[This blog is a repost from my other blog site, thought would be useful to GeomTech reader]
CAD solid models are consisting of geometry and topology. Geometry by virtue of its visualization capability is easy to understand and comprehend. Topology is on the other hand, more of pure virtual concept hence many find it difficult to understand. In this article brief introduction to manifold topology is illustrated. Intended audience is new CAD developers or students of computational geometry.
Geometry is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Many scientists have made contribution to its theory and lot has changed since Euler first laid foundation stone of Geometry. Following is brief review of the development.
Euclid 300 BC, also known as Euclid of Alexandria, was a Greek mathematician and is often referred to as the Father of Geometry. His work “Elements” is the most successful textbook in the history of mathematics. In “Elements”, the principles of what is now called Euclidean geometry were deduced from a small set of axioms. The geometrical system described in the “Elements” was long known simply as geometry, and was considered to be the only geometry possible. Today, however, that system is often referred to as Euclidean geometry to distinguish it from other so-called Non-Euclidean geometries that mathematicians discovered in the 19th century.
Euclid undertook a study of relationships among distances and angles, first in a plane (an idealized flat surface) and then in space. An example of such a relationship is that the sum of the angles in a triangle is always 180 degrees. Today these relationships are known as two- and three-dimensional Euclidean geometry. An essential property of a Euclidean space is its flatness. Important point to note is other spaces exist in geometry that are not Euclidean. For example, the surface of a sphere is not; a triangle on a sphere (suitably defined) will have angles that sum to something greater than 180 degrees.
Next major contribution in Geometry came after around 2000 year by Leonhard Paul Euler (15 April 1707 – 18 September 1783). In 1736, Euler solved the problem known as the Seven Bridges of Königsberg. The city of Königsberg, Prussia was set on the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not: there is no Eulerian circuit. This solution is considered to be the first theorem of graph theory, specifically of planar graph theory.
In the processEuler also discovered the formula V − E + F = 2 relating the number of vertices, edges, and faces of a convex polyhedron, and hence of a planar graph. The constant in this formula is now known as the Euler characteristic for the graph (or other mathematical object), and is related to the genus of the object. The study and generalization of this formula, specifically by Cauchy and L'Huillier, is at the origin of topology.
Next path breaking contribution was from Georg Friedrich Bernhard Riemann (September 17, 1826 – 
July 20, 1866) an extremely influential German mathematician who made important Non-Euclidean geometries. This discovery was a major paradigm shift in mathematics, as it freed mathematicians from the mistaken belief that Euclid's axioms were the only way to make geometry consistent and non-contradictory. Research on these geometries led to, among other things, Einstein's theory of general relativity, which describes the universe as non-Euclidean. The subject founded by his work is Riemannian geometry. Riemann found the correct way to extend into n dimensions the differential geometry of surfaces. The fundamental object is called the Riemann curvature tensor. For the surface case, this can be reduced to a number (scalar), positive, negative or zero; the non-zero and constant cases being models of the known non-Euclidean geometries. contributions to analysis and differential geometry. He was first one to discover Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives in particular local notions of angle, length of curves, surface area, and volume. From those some other global quantities can be derived by integrating local contributions. Riemannian geometry deals with a broad range of geometries categorized into two standard types of Non-Euclidean geometry,spherical geometry and hyperbolic geometry, as well as Euclidean geometry itself.
Non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line l and a point A, which is not on l, there is exactly one line through A that does not intersect l. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting l, while in elliptic geometry, any line through A intersects l.
Another way to describe the differences between these geometries is as follows: Consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line. In Euclidean geometry the lines remain at a constant distance from each other, and are known as parallels. In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultra parallels. In elliptic geometry the lines "curve toward" each other and eventually intersect.
We started discussing how we end up in geometry discussion? I felt it was necessary to create basic understanding of historical development of Euclidean Geometry and Non-Euclidian geometry to understand next topics.
As we saw earlier Euler was first one publish paper on topology in seven bridge problem demonstrating that it was impossible to find a route through the town of Königsberg that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges, nor on their distance from one another, but only on connectivity properties: which bridges are connected to which islands or riverbanks. In other words to solve many of geometric problems we do not need to know spatial information but what is requiring to be known is neighborhood or connectivity information termed as topology.
