1, y_2, ..., y_n, 1)$

Limitations of affine coordinates

• Points at infinity in projective space cannot be represented using affine coordinates, as they correspond to homogeneous coordinates with $x_n = 0$
• Affine coordinates are not well-suited for representing the intersection of parallel lines or the direction of lines
• Homogeneous coordinates provide a more unified framework for dealing with points at infinity and projective transformations

Points at infinity

Definition and properties

• Points at infinity, also called ideal points, are additional points added to Euclidean space to create projective space
• Allow parallel lines to intersect, providing a more unified treatment of geometric concepts
• In the projective plane, points at infinity lie on a special line called the line at infinity, which is the set of all points with homogeneous coordinates $(x_0, x_1, 0)$
• Each set of parallel lines in Euclidean space corresponds to a unique point at infinity in projective space, representing their common direction

Significance in projective geometry

• The inclusion of points at infinity in projective space allows for a more unified treatment of geometric concepts, such as the intersection of lines and the transformation of figures
• Points at infinity play a crucial role in the study of projective geometry and its applications, such as computer vision and computer graphics
• Understanding points at infinity is essential for working with projective transformations and analyzing the behavior of geometric objects in projective space
• Examples of applications include perspective projection, camera calibration, and 3D reconstruction from 2D images