What is projective geometry?

Projective geometry is a branch of mathematics that studies the properties of geometric figures that remain unchanged when projected onto another surface. This field originated with mathematicians like Girard Desargues in the 17th century, who explored which geometric properties were preserved under perspective mappings. Key concepts in projective geometry include: * Projections: Creating a correspondence between points of a figure and a surface, similar to how shadows are cast or images appear on a screen. * Invariants: Properties that are maintained through projection, such as collinearity (points lying on the same line). * Points at Infinity: A concept introduced to handle parallel lines, which are considered to intersect at a point at infinity. By focusing on these invariants and often disregarding measurements like distance and angle, projective geometry offers a unique perspective on geometric objects and has influenced advanced mathematical areas.

How does projective geometry differ from regular (Euclidean) geometry?

Projective geometry differs from Euclidean geometry primarily in its treatment of parallel lines and its focus on properties preserved under projection. Euclidean geometry, based on Euclid's axioms, includes the parallel postulate, stating that exactly one line parallel to a given line can be drawn through a point not on that line. Projective geometry, however, disregards measurements like distance and angles, focusing instead on properties that remain invariant when figures are projected onto another surface. In projective geometry, parallel lines are considered to meet at a point at infinity, a concept that simplifies certain geometric theorems and unifies different types of conic sections. This approach allows for a more generalized understanding of geometric objects by ignoring specific measurements that are not preserved under projection.

What is the concept of a vanishing point in projective geometry?

In projective geometry, a vanishing point is the point where receding parallel lines appear to converge when depicted in linear perspective. This concept is fundamental to understanding how three-dimensional space is represented on a two-dimensional surface, a core idea in projective geometry itself. Projective geometry studies the properties of figures that remain unchanged when projected onto another surface, essentially focusing on how shapes transform under projection. Parallel lines, which never meet in Euclidean geometry, are understood in projective geometry to meet at a point at infinity, which is then projected to a vanishing point on the picture plane.

How are parallel lines treated in projective geometry?

In projective geometry , parallel lines are treated as if they intersect at a single point at infinity. This concept is introduced to ensure that any two lines in a projective plane always have a unique point of intersection, a fundamental principle of this branch of mathematics. This approach allows for a more unified understanding of geometric properties, as theorems can be stated more simply without needing to account for special cases involving parallel lines. The introduction of points at infinity, where parallel lines are considered to meet, was a key development that led to the formulation of projective geometry. This principle is also related to the concept of duality, where "point" and "line" can be interchanged in theorems.

Where is projective geometry used in real life or art?

Projective geometry, which studies properties of figures that remain unchanged under projection, has significant applications in art and real-world scenarios. Its principles are fundamental to linear perspective , a technique developed during the Renaissance by artists like Filippo Brunelleschi and Leon Battista Alberti . This method uses sight lines to project a three-dimensional scene onto a two-dimensional picture plane, creating realistic depth and spatial relationships in paintings and drawings. Beyond art, projective geometry is crucial in fields like cartography for creating map projections, such as the Mercator projection , which represents the spherical Earth on a flat surface. It also underpins computer graphics and the way we perceive spatial relationships, as the brain interprets retinal images to understand object positions in space.

How did the study of projective geometry influence the development of perspective in painting and photography?

The study of projective geometry significantly influenced the development of perspective in painting and photography. Originating in the Italian Renaissance with artists like Filippo Brunelleschi and Leon Battista Alberti , the method of perspective drawing involves projecting a three-dimensional scene onto a two-dimensional plane using sight lines. This mathematical approach, which forms the basis of projective geometry, allowed artists to create a realistic illusion of depth and space. Similarly, the principles of central projection, fundamental to projective geometry, directly inform how cameras capture images, producing photographs that closely resemble human visual perception.

projective geometry

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Projective drawingThe sight lines drawn from the image in the reality plane (RP) to the artist's eye intersect the picture plane (PP) to form a projective, or perspective, drawing. The horizontal line drawn parallel to PP corresponds to the horizon. Early perspective experimenters sometimes used translucent paper or glass for the picture plane, which they drew on while looking through a small hole to keep their focus steady.

Projective version of the fundamental theorem of similarityIn RP, Euclid's fundamental theorem of similarity states that CD/DA = CE/EB. By introducing a scaling factor, the theorem can be saved in RP as C′D′/D′A′ = C′E′/E′B′ ∙ ΩB′/ΩA′. Note that while lines AB and DE are parallel in RP, their projections onto PP intersect at the infinitely distant horizon (Ω).

Pappus's projective theoremPappus of Alexandria (fl. ad 320) proved that the three points (x, y, z) formed by intersecting the six lines that connect two sets of three collinear points (A, B, C; and D, E, F) are also collinear.

