The work of desargues was ignored until michel chasles chanced upon a handwritten copy in 1845. In projective geometry, desargues s theorem, named after girard desargues, states. A quadrangle is a set of four points, no three of which are collinear. Desarguess 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, jeanvictor poncelet. Dec 05, 2008 a first look at projective geometry, starting with pappus theorem, desargues theorem and a fundamental relation between quadrangles and quadrilaterals. 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. Research article proving and generalizing desargues twotriangle theorem in 3dimensional projective space dimitrioskodokostas department of applied mathematics, national technical university of athens, zografou campus, athens, greece correspondence should be addressed to dimitrios kodokostas. To construct the real projective plane we need to introduce several new points and one new line which contains them all to the euclidean plane. Later, after we have studied projective geometry, it will serve a very insightful role in tying together several developments of classical euclidean geometry. Nov 29, 20 pappus and desargues finite geometries 1. It is called the desarguesian projective plane because of. Actually desargues proved the theorem because he was interested in perspective drawings and the geometry it gave rise to is however not projective geometry but what is called descriptive geometry. This is an example of a noteworthy theorem of riemannian geometry which has no analogues in more general spaces.
Proving and generalizing desargues twotriangle theorem. Pdf perspectives on projective geometry download full. Click here for a dynamic illustration of desargues theorem geogebra or javasketchpad. Desargues theorem states that if you have two triangles which are perspective to one another then.
Two triangles are in perspective axially if and only if they are in perspective centrally. The basic notions, and some of the fundamental theorems, of what would later be called projective geometry were first established in the seventeenth century, in response to questions that arose with regard to map projections and problems of perspective encountered by artists in representing threedimensional scenes on planar canvases. Meanwhile, jeanvictor poncelet had published the foundational treatise on projective geometry in 1822. For that, abc and def are each two round triangles with dual triangles abc and def, respectively, and p a point on each of the circles adad, bebe, and cfcf. Girard desargues 1591 1661, a french architect and mathematician who lived in lyons and paris, was one of the founders of projective geometry. Projective geometry 4 desargues theorem proof youtube. Desargues theorem and the invariance of the crossratio, were published in a book about perspective by bosse 1648. Drag the points x, y, z, a, a, c, or c to change the triangles. This illustrates the power and efficiency of our approach using only ranks to prove properties of the projective space. The dual of this latter characterization permits to state the projective version of menelaus theorem.
If two triangles are perspective from a line, then they are perspective from a point. Pdf a case study in formalizing projective geometry in. Girard desargues, the father of projective geometry, proved the following theorem in the 17th century. Desargues theorem if two triangles are perspective from a point, then they are perspective from a line. For readers unfamiliar with projective geometry or unfamiliar with the somewhat dated terminology in dorrie, this one is really hard to read. In two dimensions it begins with the study of configurations of points and lines.
The theorem states that the points, and at the intersections of corresponding sides lie in a line. Now reverse this process to prove desarguess theorem of. 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. Since projection preserve incidence, the figures in the plane satisfies desarguess theorem in the plane. Jan 17, 2015 desargues theorem is one of the most fundamental and beautiful results in projective geometry. A first look at projective geometry, starting with pappus theorem, desargues theorem and a fundamental relation between quadrangles and quadrilaterals. So if we prove a theorem for points in a projective plane then the dual result holds automatically for lines. Files are available under licenses specified on their description page. Any two lines l, m intersect in at least one point, denoted lm. Euclidean geometry is mistaken in portraying as coincidental certain results about euclidean points, lines and planes that in fact have a common, uni.
An application of pappus involution theorem in euclidean and. In the axiomatic development of projective geometry, desargues theorem is often taken as an axiom. The projective plane is obtained from the euclidean plane by adding the points at infinity and the line at infinity that is formed by all the points at infinity. Desargues outline finite geometry examples of problems features of desargues another example finite geometry projective geometrya. Dorrie begins by providing the reader with a short exposition of. 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. His proofs did not use linear algebra which was not developed until the 19th century and are rather more complicated. In this geometry, any two lines will meet at one point. Take desarguess theorem of two triangles in distinct planes. This is a partial version of desargues involution theorem see 3, p. I intend to explain the result, but not all the details.
The main theorem of projective geometry that we will use is. Research article proving and generalizing desargues two. Explanation, existence and natural properties in mathematics. We choose to focus on projective geometry which is a simple but powerful. Desargues theorem is one of the most fundamental and beautiful results in projective geometry.
