Projective geometry! ! ! urgent

Projective geometry is a branch of geometry that studies the projective properties of figures, that is, they remain unchanged after projective transformation. Once, it was also called projective geometry. In classical geometry, projective geometry is in a special position, through which other geometries can be linked.

Brief introduction to the development of projective geometry

/kloc-In the 7th century, when the analytic geometry founded by Descartes and Fermat came out, another geometry appeared in front of people at the same time. This geometry is closely related to painting, and some of its concepts have attracted the attention of some scholars as early as ancient Greece. The rise of perspective in European Renaissance prepared sufficient conditions for the emergence and growth of this geometry. This geometry is projective geometry.

Based on the needs of cartography and architecture, ancient Greek geometricians began to study perspective, that is, projection and silhouette. As early as around 200 BC, Apollonius studied the cone as a part of the regular cone. Pappus theorem appeared in Pappus's works in the 4th century.

During the Renaissance, people attached great importance to painting and architectural art, and made great efforts to study how to express physical graphics on a plane. At that time, it was found that the painter painted something on the canvas, just like using his own eyes as the projection center, projecting the shadow of the object onto the canvas and then depicting it. In this process, the relative size and positional relationship of each element in the depicted image has changed, while other elements remain unchanged. This prompted mathematicians to study the properties of figures under the central projection, and gradually produced many new concepts and theories that were not available in the past, forming the subject of projective geometry.

Projective geometry really became an independent discipline and an important branch of geometry, mainly in the seventeenth century. /kloc-At the beginning of the 7th century, Kepler first introduced the concept of infinity. Later, two French mathematicians, Gillard Girard Desargues and Pascal, made important contributions to the establishment of this discipline.

Gilad Girard Desargues is a self-taught mathematician. He was an officer when he was young. Later, he studied engineering technology and became an engineer and architect. He disapproves of doing theory for the sake of theory, and is determined to prove the conic theorem with new methods. 1639, he published his main work "the first draft about the intersection result of conic curve and plane", in which he introduced many new geometric concepts. His friends Descartes, Pascal and Fermat all spoke highly of his works, and Fermat even thought that he was the real founder of conic theory.

In his works, Dishag regards a straight line as a circle with infinite radius and the tangent of a curve as the limit of secant. These concepts are the basis of projective geometry. Dishag theorem named after him: "If two triangles correspond to vertices connecting points, the corresponding edges intersect with lines, and vice versa" is the basic theorem of projective geometry.

Pascal also made important contributions to the early work of projective geometry. In 164 1, he found a theorem: "The intersection of three pairs of opposite sides of a hexagon inscribed with a conic." This theorem is called Pascal's hexagon theorem, and it is also an important theorem in projective geometry. 1658 wrote The Theory of Conic Curves, in which many theorems are about projective geometry. Dishag and He are friends. He once urged him to do perspective research and suggested that he simplify many properties of conic into several basic propositions as his goal. Pascal accepted these suggestions. Later he wrote many pamphlets about projective geometry.

But these theorems of Deschag and Pascal only involve related properties, not measurement properties (length, angle, area). But they used the concept of length instead of strict projective method in their proof, and they didn't realize that their research direction would lead to a new geometric system-projective geometry. They use synthetic methods. With the establishment of analytic geometry and calculus, synthesis gave way to analytic method, and the discussion of projective geometry was interrupted.

The main founder of projective geometry is Poncelet in19th century. He is a student of gaspard monge, the founder of descriptive geometry. Gaspard monge inspired many of his students to learn geometry by comprehensive methods. Because the work of Dishag and Pascal has been neglected for a long time, they don't know much about the work of their predecessors and have to do it again.

1822, Poncelet published the first systematic work on projective geometry. He was the first mathematician to realize that projective geometry is a new branch of mathematics. He introduced the imaginary point at infinity by geometric method, studied the polar correspondence, and established the duality principle with it. Later, Steiner studied the method of generating more complex graphs from simple graphs, and he also introduced the concept of linear quadratic curve. In order to get rid of the dependence of coordinate system on the concept of measurement, Stout established the point coordinate system on a straight line through geometric drawing, and then made the cross ratio independent of the concept of length. Ignoring the necessity of continuous axiom, his method of establishing coordinate system is not perfect, but he has taken a decisive step.