Pascal's projective theoremThe 17th-century French mathematician Blaise Pascal proved that the three points (x, y, z) formed by intersecting the six lines that connect any six distinct points (A, B, C, D, E, F) on a circle are collinear.

Cross ratioAlthough distances and ratios of distances are not preserved under projection, the cross ratio, defined as AC/BC ∙ BD/AD, is preserved. That is, AC/BC ∙ BD/AD = A′C′/B′C′ ∙ B′D′/A′D′.

Diagram showing a double cone intersected by planes at different angles, producing the conic sections: a horizontal plane creates a circle, a tilted plane creates an ellipse, a plane parallel to the cone's side forms a parabola, and a steep plane produces a hyperbola. Each section is labeled, with the axis and apex of the cone indicated.

A diagram of a double cone intersected by two green planes labeled PP (picture plane) and RP (reality plane), illustrating projected conic sections in projective geometry, with the apex and a point Omega marked.

Figure 9: Use of auxiliary view to show true size and shape of an inclined surface (ABCD), which is not correctly represented in the front, top, or side view (see text).

Figure 1: Two techniques of representing an object. (A) Perspective drawing, suggesting that the object is cubical. (B) Orthographic top and front views, revealing that the object is not cubical.

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Top Questions

What is projective geometry?
Projective geometry is a branch of mathematics that studies the properties of geometric figures that remain unchanged when projected onto another surface. This field originated with mathematicians like Girard Desargues in the 17th century, who explored which geometric properties were preserved under perspective mappings.

Key concepts in projective geometry include:
* Projections: Creating a correspondence between points of a figure and a surface, similar to how shadows are cast or images appear on a screen.
* Invariants: Properties that are maintained through projection, such as collinearity (points lying on the same line).
* Points at Infinity: A concept introduced to handle parallel lines, which are considered to intersect at a point at infinity.

By focusing on these invariants and often disregarding measurements like distance and angle, projective geometry offers a unique perspective on geometric objects and has influenced advanced mathematical areas.
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How does projective geometry differ from regular (Euclidean) geometry?
Projective geometry differs from Euclidean geometry primarily in its treatment of parallel lines and its focus on properties preserved under projection. Euclidean geometry, based on Euclid's axioms, includes the parallel postulate, stating that exactly one line parallel to a given line can be drawn through a point not on that line. Projective geometry, however, disregards measurements like distance and angles, focusing instead on properties that remain invariant when figures are projected onto another surface. In projective geometry, parallel lines are considered to meet at a point at infinity, a concept that simplifies certain geometric theorems and unifies different types of conic sections. This approach allows for a more generalized understanding of geometric objects by ignoring specific measurements that are not preserved under projection.
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What is the concept of a vanishing point in projective geometry?
In projective geometry, a vanishing point is the point where receding parallel lines appear to converge when depicted in linear perspective. This concept is fundamental to understanding how three-dimensional space is represented on a two-dimensional surface, a core idea in projective geometry itself. Projective geometry studies the properties of figures that remain unchanged when projected onto another surface, essentially focusing on how shapes transform under projection. Parallel lines, which never meet in Euclidean geometry, are understood in projective geometry to meet at a point at infinity, which is then projected to a vanishing point on the picture plane.
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Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
How are parallel lines treated in projective geometry?
In projective geometry, parallel lines are treated as if they intersect at a single point at infinity. This concept is introduced to ensure that any two lines in a projective plane always have a unique point of intersection, a fundamental principle of this branch of mathematics. This approach allows for a more unified understanding of geometric properties, as theorems can be stated more simply without needing to account for special cases involving parallel lines. The introduction of points at infinity, where parallel lines are considered to meet, was a key development that led to the formulation of projective geometry. This principle is also related to the concept of duality, where "point" and "line" can be interchanged in theorems.
5 Britannica Sources
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
Where is projective geometry used in real life or art?
Projective geometry, which studies properties of figures that remain unchanged under projection, has significant applications in art and real-world scenarios. Its principles are fundamental to linear perspective, a technique developed during the Renaissance by artists like Filippo Brunelleschi and Leon Battista Alberti. This method uses sight lines to project a three-dimensional scene onto a two-dimensional picture plane, creating realistic depth and spatial relationships in paintings and drawings.

Beyond art, projective geometry is crucial in fields like cartography for creating map projections, such as the Mercator projection, which represents the spherical Earth on a flat surface. It also underpins computer graphics and the way we perceive spatial relationships, as the brain interprets retinal images to understand object positions in space.
5 Britannica Sources
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
How did the study of projective geometry influence the development of perspective in painting and photography?
The study of projective geometry significantly influenced the development of perspective in painting and photography. Originating in the Italian Renaissance with artists like Filippo Brunelleschi and Leon Battista Alberti, the method of perspective drawing involves projecting a three-dimensional scene onto a two-dimensional plane using sight lines. This mathematical approach, which forms the basis of projective geometry, allowed artists to create a realistic illusion of depth and space. Similarly, the principles of central projection, fundamental to projective geometry, directly inform how cameras capture images, producing photographs that closely resemble human visual perception.
7 Britannica Sources
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

projective geometry, branch of mathematics that deals with the relationships between geometric figures and the images, or mappings, that result from projecting them onto another surface. Common examples of projections are the shadows cast by opaque objects and motion pictures displayed on a screen.

