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Schoch Line
In geometry, the Schoch line is a line defined from an arbelos and named by Peter Woo after Thomas Schoch, who had studied it in conjunction with the Schoch circles. Construction An arbelos is a shape bounded by three mutually-tangent semicircular arcs with collinear endpoints, with the two smaller arcs nested inside the larger one; let the endpoints of these three arcs be (in order along the line containing them) ''A'', ''B'', and ''C''. Let ''K''1 and ''K''2 be two more arcs, centered at ''A'' and ''C'', respectively, with radii ''AB'' and ''CB'', so that these two arcs are tangent at ''B''; let ''K''3 be the largest of the three arcs of the arbelos. A circle, with the center ''A''1, is then created tangent to the arcs ''K''1, ''K''2, and ''K''3. This circle is congruent with Archimedes' twin circles, making it an Archimedean circle; it is one of the Schoch circles. The Schoch line is perpendicular In geometry, two geometric objects are perpendicular if they intersec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line (geometry)
In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray (optics), ray of light. Lines are space (mathematics), spaces of dimension one, which may be Embedding, embedded in spaces of dimension two, three, or higher. The word ''line'' may also refer, in everyday life, to a line segment, which is a part of a line delimited by two Point (geometry), points (its ''endpoints''). Euclid's Elements, Euclid's ''Elements'' defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. ''Euclidean line'' and ''Euclidean geometry'' are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as Non-Euclidean geometry, non-Euclidean, Project ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arbelos
In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line (the ''baseline'') that contains their diameters. The earliest known reference to this figure is in Archimedes's ''Book of Lemmas'', where some of its mathematical properties are stated as Propositions 4 through 8. The word ''arbelos'' is Greek for 'shoemaker's knife'. The figure is closely related to the Pappus chain. Properties Two of the semicircles are necessarily concave, with arbitrary diameters and ; the third semicircle is Convex curve, convex, with diameter Let the diameters of the smaller semicircles be and ; then the diameter of the larger semircle is . Area Let be the intersection of the larger semicircle with the line perpendicular to at . Then the area (geometry), area of the arbelos is equal to the area of a circle with diameter . Proof: For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schoch Circles
In geometry, the Schoch circles are twelve Archimedean circles constructed by Thomas Schoch. History In 1979, Thomas Schoch discovered a dozen new Archimedean circles; he sent his discoveries to ''Scientific Americans "Mathematical Games" editor Martin Gardner. The manuscript was forwarded to Leon Bankoff. Bankoff gave a copy of the manuscript to Professor Clayton Dodge of the University of Maine in 1996. The two were planning to write an article about the Arbelos, in which the Schoch circles would be included; however, Bankoff died the year after. In 1998, Peter Y. Woo of Biola University Biola University () is a private, nondenominational, evangelical Christian university in La Mirada, California. It was founded in 1908 as the Bible Institute of Los Angeles. It has over 150 programs of study in nine schools offering bachelor' ... published Schoch's findings on his website. By generalizing two of Schoch's circles, Woo discovered an infinite family of Archimedean circles n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is tangent to the curve at a point if the line passes through the point on the curve and has slope , where ''f'' is the derivative of ''f''. A similar definition applies to space curves and curves in ''n''-dimensional Euclidean space. The point where the tangent line and the curve meet or intersect is called the ''point of tangency''. The tangent line is said to be "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. The tangent line to a point on a differentiable curve can also be thought of as a '' tangent line approximation'', the graph of the affine function that best approximates the original function at the given point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedes' Circles
In geometry, the twin circles are two special circles associated with an arbelos. An arbelos is determined by three collinear points , , and , and is the curvilinear triangular region between the three semicircles that have , , and as their diameters. If the arbelos is partitioned into two smaller regions by a line segment through the middle point of , , and , perpendicular to line , then each of the two twin circles lies within one of these two regions, tangent to its two semicircular sides and to the splitting segment. These circles first appeared in the '' Book of Lemmas'', which showed (Proposition V) that the two circles are congruent. Thābit ibn Qurra, who translated this book into Arabic, attributed it to Greek mathematician Archimedes. Based on this claim the twin circles, and several other circles in the Arbelos congruent to them, have also been called Archimedes's circles. However, this attribution has been questioned by later scholarship. Construction Specifically ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedean Circle
In geometry, an Archimedean circle is any circle constructed from an arbelos that has the same radius as each of Archimedes' twin circles. If the arbelos is normed such that the diameter of its outer (largest) half circle has a length of 1 and ''r'' denotes the radius of any of the inner half circles, then the radius ''ρ'' of such an Archimedean circle is given by :\rho=\fracr\left(1-r\right), There are over fifty different known ways to construct Archimedean circles. Origin An Archimedean circle was first constructed by Archimedes in his '' Book of Lemmas''. In his book, he constructed what is now known as Archimedes' twin circles. Radius If a and b are the radii of the small semicircles of the arbelos, the radius of an Archimedean circle is equal to :R = \frac This radius is thus \frac 1R = \frac 1a + \frac 1b. The Archimedean circle with center C (as in the figure at right) is tangent to the tangents from the centers of the small semicircles to the other small semi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perpendicular
In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. Perpendicular intersections can happen between two lines (or two line segments), between a line and a plane, and between two planes. ''Perpendicular'' is also used as a noun: a perpendicular is a line which is perpendicular to a given line or plane. Perpendicularity is one particular instance of the more general mathematical concept of '' orthogonality''; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its '' normal vector''. A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinity
Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophical nature of infinity has been the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including Guillaume de l'Hôpital, l'Hôpital and Johann Bernoulli, Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or Magnitude (mathematics), magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Woo Circles
In geometry, the Woo circles, introduced by Peter Y. Woo, are a set of infinitely many Archimedean circles. Construction Form an arbelos with the two inner semicircles tangent at point ''C''. Let ''m'' denote any nonnegative real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re .... Draw two circles, with radii ''m'' times the radii of the smaller two arbelos semicircles, centered on the arbelos ground line, also tangent to each other at point ''C'' and with radius ''m'' times the radius of the corresponding small arbelos arc. Any circle centered on the Schoch line and externally tangent to the circles is a Woo circle. See also * Schoch circles References Arbelos Circles {{Elementary-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arbelos
In geometry, an arbelos is a plane region bounded by three semicircles with three apexes such that each corner of each semicircle is shared with one of the others (connected), all on the same side of a straight line (the ''baseline'') that contains their diameters. The earliest known reference to this figure is in Archimedes's ''Book of Lemmas'', where some of its mathematical properties are stated as Propositions 4 through 8. The word ''arbelos'' is Greek for 'shoemaker's knife'. The figure is closely related to the Pappus chain. Properties Two of the semicircles are necessarily concave, with arbitrary diameters and ; the third semicircle is Convex curve, convex, with diameter Let the diameters of the smaller semicircles be and ; then the diameter of the larger semircle is . Area Let be the intersection of the larger semicircle with the line perpendicular to at . Then the area (geometry), area of the arbelos is equal to the area of a circle with diameter . Proof: For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
