PROJECTIVE GEOMETRY Studio Shodwe
INTRODUCTIVE PROJECTIVE GEOMETRY Projective geometry is a branch of mathematics that deals with the properties of geometric figures under projection. In projective geometry, the focus is on properties that are invariant under projective transformations, which include projections, rotations, translations, and scalings. The key concept in projective geometry is the notion of a projective space, which is a mathematical space where lines are considered as fundamental objects, rather than points. In projective space, any two distinct lines intersect at exactly one point, and any two distinct points determine exactly one line. This property is known as the projective duality.
PROJECTIVE INVARIANTS Projective invariants are geometric properties or quantities of objects in projective geometry that remain unchanged under projective transformations. In other words, if you apply a projective transformation (such as a projection, rotation, translation, or scaling) to a geometric figure, its projective invariants will remain the same. . Projective invariants play a crucial role in the study of projective geometry because they provide a way to characterize and classify geometric figures based on properties that are independent of the specific configuration or orientation.
A significant property of the projective transformations is that certain measurements are invariant under these transformations. . In this case, the transformations are rotation and translation and the most important invariants are distance and angle - consequently, distances and angles are key concepts in a Euclidian analysis. For projective transformations, the most fundamental invariant is called the cross-ratio.
PROJECTIVE INFINITY In projective geometry, the concept of " projective infinity " refers to a point at infinity that is introduced to extend the Euclidean plane into the projective plane. In the Euclidean plane, parallel lines do not intersect. However, in projective geometry, parallel lines are considered to intersect at a point at infinity. This concept allows projective geometry to handle parallel lines in a unified way, without having to treat them as special cases. The notion of projective infinity is fundamental in projective geometry and is used to define various properties and relationships, such as the vanishing points in perspective drawing or the direction of parallel lines.
PROJECTIVE COLLINEARITY In projective geometry, collinearity refers to the property of three or more points lying on the same straight line. However, in the context of projective geometry, collinearity is defined more broadly to include points lying on the same projective line. A projective line is a set of points that can be mapped to each other by a projective transformation. In projective geometry, any two distinct points determine a unique projective line, and any two distinct projective lines intersect at a unique point. This is known as the projective duality.So, when we say that points are collinear in projective geometry, it means that they lie on the same projective line.
PROJECTIVE CROSS RATIO Cross ratio, in projective geometry, ratio that is of fundamental importance in characterizing projections. In a projection of one line onto another from a central point (see Figure), the double ratio of lengths on the first line (AC/AD)/(BC/BD) is equal to the corresponding ratio on the other line. Such a ratio is significant because projections distort most metric relationships (i.e., those involving the measured quantities of length and angle), while the study of projective geometry centres on finding those properties that remain invariant. Although the cross ratio was used extensively by early 19th-century projective geometers in formulating theorems, it was felt to be a somewhat unsatisfactory concept because its definition depended upon the Euclidean concept of length, a concept from which projective geometers wanted to free the subject altogether.
THE AXIOMS OF PROJECTIVE GEOMETRY The axioms of projective geometry provide the foundational principles upon which the subject is built. While different formulations of projective geometry may vary slightly, a typical set of axioms includes: Incidence Axioms: Any two distinct points lie on a unique line. Any two distinct lines intersect at a unique point. There exist at least four points, no three of which are collinear (i.e., not lying on the same line). Order Axioms: If three points A, B, and C are collinear, and B lies between A and C, then B is also between C and A. Given any line segment AB, there exists a point C on the line containing AB such that A is between C and B.
THE AXIOMS OF PROJECTIVE GEOMETRY Desargues ' Axiom (Optional, not always included): If two triangles are perspective from a point (i.e., corresponding lines intersect at a point), then they are perspective from a line (i.e., corresponding vertices lie on a line). Pappus ' Axiom (Optional, not always included): Given two lines and their intersections with a third line, the intersections of corresponding lines determined by pairs of points on the two lines lie on a straight line. These axioms provide a basis for reasoning about points, lines, and their relationships in projective space. From these axioms, various theorems and properties of projective geometry can be derived.
PRINCIPLE OF DUALITY The principle of duality is a fundamental concept in projective geometry that establishes a correspondence between points and lines. It states that in a projective plane, any theorem involving only points and lines can be transformed into another valid theorem by interchanging the words " point" and "line, " as well as the words "lie on " and " pass through." In other words, if you have a true statement about points and lines in a projective plane, you can obtain another true statement by replacing every occurrence of " point" with "line " and vice versa, and replacing "lie on " with " pass through" and vice versa. This principle highlights the symmetry between points and lines in projective geometry. It allows geometric properties and relationships to be analyzed and understood from different perspectives, which can often simplify proofs and facilitate the discovery of new theorems. .
PAPPUS'S THEOREM
DESARGUES’S THEOREM Desargues’s theorem, in geometry, mathematical statement discovered by the French mathematician Girard Desargues in 1639 that motivated the development, in the first quarter of the 19th century, of projective geometry by another French mathematician, Jean-Victor Poncelet. The theorem states that if two triangles ABC and A′B′C′ , situated in three-dimensional space, are related to each other in such a way that they can be seen perspectively from one point (i.e., the lines AA′ , BB′ , and CC′ all intersect in one point), then the points of intersection of corresponding sides all lie on one line (see Figure), provided that no two corresponding sides are parallel. Rather than modify the theorem to cover this special case, After Poncelet discovered that Desargues’s theorem could be more simply formulated in projective space, other theorems followed within this framework that could be stated more simply in terms of only intersections of lines and collinearity of points, with no need for reference to measures of distance, angle, congruence, or similarity.
PROJECTIVE CONIC SECTIONS Conic sections can be regarded as plane sections of a right circular cone (see the figure). By regarding a plane perpendicular to the cone’s axis as the reality plane (RP), a “cutting” plane as the picture plane (PP), and the cone’s apex as the projective “eye, ” each conic section can be seen to correspond to a projective image of a circle (see the figure). A plane Ω passing through the apex and parallel to PP defines the line at infinity in the projective plane PP. The situation of Ω relative to RP determines the conic section in PP: If Ω intersects RP outside the base circle (the circle formed by the intersection of the cone and RP), the image of the circle will be an ellipse (as shown in the figure). If Ω is tangent to the base circle (in effect, tangent to the cone), the image will be a parabola. If Ω intersects the base circle (thus, cutting the circle in two), a hyperbola will result. . Now project all nine points back to the conic section. Since collinear points (the three intersection points from the circle) are mapped onto collinear points, the theorem holds for any conic section. In this way the projective point of view unites the three different types of conics.
Pappus's Theorem: Pappus's theorem highlights the principle of projective duality, as it involves the intersection of linesand points. The theorem establishes a relationship between two sets of collinear points on two distinctlines and the intersections of lines connecting corresponding points. This demonstrates how projectivegeometry considers both points and lines on an equal footing, as they are interchangeable underprojective transformations. Desargues's Theorem: Desargues's theorem is another example of the principle of projective duality. It states that if twotriangles are perspective from a point, they are also perspective from a line. This theorem emphasizesthe symmetrical relationship between points and lines in projective geometry. Additionally, itdemonstrates the concept of perspective, which is crucial in projective transformations andmappings. ANALYSIS OF STATEMENTS RELATED TO PROJECTIVE GEOMETRY