In these properties of having centers on an ellipse and tangencies on a circle, the Pappus chain is analogous to the Steiner chain, in which finitely many circles are tangent to two circles.
|
|
Since it preserves tangencies, angles and circles, inversion transforms one Steiner chain into another of the same number of circles. One particular choice of inversion transforms the given circles α and β into concentric circles; in this case, all the circles of the Steiner chain have the same size and can "roll" around in the annulus between the circles similar to ball bearings. This standard configuration allows several properties of Steiner chains to be derived, e.g., its points of tangencies always lie on a circle.
|
|
All eight general solutions can be obtained by shrinking and swelling the circles according to the differing internal and external tangencies of each solution; however, different given circles may be shrunk to a point for different solutions.
|
|
While all triangles are tangential to some circle, a triangle is called the tangential triangle of a reference triangle if the tangencies of the tangential triangle with the circle are also the vertices of the reference triangle.
|
|
However, between each pair of tangencies, the curvature must decrease to less than that of the circle, for instance at a point obtained by translating the circle until it no longer contains any part of the curve between the two points of tangency and considering the last point of contact between the translated circle and the curve. Therefore, there is a local minimum of curvature between each pair of tangencies, giving two of the four vertices. There must be a local maximum of curvature between each pair of local minima, giving the other two vertices..
|
|
He met Kepler, and discussed with François Viète two questions about equations and tangencies. He then spent some time in Italy, particularly with Clavius in Rome in 1585. He was ordained a priest in 1604. After 1610 he tutored mathematics in Poland.
|
|
The first n circles of this construction form a ring, whose minimum radius can be calculated by Descartes' theorem to be the same as the radius specified in the ring lemma. This construction can be perturbed to a ring of n finite circles, without additional tangencies, whose minimum radius is arbitrarily close to this bound.
|
|
The chain of six spheres can be rotated about the central sphere without affecting their tangencies, showing that there is an infinite family of solutions for this case. As they are rotated, the spheres of the hexlet trace out a torus (a doughnut-shaped surface); in other words, a torus is the envelope of this family of hexlets.
|
|
Notably, presented a simple geometric construction based on bitangents; other authors have since claimed that Steiner's presentation lacked a proof, which was later supplied by , but Guy points to the proof scattered within two of Steiner's own papers from that time. Solutions based on algebraic formulations of the problem include those by , , , , and . The algebraic solutions do not distinguish between internal and external tangencies among the circles and the given triangle; if the problem is generalized to allow tangencies of either kind, then a given triangle will have 32 different solutions and conversely a triple of mutually tangent circles will be a solution for eight different triangles. credits the enumeration of these solutions to , but notes that this count of the number of solutions was already given in a remark by .
|
|
The Hanoi graph H^5_3 as a penny graph (the contact graph of the black disks) Penny graphs are a special case of the coin graphs (graphs that can be represented by tangencies of non-crossing circles of arbitrary radii). Because the coin graphs are the same as the planar graphs,, Theorem 8.4.2, p. 173. all penny graphs are planar.
|
|
Every matchstick graph is a unit distance graph. Penny graphs are the graphs that can be represented by tangencies of non-overlapping unit circles. Every penny graph is a matchstick graph. However, some matchstick graphs (such as the eight-vertex cubic matchstick graph of the first illustration) are not penny graphs, because realizing them as a matchstick graph causes some non-adjacent vertices to be closer than unit distance to each other.
|
|
The usefulness of inversion can be increased significantly by resizing. As noted in Viète's reconstruction, the three given circles and the solution circle can be resized in tandem while preserving their tangencies. Thus, the initial Apollonius problem is transformed into another problem that may be easier to solve. For example, the four circles can be resized so that one given circle is shrunk to a point; alternatively, two given circles can often be resized so that they are tangent to one another.
|
|
Right circular and oblique circular cones Democritus was also a pioneer of mathematics and geometry in particular. We only know this through citations of his works (titled On Numbers, On Geometrics, On Tangencies, On Mapping, and On Irrationals) in other writings, since most of Democritus's body of work did not survive the Middle Ages. Democritus was among the first to observe that a cone and pyramid with the same base area and height has one-third the volume of a cylinder or prism respectively.
|
|
Thus, as the solution circle swells, the internally tangent given circles must swell in tandem, whereas the externally tangent given circles must shrink, to maintain their tangencies. Viète used this approach to shrink one of the given circles to a point, thus reducing the problem to a simpler, already solved case. He first solved the CLL case (a circle and two lines) by shrinking the circle into a point, rendering it a LLP case. He then solved the CLP case (a circle, a line and a point) using three lemmas.