One needs to understand a few definitions to start with. I understand this is figure 4 desargues theorem in 3 dimen sions. That is, desargues theorem can be proven from the other axioms only in a projective geometry of more than two dimensions. This is a theorem in projective geometry, more specifically in the augmented or extended euclidean plane. Projective geometry in a plane fundamental concepts undefined concepts. It is the study of geometric properties that are invariant with respect to projective transformations. The axiomatic destiny of the theorems of pappus and desargues. In this article, we investigate formalizing projective plane geometry. This is an immediate consequence of desargues s twotriangle theorem itself, as applied to the triangle aqq and bpp, whose joins of corresponding vertices all pass through c, while their intersections of corresponding sides are o, r, r.
A case study in formalizing projective geometry in coq. An application of pappus involution theorem in euclidean. Narboux has formalized in coq the area method of chou, gao and zhang 6, 15, 23 and applied it to obtain a proof of desargues theorem in a. Common examples of projections are the shadows cast by opaque objects and motion pictures displayed on a screen.
Two triangles that are perspective from a point are perspective from a line, and converseley, two triangles that are perspective from a line are perspective from a point. The usual euclidean plane is contained in what we call the real projective plane. Nine proofs and three variations x y z a b c a b z y c x b a z x c y fig. The works of gaspard monge at the end of 18th and beginning of 19th century were important for the subsequent development of projective geometry. All structured data from the file and property namespaces is available under the creative commons cc0 license. In the riemannian case the only desargues geometries are the euclidean, hyperbolic and elliptic geometries, i.
The elegance of their proofs testifies to the power of the method of homogeneous coordinates. This theorem is an important milestone toward obtaining the arithmetization of geometry which will allow us to provide a connection between analytic and synthetic geometry. Perhaps the most important proposition deduced from the axioms of incidence in projective geometry for the projective space is desargues twotriangle theorem usually stated quite concisely as follows. Desargues theorem article pdf available in computational geometry 458 october 2012 with 79 reads how we measure reads. This approach allows to carry out proofs in a more systematic way and was successfully used to fairly easily formalize desargues theorem in coq. Timothy peil, 5 february 20, created with geogebra. In addition, when considering higherdimensions, the amount of incidence relations e.
Two triangles in the real projective plane are in perspective centrally if and only if they are in perspective axially. Projective geometry in projective geometry there are no parallel lines. Let v be a point and let two triangles be given so that their vertices are distinct from v. In other words, we can say the triangles are in perspective from the point p. Axial perspectivity means that lines ab and ab meet in a point, lines ac and ac meet in a second point, and lines bc and bc meet in a third point, and that these three points all lie on a common line called the axis of perspectivit. Pdf a case study in formalizing projective geometry in coq.
Kusak has formalized in mizar desargues theorem in. Guilhot has formalized in coq a proof of desargues theorem in affine geometry. Any two points p, q lie on exactly one line, denoted pq. In projective geometry, desarguess theorem, named after girard desargues, states. Chapter 3 solid geometry and desargues theorem math 4520, fall 2017 3. In plane projective geometry, desargues theorem cannot be proven from the other axioms. Triangles d abc and d au bu cu are perspective from a point o if lines aau, bbu and ccu. That there is indeed some geometric interest in this sparse setting was first established by desargues and others in their exploration of the principles of perspective art. We say that the two triangle are in perspective from v if the. Denote the three vertices of one triangle by a, b and c, and those of the other by a, b and c. Imo training 2010 projective geometry part 2 alexander remorov 1. Drag the points l 1 or l 2 of the line to change the axis i. A projective point is a line in ir3 that passes through the origin. 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.
This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. 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. 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. Two triangles are said to be perspective from a line if the three points of intersection of. Narboux has formalized in coq the area method of chou, gao and zhang, and applied it to obtain a proof of desargues theorem in affine geometry. Chapter 1 of tondeurs textbook closes with a proof of pappus theorem which we can be stated very compactly, but which we will not prove just now. Desargues theorem working toward a proof in what follows, we will use four axioms of projective geometry that happen to hold true for rp3. In the plane, proofs are constructed in a traditional way using points and lines. This article deals with formalizing projective geometry in the coq proof assistant, and studies desargues property both in the plane and in an at least threedimensional setting noted. Kusak has formalized in mizar desargues theorem in the fanoian projective. Chasles et m obius study the most general grenoble universities 3. In the longer term, the underlying objective of the presented work consists in designing a formal geometry prover able to handle the nondegeneracy conditions, and especially in geometric constraint solving 14, 16. Objects points, lines, planes, etc incidence relation antire.
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