Problems like a Möbius strip, an object with only one surface and one edge; such shapes are alos an object of study in topology.
Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician and theoretical physicist, and a philosopher of science.As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. He was responsible for formulating the Poincaré conjecture, one of the most famous problems in mathematics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory.
He was a founder of topology, also known as “rubber-sheet geometry,” for its focus on the intrinsic properties of spaces.
From a topologist’s perspective, there is no difference between a bagel (shape like torus or from Indian menu medu wada J ) and a coffee cup with a handle. Each has a single hole and can be manipulated to resemble the other without being torn or cut. Poincaré used the term “manifold” to describe such an abstract topological space.
The simplest possible two-dimensional manifold is the surface of a soccer ball, which, to a topologist, is a sphere—even when it is stomped on, stretched, or crumpled. The proof that an object is a so-called two-sphere, since it can take on any number of shapes, is that it is “simply connected,” meaning that no holes puncture it. Unlike a soccer ball, a bagel is not a true sphere. If you tie a slipknot around a soccer ball, you can easily pull the slipknot closed by sliding it along the surface of the ball. But if you tie a slipknot around a bagel through the hole in its middle you cannot pull the slipknot closed without tearing the bagel.
Euler- Poincaré equation
Euler’s polyhedron formula was applicable to only simple polyhedron. This new equation from Poincaré provides relationship between topological elements for any single two-manifold body.
V -E + F - Li = 2(1 - G)
where,
V : Number of vertices.
E: Number of edges.
F: Number of faces.
Li: Number of interior loops.
G: Genus, the number of closed paths on a surface which do not separate the surface into more than one region. Or, genus is the number of handles to be added to a sphere to make it homeomorphic to the object.
Classification of manifolds
· 0-manifold is just a discrete space. Eg Point in Cartesian space corresponds to vertex in topological space. Point or set of points are zero dimensional manifolds
· 1-manifold is curve in Cartesian space. Eg circle, line, parabole b-spline curve (non intersecting)
· 2-manfold is surface Cartesian space. Sphere (empty inside), torus, plane, cylinder, b-spline surface (if closed empty inside, orientable and non self-intersecting ) , circular disc
· 3-manfold is 3 dimensional manifold. Eg Solid Objects, Solid Sphere, Solid Cylinder, Our universe J
Two-dimensional manifolds were well understood by the mid-nineteenth century. But it remained unclear whether what was true for two dimensions was also true for three. Poincaré proposed that all closed, simply connected, three-dimensional manifolds—those which lack holes and are of finite extent—were spheres. The conjecture was potentially important for scientists studying the largest known three-dimensional manifold: the universe. Proving it mathematically, however, was far from easy. Most attempts were merely embarrassing, but some led to important mathematical discoveries, including proofs of Dehn’s Lemma, the Sphere Theorem, and the Loop Theorem, which are now fundamental concepts in topology.
By the nineteen-sixties, topology had become one of the most productive areas of mathematics, and young topologists were launching regular attacks on the Poincaré. To the astonishment of most mathematicians, it turned out that manifolds of the fourth, fifth, and higher dimensions were more tractable than those of the third dimension. By 1982, Poincaré’s conjecture had been proved in all dimensions except the third. In 2000, the Clay Mathematics Institute, a private foundation that promotes mathematical research, named the Poincaré one of the seven most important outstanding problems in mathematics and offered a million dollars to anyone who could prove it.
After nearly a century of effort by mathematicians, Grigori Perelman sketched a proof of the conjecture in a series of papers made available in 2002 and 2003. The proof followed the program of Richard Hamilton. Several high-profile teams of mathematicians have since verified the correctness of Perelman's proof.
The Poincaré conjecture was, before being proven, one of the most important open questions in topology. It is one of the seven Millennium Prize Problems, for which the Clay Mathematics Institute offered a $1,000,000 prize for the first correct solution. Perelman's work survived review and was confirmed in 2006, leading to him being offered a Fields Medal, which he declined. The Poincaré conjecture remains the only solved Millennium problem.
This article was written with the help of various sources of web sources, mainly Wikipedia. Intention is to explain interesting subject such as manifold which carries paramount importance in CAD development in less mathematical and elaborative manner. Research in Manifolds is as recent as 2006 and is one of the most studied area in mathematics.