|
|
Pappus mentions other treatises of Apollonius: # Λόγου ἀποτομή, De Rationis Sectione ("Cutting of a Ratio") # Χωρίου ἀποτομή, De Spatii Sectione ("Cutting of an Area") # Διωρισμένη τομή, De Sectione Determinata ("Determinate Section") # Ἐπαφαί, De Tactionibus ("Tangencies") # Νεύσεις, De Inclinationibus ("Inclinations") # Τόποι ἐπίπεδοι, De Locis Planis ("Plane Loci"). Each of these was divided into two books, and—with the Data, the Porisms, and Surface-Loci of Euclid and the Conics of Apollonius—were, according to Pappus, included in the body of the ancient analysis. Descriptions follow of the six works mentioned above.
|
|
This conjecture can be traced back to Fatou in the 1920s, and later Smale proposed him in the 1960s. Axiom A, and guess that the hyperbolic system should be dense in any system, but this is not true when the dimension is greater than or equal to 2, because there is homoclinic tangencies. The work of Shen Weixiao and others is equivalent to confirming that Smale's conjecture is correct in one dimension. The proof of Real Fatou conjecture is one of the most important developments in conformal dynamics in the past decade.
|
|
Specifically, let A, B, and C be the three corners of the arbelos, with B between A and C. Let H be the point where the larger semicircle intercepts the line perpendicular to the AC through the point B. The segment BH divides the arbelos in two parts. The twin circles are the two circles inscribed in these parts, each tangent to one of the two smaller semicircles, to the segment BH, and to the largest semicircle. Each of the two circles is uniquely determined by its three tangencies. Constructing it is a special case of the Problem of Apollonius.
|
|
Blocked-in poses may also include important in-betweens, extremes, and breakdowns necessary to establishing the flow and timing of a particular shot. In 3D, the animation curves of a blocked shot are often created using "stepped" or "square" tangencies, which provides no interpolation between animation poses. This allows the animator to see the poses of the animation without any strange and/or unintentional automatic interpolation. While this is sometimes problematic due to gimbal lock, seeing the poses in this way allows the animator to adjust the timing of an animation quickly, without the distraction of the software's automatic interpolation.
|
|
Apollonius of Perga (c. 262 190 BC) posed and solved this famous problem in his work (', "Tangencies"); this work has been lost, but a 4th-century AD report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them (Figure 2), a pair of solutions for each way to divide the three given circles in two subsets (there are 4 ways to divide a set of cardinality 3 in 2 parts). In the 16th century, Adriaan van Roomen solved the problem using intersecting hyperbolas, but this solution does not use only straightedge and compass constructions.
|
|
Viète first solved some simple special cases of Apollonius' problem, such as finding a circle that passes through three given points which has only one solution if the points are distinct; he then built up to solving more complicated special cases, in some cases by shrinking or swelling the given circles. According to the 4th-century report of Pappus, Apollonius' own book on this problem—entitled (', "Tangencies"; Latin: De tactionibus, De contactibus)—followed a similar progressive approach. Hence, Viète's solution is considered to be a plausible reconstruction of Apollonius' solution, although other reconstructions have been published independently by three different authors.Simson R (1734) Mathematical Collection, volume VII, p. 117.
|
|
First Ajima–Malfatti point Given a triangle ABC and its three Malfatti circles, let D, E, and F be the points where two of the circles touch each other, opposite vertices A, B, and C respectively. Then the three lines AD, BE, and CF meet in a single triangle center known as the first Ajima–Malfatti point after the contributions of Ajima and Malfatti to the circle problem. The second Ajima–Malfatti point is the meeting point of three lines connecting the tangencies of the Malfatti circles with the centers of the excircles of the triangle..C. Kimberling, Encyclopedia of Triangle Centers , X(179) and X(180).
|
|
He then derived a lemma for constructing the line perpendicular to an angle bisector that passes through a point, which he used to solve the LLP problem (two lines and a point). This accounts for the first four cases of Apollonius' problem, those that do not involve circles. To solve the remaining problems, Viète exploited the fact that the given circles and the solution circle may be re-sized in tandem while preserving their tangencies (Figure 4). If the solution-circle radius is changed by an amount Δr, the radius of its internally tangent given circles must be likewise changed by Δr, whereas the radius of its externally tangent given circles must be changed by −Δr.
|
|
An intersection of these two hyperbolas (if any) gives the center of a solution circle that has the chosen internal and external tangencies to the three given circles. The full set of solutions to Apollonius' problem can be found by considering all possible combinations of internal and external tangency of the solution circle to the three given circles. Isaac Newton (1687) refined van Roomen's solution, so that the solution-circle centers were located at the intersections of a line with a circle. Newton formulates Apollonius' problem as a problem in trilateration: to locate a point Z from three given points A, B and C, such that the differences in distances from Z to the three given points have known values.
|
|
An infinite sequence of circles can be constructed, containing rings for each n that exactly meet the bound of the ring lemma, showing that it is tight. The construction allows halfplanes to be considered as degenerate circles with infinite radius, and includes additional tangencies between the circles beyond those required in the statement of the lemma. It begins by sandwiching the unit circle between two parallel halfplanes; in the geometry of circles, these are considered to be tangent to each other at the point at infinity. Each successive circle after these first two is tangent to the central unit circle and to the two most recently added circles; see the illustration for the first six circles (including the two halfplanes) constructed in this way.