On the other hand, great progress has been made in the study of projective geometry by analytical methods. Firstly, Mobius established a homogeneous coordinate system, divided the transformations into congruence, similarity, affine, direct and other types, and gave the measurement formula of the intersection ratio of the four lines in the harness. Then, Pruck introduced another homogeneous coordinate system, and obtained the equation of infinite straight line on the plane and the coordinates of infinite point. He also introduced the concept of line coordinates, so he naturally got the duality principle from the angle of algebra and got some concepts about general line prime curves.

/kloc-in the geometric research in the first half of the 0/9th century, the argument between the comprehensive method and the analytical method was extremely fierce. Some mathematicians completely deny the synthesis method and think it has no future, while some geometricians, such as Scheler, Stody and Steiner, insist on the synthesis method and refuse the analytical method. Others, such as Peng Selie, always use the synthesis method to demonstrate in their works, although they admit that the synthesis method has its limitations and algebra is inevitably used in the research process. Their efforts make the synthetic projective geometry form a beautiful system, and the synthetic method is really vivid, and some problems are directly demonstrated? Yo? In 882, Pasch established the first strict deductive system of projective geometry.

The development of projective geometry is closely related to the development of other branches of mathematics, especially after the concept of "group" came into being, it was also introduced into projective geometry, which promoted the study of this geometry.

It is Klein who uses transformation groups to relate various geometries. He put forward this viewpoint in Herun Root Program, and regarded several classical geometries as sub-geometries of projective geometry, which made the relationship between these geometries very clear. This program has a great influence. But some geometries, such as Riemannian geometry, can't be classified into this category. Later, Gadang and others made new contributions in expanding geometric classification methods.

Content of projective geometry

Generally speaking, projective geometry is an important branch of geometry. It is a science that studies the positional relationship of graphics and discusses the invariance of graphics when points are projected to a straight line or plane.

In projective geometry, infinity is regarded as an "ideal point". The usual straight line plus an infinity point is an infinity straight line. If two straight lines on a plane are parallel, they intersect at infinity. All straight lines passing through the same point at infinity are parallel.

After introducing infinite points and infinite straight lines, the original combination relationship between ordinary points and ordinary straight lines still holds, while the restriction that intersection points can be found only when two straight lines are not parallel in the past disappears.

Because the straight lines passing through the same infinity point are parallel, the central projection and parallel projection can be unified. Parallel projection can be regarded as a central projection through infinity. In this way, any mapping that uses central projection or parallel projection to map one graph to another graph can be called projective transformation.

Projective transformation has two important properties: first, projective transformation makes a point sequence into a point sequence, a straight line into a straight line, a harness into a harness, and the combination of points and straight lines is invariant to projective transformation; Secondly, under the projective transformation, the cross ratio is unchanged. Cross ratio is an important concept in projective geometry, which can be used to explain the projective correspondence between two points in a plane.

In projective geometry, points and straight lines are called dual elements, and "crossing a point to make a straight line" and "taking a point on a straight line" are called dual operations. If both graphs are composed of points and straight lines, replace each element in one graph with its dual element and each operation with its dual operation, and the result is another graph. These two graphs are called dual graphs. What a proposition describes is only about the position of points, lines and surfaces. Every element can become its dual element, and when every operation becomes its dual operation, another proposition will be obtained as the result. These two propositions are called dual propositions.

This is the unique duality principle of projective geometry. On the projective plane, if a proposition holds, then its dual proposition also holds, which is called the plane duality principle. Similarly, in projective space, if a proposition holds, then its dual proposition also holds, which is the so-called spatial duality principle.

It is also an important content of projective geometry to study the invariant properties of conic under projective transformation.

If the content of geometry is concerned, projective geometry

1872, the German mathematician Klein put forward the famous Herun Root Plan at the University of Herun Root, and proposed to classify geometry with transformation groups, that is, each transformation can form a "group". In each geometry, the invariance and invariance under the corresponding transformation are mainly studied.