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Four-vertex Theorem
The four-vertex theorem of geometry states that the curvature along a simple, closed, smooth plane curve has at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives from the convention of calling an extreme point of the curvature function a vertex. This theorem has many generalizations, including a version for space curves where a vertex is defined as a point of vanishing torsion. Definition and examples The curvature at any point of a smooth curve in the plane can be defined as the reciprocal of the radius of an osculating circle at that point, or as the norm of the second derivative of a parametric representation of the curve, parameterized consistently with the length along the curve. For the vertices of a curve to be well-defined, the curvature itself should vary continuously, as happens for curves of smoothness C^2. A vertex is then a local maximum or local minimum of curvature. If the curvature i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ellipse Evolute
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity e, a number ranging from e = 0 (the limiting case of a circle) to e = 1 (the limiting case of infinite elongation, no longer an ellipse but a parabola). An ellipse has a simple algebraic solution for its area, but only approximations for its perimeter (also known as circumference), for which integration is required to obtain an exact solution. Analytically, the equation of a standard ellipse centered at the origin with width 2a and height 2b is: : \frac+\frac = 1 . Assuming a \ge b, the foci are (\pm c, 0) for c = \sqrt. The standard parametric equation is: : (x,y) = (a\cos(t),b\sin(t)) \quad \text \quad 0\leq t\leq 2\pi. Ellipses a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Minimum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the ''local'' or ''relative'' extrema), or on the entire domain (the ''global'' or ''absolute'' extrema). Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. Definition A real-valued function ''f'' defined on a domain ''X'' has a global (or absolute) maximum point at ''x''∗, if for all ''x'' in ''X''. Similarly, the function has a global (or absolute) minimum point at ''x''∗, if for all ''x'' in ''X''. The value of the function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are t ... [...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 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 said to be a tangent of a 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. As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "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. Similarly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smallest-circle Problem
The smallest-circle problem (also known as minimum covering circle problem, bounding circle problem, least bounding circle problem, smallest enclosing circle problem) is a mathematical problem of computing the smallest circle that contains all of a given set of points in the Euclidean plane. The corresponding problem in ''n''-dimensional space, the smallest bounding sphere problem, is to compute the smallest ''n''-sphere that contains all of a given set of points. The smallest-circle problem was initially proposed by the English mathematician James Joseph Sylvester in 1857. The smallest-circle problem in the plane is an example of a facility location problem (the 1-center problem) in which the location of a new facility must be chosen to provide service to a number of customers, minimizing the farthest distance that any customer must travel to reach the new facility. Both the smallest circle problem in the plane, and the smallest bounding sphere problem in any higher-dimensio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adolf Kneser
Adolf Kneser (19 March 1862 – 24 January 1930) was a German mathematician. He was born in Grüssow, Mecklenburg, Germany and died in Breslau, Germany (now Wrocław, Poland). He is the father of the mathematician Hellmuth Kneser and the grandfather of the mathematician Martin Kneser. Kneser is known for the first proof of the four-vertex theorem that applied in general to non-convex curves. Kneser's theorem on differential equations is named after him, and provides criteria to decide whether a differential equation is oscillating. He is also one of the namesakes of the Tait–Kneser theorem In differential geometry, the Tait–Kneser theorem states that, if a smooth plane curve has monotonic curvature, then the osculating circles of the curve are disjoint and nested within each other. The logarithmic spiral or the pictured Archim ... on osculating circles. Selected publications * *; *; * * References External links * * 1862 births 1930 deaths 19th-ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contact (mathematics)
In mathematics, two functions have a contact of order ''k'' if, at a point ''P'', they have the same value and ''k'' equal derivatives. This is an equivalence relation, whose equivalence classes are generally called jets. The point of osculation is also called the double cusp. Contact is a geometric notion; it can be defined algebraically as a valuation. One speaks also of curves and geometric objects having ''k''-th order contact at a point: this is also called ''osculation'' (i.e. kissing), generalising the property of being tangent. (Here the derivatives are considered with respect to arc length.) An osculating curve from a given family of curves is a curve that has the highest possible order of contact with a given curve at a given point; for instance a tangent line is an osculating curve from the family of lines, and has first-order contact with the given curve; an osculating circle is an osculating curve from the family of circles, and has second-order contact (sam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Syamadas Mukhopadhyaya
Syamadas Mukhopadhyaya (22 June 1866 – 8 May 1937) was an Indian mathematician who introduced the four-vertex theorem and Mukhopadhyaya's theorem in plane geometry. Biography Syamadas Mukhopadhyaya was born at Haripal, Hooghly district, in the West Bengal, India. He graduated from Hooghly College, received his M.A. degree from Presidency College in Calcutta, and his Ph.D. degree from Calcutta University in 1910. He also took classes from the Indian Association for the Cultivation of Science. Mukhopadhyaya was appointed by Asustosh Mookerjee as professor of mathematics in the Rajabazar Science College, University of Calcutta. Jacques Hadamard communicated with Mukhopadyaya about the latter's work on the geometry of a plane arc and Wilhelm Blaschke's book on geometry had a reference to Mukhopadhyaya. He worked at Bangabasi College and then at Bethune College in Calcutta, where he lectured in Mathematics, English Literature, and Philosophy. In 1932, he was elected presi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Curve
In geometry, a convex curve is a plane curve that has a supporting line through each of its points. There are many other equivalent definitions of these curves, going back to Archimedes. Examples of convex curves include the convex polygons, the boundaries of convex sets, and the graphs of convex functions. Important subclasses of convex curves include the closed convex curves (the boundaries of bounded convex sets), the smooth curves that are convex, and the strictly convex curves, which have the additional property that each supporting line passes through a unique point of the curve. Combinations of these properties have also been considered. Bounded convex curves have a well-defined length, which can be obtained by approximating them with polygons, or from the average length of their projections onto a line. The maximum number of grid points that can belong to a single curve is controlled by its length. The points at which a convex curve has a unique supporting line ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reuleaux Triangle
A Reuleaux triangle is a curved triangle with constant width, the simplest and best known curve of constant width other than the circle. It is formed from the intersection of three circular disks, each having its center on the boundary of the other two. Constant width means that the separation of every two parallel supporting lines is the same, independent of their orientation. Because its width is constant, the Reuleaux triangle is one answer to the question "Other than a circle, what shape can a manhole cover be made so that it cannot fall down through the hole?" Reuleaux triangles have also been called spherical triangles, but that term more properly refers to triangles on the curved surface of a sphere. They are named after Franz Reuleaux,. a 19th-century German engineer who pioneered the study of machines for translating one type of motion into another, and who used Reuleaux triangles in his designs. However, these shapes were known before his time, for instance by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curve Of Constant Width
In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body of constant width or an orbiform, the name given to these shapes by Leonhard Euler. Standard examples are the circle and the Reuleaux triangle. These curves can also be constructed using circular arcs centered at crossings of an arrangement of lines, as the involutes of certain curves, or by intersecting circles centered on a partial curve. Every body of constant width is a convex set, its boundary crossed at most twice by any line, and if the line crosses perpendicularly it does so at both crossings, separated by the width. By Barbier's theorem, the body's perimeter is exactly times its width, but its area depends on its shape, with the Reuleaux triangle having the smallest possible area for its width and the circle the largest. Every superset of a bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
