Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Geometry ↓


Projective Geometry


Projective geometry is a fascinating area of mathematics that extends the concepts of geometry by considering points located at infinity. It is the study of geometric properties that are invariant under projection. Projective geometry can be thought of as a natural development of ordinary Euclidean geometry.

Basic concepts

To understand projective geometry, we need to start by thinking about the projective plane. The projective plane can be thought of as an extension of the normal Euclidean plane. Here, we add "points at infinity" where parallel lines meet. This addition allows us to handle parallel lines easily.

Consider the following illustration, which conceptually represents the projective plane:

In this illustration, the two red and blue dashed lines do not appear to meet on the Euclidean plane. However, in projective geometry, they meet at a point at infinity, in the direction in which they trend. This extra point is one of the key principles of projective geometry.

Homogeneous coordinates

In projective geometry, points are often expressed in terms of homogeneous coordinates. In the Euclidean plane, a point is typically defined as (x, y) In the projective plane, a point is represented using three coordinates (x, y, z), with an equivalence relation such that (x, y, z) is equivalent to (kx, ky, kz) for any non-zero scalar k.

x = x/z
y = y/z

These are the homogeneous coordinates of the point. The use of homogeneous coordinates has a profound impact because it simplifies the mathematics involved in dealing with the projection and intersection of lines.

Lines in projective geometry

Just like in Euclidean geometry, in projective geometry a line is described by an equation. However, because of the homogeneous coordinates, the equation of the line takes a slightly different form:

ax + by + cz = 0

In this equation, (a, b, c) are the coefficients that define the line. Note that the equation is defined in such a way that it allows intersection at infinity. In fact, two lines will always meet at a point in projective geometry, whether that point is on the plane or at infinity.

Duel

Projective geometry is unique because of its duality. The principle of duality states that theorems in projective geometry remain valid even when points and lines are interchanged. This means that every statement or theorem has a dual counterpart.

Consider the following dual statements:

  • Two points are given and the same line falls on both of them.
  • Two lines are given, the same point is incident on both of them.

These statements demonstrate the dual nature of projective geometry and lead to a powerful analogy between points and lines.

Cross ratio

The cross-ratio is an important invariant in projective geometry. For four collinear points A, B, C, and D, their cross-ratio is given by:

(a, b; c, d) = (AC * BD) / (AD * BC)

This ratio remains constant under projection and is an important tool for the analysis of projective transformations, since it captures intrinsic properties of collinear point quadrilaterals.

Visual example

Let us consider a visual illustration to understand the idea of cross-ratio:

A B C D

In this illustrative diagram, points A, B, C, D are shown. The cross-ratio of these points will remain constant under any projection transformation.

Projectile transformations

A projective transformation, also known as homography, is a transformation of the projective plane that maps points to points and lines to lines, maintaining the cross-ratio of the points. It can be described using the matrix form:

| X' | | A11 A12 A13 | | X |
| y' | = | a21 a22 a23 | * | y |
| z' | | a31 a32 a33 | | z |

where (x, y, z) are the homogeneous coordinates of the origin and (x', y', z') are the coordinates after the transformation, and the matrix is a non-singular 3x3 matrix.

Application

Projective geometry has important applications in various fields. Here are some of the major areas where it plays an important role:

Computer graphics

In computer graphics, projective geometry is used to render 3D scenes on a 2D screen in which objects appear to diminish in size with distance, a technique known as perspective projection.

Photography

In photography, understanding perspective requires knowledge of projective transformations. Certain lenses produce specific transformative effects on photos that can be analyzed with projective geometry.

Architecture

Projective geometry is important in architectural drawing and design, especially in perspective drawing. Architects use it to estimate and visualize what buildings will look like from different viewpoints.

Conclusion

Projective geometry beautifully extends the logic of Euclidean geometry by addressing the behavior of parallel lines and providing a comprehensive system where every pair of lines intersects. With its applications spanning from art to technology and its fundamental principle of duality, projective geometry remains an essential concept in mathematics and its related fields. The study of projective geometry not only serves as a powerful mathematical tool but also enriches our view and understanding of spatial relationships.


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