Glossary

3.1 Projective space and homogeneous coordinates

4 min read•july 30, 2024

Projective space and 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 or an ideal point ()
  • 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 RP2RP^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 (x0,x1,...,xn)(x_0, x_1, ..., x_n) defined up to a non-zero scalar multiple
  • In the projective plane (RP2RP^2), a point is represented by a triple (x0,x1,x2)(x_0, x_1, x_2), where at least one coordinate is non-zero
  • Two sets of homogeneous coordinates (x0,x1,...,xn)(x_0, x_1, ..., x_n) and (λx0,λx1,...,λxn)(\lambda x_0, \lambda x_1, ..., \lambda x_n) represent the same point in projective space for any non-zero scalar λ\lambda

Lines and points at infinity

  • Points at infinity in projective space are represented by homogeneous coordinates where the last coordinate (xn)(x_n) is equal to zero
  • Lines in the projective plane are represented by linear equations in homogeneous coordinates, such as ax0+bx1+cx2=0ax_0 + bx_1 + cx_2 = 0
  • The line at infinity in the projective plane is given by the equation x2=0x_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 (x0,x1,...,xn)(x_0, x_1, ..., x_n) to affine coordinates (y1,y2,...,yn)(y_1, y_2, ..., y_n), divide each coordinate by the last non-zero coordinate (usually xnx_n): yi=xi/xny_i = x_i / x_n for i=1,2,...,n−1i = 1, 2, ..., n-1
  • To convert from affine coordinates (y1,y2,...,yn)(y_1, y_2, ..., y_n) to homogeneous coordinates (x0,x1,...,xn)(x_0, x_1, ..., x_n), append a 1 as the last coordinate: (x0,x1,...,xn)=(y1,y2,...,yn,1)(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 xn=0x_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 (x0,x1,0)(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 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

Key Terms to Review (13)

Bézout's Theorem: Bézout's Theorem is a fundamental result in algebraic geometry that states that the number of intersection points of two projective plane curves, counted with multiplicities, is equal to the product of their degrees. This theorem highlights the relationship between geometry and algebra and connects projective varieties with their intersections, making it essential for understanding various concepts like projective space, affine varieties, and singularities in plane curves.
Collineation: Collineation is a geometric transformation that maps points to points in a projective space while preserving the incidence structure, meaning that if three points are collinear before the transformation, they remain collinear afterward. This concept is vital as it helps in understanding how objects relate to each other within projective geometry, particularly in the context of homogeneous coordinates, where it can be represented as a linear transformation of coordinate vectors. It embodies the idea of maintaining relationships and properties under transformation, making it essential for studying geometric properties in a projective setting.
Complex Projective Space: Complex projective space, denoted as $$ ext{CP}^n$$, is a mathematical space that consists of lines through the origin in complex Euclidean space $$ ext{C}^{n+1}$$. Each point in this space represents a line of complex vectors, allowing for the study of properties that are invariant under projective transformations. This structure is essential in various areas of geometry and algebra because it enables us to analyze geometric objects using homogeneous coordinates.
Computer Vision: Computer vision is a field of artificial intelligence that enables computers to interpret and understand visual information from the world, similar to the way humans do. It involves using algorithms and mathematical models to analyze images and videos, recognizing patterns, objects, and even actions. This technology plays a vital role in various applications, such as image processing, robotics, and autonomous vehicles, by transforming visual data into actionable insights.
Desargues' Theorem: Desargues' Theorem states that if two triangles are situated in such a way that their corresponding sides meet at points that are collinear, then the triangles are in perspective from a point. This theorem connects deeply to projective geometry, where the concepts of points and lines are more abstract, allowing for insights into the relationships between geometric figures through the use of projective space and homogeneous coordinates.
Homogeneous coordinates: Homogeneous coordinates are a system of coordinates used in projective geometry that allows for the representation of points in a projective space. By adding an extra dimension, points can be expressed in a way that makes it easier to handle concepts such as infinity and intersections in geometric contexts. This representation plays a crucial role in various mathematical topics, connecting projective spaces, affine spaces, and the properties of projective varieties.
Incidence Relation: An incidence relation is a mathematical concept that describes the relationship between geometric objects, such as points, lines, and planes, particularly in projective geometry. This relation determines whether a particular geometric object is incident to another, meaning they meet or intersect in some way. Incidence relations are fundamental in understanding the structure and properties of projective spaces and how different objects relate to one another.
Line at infinity: The line at infinity is a concept in projective geometry that represents a boundary for parallel lines, where these lines are said to meet. This idea allows us to extend the notion of lines in the Euclidean plane to include points at infinity, enabling a more comprehensive understanding of geometric properties and relationships in projective space.
Poincaré Disk Model: The Poincaré Disk Model is a representation of hyperbolic geometry in which the entire hyperbolic plane is represented within the unit disk. This model allows for the visualization of points, lines, and angles in hyperbolic space while maintaining the unique properties of this non-Euclidean geometry. It connects to projective space and homogeneous coordinates by providing a way to represent points at infinity and projective transformations within a bounded region, thereby offering insights into the relationships between different geometric constructs.
Point at Infinity: A point at infinity is a concept in projective geometry that represents an idealized location where parallel lines intersect. In the context of projective space, these points are crucial for providing a complete understanding of geometric properties, as they allow us to treat parallel lines as if they meet at a specific point. This idea helps in simplifying many theorems and definitions in geometry, giving rise to the use of homogeneous coordinates to describe points in this extended space.
Projective Coordinates: Projective coordinates are a system of coordinates used in projective geometry to represent points in projective space. They extend the concept of ordinary coordinates by allowing for a more unified treatment of geometric objects, including points at infinity, thereby facilitating the study of properties that remain invariant under projection. This system is crucial for understanding how geometric shapes behave when viewed from different perspectives.
Projective Transformation: A projective transformation is a geometric mapping that preserves the properties of collinearity and incidence among points, lines, and planes in projective space. This transformation can be represented using homogeneous coordinates, allowing for a unified way to handle points at infinity and transformations such as perspective projection, which is crucial for understanding how shapes appear under various viewing conditions.
Real Projective Space: Real projective space, denoted as $$ ext{RP}^n$$, is a geometric space that represents the set of all lines through the origin in $$ ext{R}^{n+1}$$. It captures the idea of projective geometry by treating lines as equivalent classes of points, where each line corresponds to a direction in space, allowing for the incorporation of points at infinity. This concept is essential for understanding homogeneous coordinates and their application in projective transformations.
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