A first look at projective geometry, starting with pappus theorem, desargues theorem and a fundamental relation between quadrangles and quadrilaterals. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. Pascals theorem is in turn a special case of the cayleybacharach theorem. Theorem asserts that the points a, b, c lie on a straight line. Download pdf projective geometry free online new books.
The theorems are attributed to pappus of alexandria and paul guldin. These two approaches are carried along independently, until the. It is the study of geometric properties that are invariant with respect to projective transformations. Bundles of parallel lines meet at an infinite point. Theorem 2 is a consequence of the fundamental theorem of projective geometry see section 6 and is the key to our proof of pappus theorem. This is an important theorem of projective geometry. Hence a classical projective plane can be defined as such a. In 1639, blaise pascal discovered a generalization of pappuss theorem. Download projective geometry ebook pdf or read online books in pdf, epub.
The main theorem of projective geometry that we will use is. Pdf download affine and projective geometry free unquote. An analytic proof of the theorems of pappus and desargues. Pappus theorem and the cross ratio universal hyperbolic. We prove several theorems on orthopoles using the pappus theorem, a fundamental result of projective geometry. The axiomatic destiny of the theorems of pappus and. Fanos geometry consists of exactly seven points and seven lines.
Projective geometry exists in any number of dimensions, just like euclidean geometry. The concept of projectivity lies at the very heart. Higher geometry mathematical and statistical sciences. The dual of this latter characterization permits to state the projective version of menelaus theorem. Geometry, this very ancient field of study of mathematics, frequently remains too little familiar to students. It is an important milestone in the arithmetization of geometry. Another of his significant contributions was the notion of cross ratio of four points on a. Thus each equivalence class of parallel lines contains one of these ideal points, which is defined in projective geometry as the intersection of these parallel lines.
Pdf orthopoles and the pappus theorem semantic scholar. Chapters 5 and 6 make use of projectivities on a line and plane, respectively. Pappus and desargues finite geometries linkedin slideshare. Dec 05, 2008 a first look at projective geometry, starting with pappus theorem, desargues theorem and a fundamental relation between quadrangles and quadrilaterals. His great work a mathematical collection is an important source of information about ancient greek mathematics. A simple proof for the theorems of pascal and pappus. Nov 29, 20 the pappus geometry configuration has 9 points and 9 lines. Pappuss theorem is a special case of pascals theorem for a conicthe limiting case when the conic degenerates into 2 straight lines. The theorem of pascal concerning a hexagon inscribed in a conic. The third and fourth chapters introduce the famous theorems of desargues and pappus. Girard desargues 1591 1661 father of projective geometry 6.
Download pdf projective geometry free online new books in. No distances, no angles, no right angles, no parallel lines. A projective line lis a plane passing through o, and a projective point p is a line passing through o. Section 6 and is the key to our proof of pappus theorem. Both subjects were discussed with pickert in the last year of his life. What other properties are preserved under the allowed transformations. Furthermore we add a projective butterfly theorem which covers all known affine cases. In mathematics, pappuss centroid theorem also known as the guldinus theorem, pappusguldinus theorem or pappuss theorem is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. Nine proofs and three variations x y z a b c a b z y c x b a z x c y fig. Let s be the surface generated by revolving this curve about the xaxis. In modern axiomatic projective plane geometry, the theorems of pappus and desargues are not equivalent. We then formalize pappus property as well as hexamys in order to prove hessenberg theorem, which states that pappus property entails desargues property in projective plane geometry.
Pappus theorem is the first and foremost result in projective geometry. Every line of the geometry has exactly 3 points on it. Given three collinear points a, b, c or concurrent lines a, b, c and the corresponding three collinear points a, b, c or concurrent lines a, b, c, there is a unique projectivity relating abc or abc to abc or abc. Another of his significant contributions was the notion of cross ratio of four points on a line, or of four lines through a. Theorem 2 is a consequence of the fundamental theorem of projective geometry see. Chasles et m obius study the most general grenoble universities 3. Many books on projective geometry discuss the topic. Given any figures drawn on a flat plane surface s, we can imagine this plane embedded in threedimensional space, and we can select an arbitrary point p in the space not on s and some other flat plane surface s, and we can project map every point p.
An application of pappus involution theorem in euclidean and. Theorems on orthopoles are often proved with the help. Call pappian an affine or projective plane satis fying the relevant version of pappus theorem. A simple proof for the theorems of pascal and pappus marian palej geometry and engineering graphics centre, the silesian technical university of gliwice ul.
Desargues theorem 1 two triangles said to be perspective from a point if three lines joining vertices of the triangles meet at a corresponding common point called the center or polar point. If the pappus line u \displaystyle u and the lines g, h \displaystyle g,h have a point in common, one gets the socalled little version of pappus s theorem 2. Here are some other wellknown theorems from projective geometry. In a projective plane, two triangles are said to be perspective from a point if the three lines joining corresponding vertices of the triangles meet at a common point called the center. If the vertices of a triangle are projected onto a given line, the per pendiculars from the projections to the corresponding sidelines of the triangle intersect at one point, the orthopole of the line with respect to the triangle. Original proof of pappus hexagon theorem mathoverflow. Projective geometry and pappus theorem kelly mckinnie history pappus theorem geometries picturing the projective plane lines in projective geometry back to pappus theorem proof of pappus theorem pappus of alexandria pappus of alexandria was a greek mathematician. Not all points of the geometry are on the same line. The pappus configuration is the configuration of 9 lines and 9 points that occurs in pappuss.