Projective geometry has its origins in the early Italian Renaissance, particularly in the architectural drawings of Filippo Brunelleschi (1377–1446) and Leon Battista Alberti (1404–72), who invented the method of perspective drawing. By this method, as shown in the figure, the eye of the painter is connected to points on the landscape (the horizontal reality plane, RP) by so-called sight lines. The intersection of these sight lines with the vertical picture plane (PP) generates the drawing. Thus, the reality plane is projected onto the picture plane, hence the name projective geometry. See also geometry: Linear perspective.

Although some isolated properties concerning projections were known in antiquity, particularly in the study of optics, it was not until the 17th century that mathematicians returned to the subject. The French mathematicians Girard Desargues (1591–1661) and Blaise Pascal (1623–62) took the first significant steps by examining what properties of figures were preserved (or invariant) under perspective mappings. The subject’s real importance, however, became clear only after 1800 in the works of several other French mathematicians, notably Jean-Victor Poncelet (1788–1867). In general, by ignoring geometric measurements such as distances and angles, projective geometry enables a clearer understanding of some more generic properties of geometric objects. Such insights have since been incorporated in many more advanced areas of mathematics.

Parallel lines and the projection of infinity

A theorem from Euclid’s Elements (c. 300 bc) states that if a line is drawn through a triangle such that it is parallel to one side (see the figure), then the line will divide the other two sides proportionately; that is, the ratio of segments on each side will be equal. This is known as the proportional segments theorem, or the fundamental theorem of similarity, and for triangle ABC, shown in the diagram, with line segment DE parallel to side AB, the theorem corresponds to the mathematical expression CD/DA = CE/EB.

Now consider the effect produced by projecting these line segments onto another plane as shown in the figure. The first thing to note is that the projected line segments A′B′ and D′E′ are not parallel; i.e., angles are not preserved. From the point of view of the projection, the parallel lines AB and DE appear to converge at the horizon, or at infinity, whose projection in the picture plane is labeled Ω. (It was Desargues who first introduced a single point at infinity to represent the projected intersection of parallel lines. Furthermore, he collected all the points along the horizon in one line at infinity.) With the introduction of Ω, the projected figure corresponds to a theorem discovered by Menelaus of Alexandria in the 1st century ad: C′D′/D′A′ = C′E′/E′B′ ∙ ΩB′/ΩA′. Since the factor ΩB′/ΩA′ corrects for the projective distortion in lengths, Menelaus’s theorem can be seen as a projective variant of the proportional segments theorem.

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Projective invariants

With Desargues’s provision of infinitely distant points for parallels, the reality plane and the projective plane are essentially interchangeable—that is, ignoring distances and directions (angles), which are not preserved in the projection. Other properties are preserved, however. For instance, two different points have a unique connecting line, and two different lines have a unique point of intersection. Although almost nothing else seems to be invariant under projective mappings, one should note that lines are mapped onto lines. This means that if three points are collinear (share a common line), then the same will be true for their projections. Thus, collinearity is another invariant property. Similarly, if three lines meet in a common point, so will their projections.

The following theorem is of fundamental importance for projective geometry. In its first variant, by Pappus of Alexandria (fl. ad 320) as shown in the figure, it only uses collinearity:

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Let the distinct points A, B, C and D, E, F be on two different lines. Then the three intersection points—x of AE and BD, y of AF and CD, and z of BF and CE—are collinear.

The second variant, by Pascal, as shown in the figure, uses certain properties of circles:

If the distinct points A, B, C, D, E, and F are on one circle, then the three intersection points x, y, and z (defined as above) are collinear.

There is one more important invariant under projective mappings, known as the cross ratio (see the figure). Given four distinct collinear points A, B, C, and D, the cross ratio is defined as CRat(A, B, C, D) = AC/BC ∙ BD/AD. It may also be written as the quotient of two ratios: CRat(A, B, C, D) = AC/BC : AD/BD.

The latter formulation reveals the cross ratio as a ratio of ratios of distances. And while neither distance nor the ratio of distance is preserved under projection, Pappus first proved the startling fact that the cross ratio was invariant—that is, CRat(A, B, C, D) = CRat(A′, B′, C′, D′). However, this result remained a mere curiosity until its real significance became gradually clear in the 19th century as mappings became more and more important for transforming problems from one mathematical domain to another.

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Parallel lines and the projection of infinity

Projective invariants