|
|
Constructing a penny graph from the locations of its circles can be performed as an instance of the closest pair of points problem, taking worst-case time or (with randomized time and with the use of the floor function) expected time .. An alternative method with the same worst-case time is to construct the Delaunay triangulation or nearest neighbor graph of the circle centers (both of which contain the penny graph as a subgraph) and then test which edges correspond to circle tangencies. However, testing whether a given graph is a penny graph is NP-hard, even when the given graph is a tree.. Similarly, testing whether a graph is a three-dimensional mutual nearest neighbor graph is also NP-hard.
|
|
The circle packing theorem implies that every polyhedral graph can be represented as the graph of a polyhedron that has a midsphere. A stronger form of the circle packing theorem asserts that any polyhedral graph and its dual graph can be represented by two circle packings, such that the two tangent circles representing a primal graph edge and the two tangent circles representing the dual of the same edge always have their tangencies at right angles to each other at the same point of the plane. A packing of this type can be used to construct a convex polyhedron that represents the given graph and that has a midsphere, a sphere tangent to all of the edges of the polyhedron. Conversely, if a polyhedron has a midsphere, then the circles formed by the intersections of the sphere with the polyhedron faces and the circles formed by the horizons on the sphere as viewed from each polyhedron vertex form a dual packing of this type.
|
|
An example of an Apollonian gasket Construction of an Apollonian network from a circle packing Apollonian networks are named after Apollonius of Perga, who studied the Problem of Apollonius of constructing a circle tangent to three other circles. One method of constructing Apollonian networks is to start with three mutually-tangent circles and then repeatedly inscribe another circle within the gap formed by three previously-drawn circles. The fractal collection of circles produced in this way is called an Apollonian gasket. If the process of producing an Apollonian gasket is stopped early, with only a finite set of circles, then the graph that has one vertex for each circle and one edge for each pair of tangent circles is an Apollonian network.. The existence of a set of tangent circles whose tangencies represent a given Apollonian network forms a simple instance of the Koebe–Andreev–Thurston circle-packing theorem, which states that any planar graph can be represented by tangent circles in the same way..
|
|
From there they could agree to a mutually beneficial trade to anywhere in the lens formed by these indifference curves. But the only points from which no mutually beneficial trade exists are the points of tangency between the two people's indifference curves, such as point E. The contract curve is the set of these indifference curve tangencies within the lens—it is a curve that slopes upward to the right and goes through point E. right In microeconomics, the contract curve is the set of points representing final allocations of two goods between two people that could occur as a result of mutually beneficial trading between those people given their initial allocations of the goods. All the points on this locus are Pareto efficient allocations, meaning that from any one of these points there is no reallocation that could make one of the people more satisfied with his or her allocation without making the other person less satisfied. The contract curve is the subset of the Pareto efficient points that could be reached by trading from the people's initial holdings of the two goods.
|
|
This shape was called a Siamese dodecahedron in the paper by Hans Freudenthal and B. L. van der Waerden (1947) which first described the set of eight convex deltahedra.. The dodecadeltahedron name was given to the same shape by , referring to the fact that it is a 12-sided deltahedron. There are other simplicial dodecahedra, such as the hexagonal bipyramid, but this is the only one that can be realized with equilateral faces. Bernal was interested in the shapes of holes left in irregular close-packed arrangements of spheres, so he used a restrictive definition of deltahedra, in which a deltahedron is a convex polyhedron with triangular faces that can be formed by the centers of a collection of congruent spheres, whose tangencies represent polyhedron edges, and such that there is no room to pack another sphere inside the cage created by this system of spheres. This restrictive definition disallows the triangular bipyramid (as forming two tetrahedral holes rather than a single hole), pentagonal bipyramid (because the spheres for its apexes interpenetrate, so it cannot occur in sphere packings), and icosahedron (because it has interior room for another sphere).
|
|
Alvord was interested in the classical problem of Apollonius, to find a circle tangent to three given circles, and the special cases of Apollonius' problem, as well as the generalization to spheres. In 1855 he published in Smithsonian Contributions to Knowledge.B. Alvord (1855) Tangencies of Circles and of Spheres, Smithsonian Contributions, volume 8, from Google Books Posted to the remote Fort Vancouver, he continued his investigations and submitted his findings in 1860 but was frustrated by a fire. In 1882, when he found that there are 96 circles which cut four given circles at a fixed angle and there are 640 spheres which cut five given spheres at a fixed angle, he assembled all his results for an article in American Journal of Mathematics.B. Alvord (1882) The Intersection of Circles and the Intersection of Spheres, American Journal of Mathematics 5(1): 25–44 where he explained the delay: :All of this memoir, except the last two problems, were completed and sent to the Smithsonian Institute in January, 1860, from Fort Vancouver, Washington Territory, but the manuscript was burned in January 1865 when the upper story of the Smithsonian building was on fire.
|
|