Pappuss theorem, in mathematics, theorem named for the 4thcentury greek geometer pappus of alexandria that describes the volume of a solid, obtained by revolving a plane region d about a line l not intersecting d, as the product of the area of d and the length of the circular path traversed by the centroid of d during the revolution. Request pdf the axiomatic destiny of the theorems of pappus and desargues we present the largely twentieth century history of the discovery of the significance of pappus and desargues for the. That is a central topic in projective geometry, and in fact, of any type of geometry. Dorrie begins by providing the reader with a short exposition of. It has now been four decades since david mumford wrote that algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and. Apr 20, 2011 pappus theorem is the first and foremost result in projective geometry. Given any figures drawn on a flat plane surface s, we can imagine this plane embedded in threedimensional space, and we can select an arbitrary point p in the space not on s and some other flat plane surface s. Geometry, projective introduction to projective geometry. The usual method of proving pappus theorem today is in the context of projective geometry. The points in the hyperbolic plane are the interior points of the conic. An almost parallel bundle of lines which meets at a point far on the right.
Projective geometry has its origins in the early italian renaissance, particularly in the architectural drawings of filippo brunelleschi 771446 and leon battista alberti 140472, who invented the method of perspective drawing. The essence of real projective geometry may be summarized in the fol lowing two sentences. If one restricts the projective plane such that the pappus line is the line at infinity, one gets the affine version of pappus s theorem shown in the second diagram. We then formalize pappus property as well as hexamys in order to prove hessenberg theorem, which states that pappus property entails desargues property in projective plane. Pappus theorem the cross ratio of four lines of a pencil of lines equals the cross ratio. In pascals theorem, the 6 white points are contained in a conic section, as shown on the left hand side of figure 2. Desargues theorem, pappuss theorem, pascals theorem, brianchons theorem 1.
Remarks on orthocenters, pappus theorem and butterfly. Since we have not listed the axioms for a projective geometry in 3space, we will not discuss the proof of the theorem here, but the proof is similar to the argument made in the illustration above. Areas of surfaces of revolution, pappuss theorems let f. Dec 11, 2015 we present a generalization of the notion of the orthocenter of a triangle and of pappus theorem. Theorem 1 fundamental theorem of projective geometry.
This is a theorem in projective geometry, more specifically in the augmented or extended euclidean plane. Reference projective geometry by veblen and young, 1938 dual of desargues theorem. A case study in formalizing projective geometry in coq. In the axiomatic development of projective geometry, desargues theorem is often taken as an axiom.
In modern axiomatic projective plane geometry, the theo. Theorem 2 pappus involution theorem the three pairs of opposite sides of a complete quadrangle meet any line not through a vertex in three pairs of an involution. Does anyone know where i can find an english translation, preferably online or in a book the library of a small liberal arts college would be likely to have, of the original proof of pappus hexagon theorem from projective geometry. Theorem 2 pappus involution theorem the three pairs of oppo site sides of a complete quadrangle meet any line not through a vertex in three pairs of an involution. You can draw it with a straightedge with no compass. Prove that the axioms are dual in the concepts of a point and a line, i. A synthetic proof of pappus theorem in tarskis geometry. Recall that the points of the dual projective plane are the lines of the. To get hyperbolic geometry from projective geometry with betweenness axioms, pick a conic corresponding to a hyperbolic polarity e. Pappus theorem the theorem has only to do with points lying on lines.
Pappus theorem, indicates collinearity of the three intersection points. For projective plane geometry, we use a traditional approach dealing with points, lines and an incidence relation to formally prove the independence of desargues property. A geometry which begins with the ordinary points and lines of euclidean plane geometry, and adds an ideal line consisting of ideal points which are considered the intersections of parallel lines. Michle audin, professor at the university of strasbourg, has written a book allowing them to remedy this situation and, starting from linear algebra, extend their knowledge of affine, euclidean and projective geometry, conic sections and quadrics, curves and surfaces. A course in projective geometry matematik bolumu, mimar. Request pdf the axiomatic destiny of the theorems of pappus and desargues we present the largely twentieth century history of the discovery of the. A course in projective geometry matematik bolumu mimar sinan. Later, this theorem will play a central role in modern projective geometry. We present a generalization of the notion of the orthocenter of a triangle and of pappus theorem. In 1640, blaise pascal, in his work essays pour les coniques, obtained a result about a hexagon inscribed in a conic that generalizes pappus theorem.
In class we proved, not exactly their equivalence with thaless theorem, but simply their truth in the geometry of book i of euclids elements. The axiomatic destiny of the theorems of pappus and desargues. Pappuss theorem, in mathematics, theorem named for the 4thcentury greek geometer pappus of alexandria that describes the volume of a solid, obtained by revolving a plane region d about a line l not intersecting d, as the product of the area of d and the length of the circular path traversed by. The projective plane p2 is the set of lines through an observation point oin three dimensional space. Pascals result is proved using projective methods, in particular, using desargues idea of points at. Desarguesian plane which the pappuss theorem is valid. An application of pappus involution theorem in euclidean.