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Inellipse
In triangle geometry, an inellipse is an ellipse that touches the three sides of a triangle. The simplest example is the incircle. Further important inellipses are the Steiner inellipse, which touches the triangle at the midpoints of its sides, the Mandart inellipse and Brocard inellipse (see examples section). For any triangle there exist an infinite number of inellipses. The Steiner inellipse plays a special role: Its area is the greatest of all inellipses. Because a non-degenerate conic section is uniquely determined by five items out of the sets of vertices and tangents, in a triangle whose three sides are given as tangents one can specify only the points of contact on two sides. The third point of contact is then uniquely determined. Parametric representations, center, conjugate diameters The inellipse of the triangle with vertices :O=(0,0), \; A=(a_1,a_2), \; B=(b_1,b_2) and points of contact :U=(u_1,u_2) ,\; V=(v_1,v_2) on OA and OB respectively can by described ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steiner Inellipse
In geometry, the Steiner inellipse,Weisstein, E. "Steiner Inellipse" — From MathWorld, A Wolfram Web Resource, http://mathworld.wolfram.com/SteinerInellipse.html. midpoint inellipse, or midpoint ellipse of a triangle is the unique ellipse inscribed in the triangle and tangent to the sides at their midpoints. It is an example of an inellipse. By comparison the inscribed circle and Mandart inellipse of a triangle are other inconics that are tangent to the sides, but not at the midpoints unless the triangle is equilateral. The Steiner inellipse is attributed by Dörrie to Jakob Steiner, and a proof of its uniqueness is given by Dan Kalman.. The Steiner inellipse contrasts with the Steiner circumellipse, also called simply the Steiner ellipse, which is the unique ellipse that passes through the vertices of a given triangle and whose center is the triangle's centroid. Definition and properties ;Definition An ellipse that is tangent to the sides of a triangle at its midpoin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mandart Inellipse
In geometry, the Mandart inellipse of a triangle is an ellipse inscribed within the triangle, tangent to its sides at the contact points of its excircles (which are also the vertices of the extouch triangle and the endpoints of the splitters). The Mandart inellipse is named after H. Mandart, who studied it in two papers published in the late 19th century..; . As cited by . Parameters As an inconic, the Mandart inellipse is described by the parameters :x:y:z=\frac:\frac:\frac where ''a'', ''b'', and ''c'' are sides of the given triangle. Related points The center of the Mandart inellipse is the mittenpunkt of the triangle. The three lines connecting the triangle vertices to the opposite points of tangency all meet in a single point, the Nagel point of the triangle. See also *Steiner inellipse In geometry, the Steiner inellipse,Weisstein, E. "Steiner Inellipse" — From MathWorld, A Wolfram Web Resource, http://mathworld.wolfram.com/SteinerInellipse.html. midpoint inellips ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trilinear Coordinates
In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices and respectively; the ratio is the ratio of the perpendicular distances from the point to the sidelines opposite vertices and respectively; and likewise for and vertices and . In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (, , ), or equivalently in ratio form, for any positive constant . If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0. If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative. It is impossible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumconic And Inconic
In Euclidean geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle.Weisstein, Eric W. "Inconic." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Inconic.html Suppose are distinct non-collinear points, and let denote the triangle whose vertices are . Following common practice, denotes not only the vertex but also the angle at vertex , and similarly for and as angles in . Let a= , BC, , b=, CA, , c=, AB, , the sidelengths of . In trilinear coordinates, the general circumconic is the locus of a variable point X = x:y:z satisfying an equation :uyz + vzx + wxy = 0, for some point . The isogonal conjugate of each point on the circumconic, other than , is a point on the line :ux + vy + wz = 0. This line meets the circumcircle of in 0,1, or 2 points according as the circumconic is an ellipse, parabola, or hyperbola. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maxima And Minima
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 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steiner Ellipse
In geometry, the Steiner ellipse of a triangle, also called the Steiner circumellipse to distinguish it from the Steiner inellipse, is the unique circumellipse (ellipse that touches the triangle at its vertices) whose center is the triangle's centroid.Weisstein, Eric W. "Steiner Circumellipse." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/SteinerCircumellipse.html Named after Jakob Steiner, it is an example of a circumconic. By comparison the circumcircle of a triangle is another circumconic that touches the triangle at its vertices, but is not centered at the triangle's centroid unless the triangle is equilateral. The area of the Steiner ellipse equals the area of the triangle times \frac, and hence is 4 times the area of the Steiner inellipse. The Steiner ellipse has the least area of any ellipse circumscribed about the triangle. The Steiner ellipse is the scaled Steiner inellipse (factor 2, center is the centroid). Hence both ellipses are similar (h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix. If homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. Since homogeneous coordinates are also given to points at infinity, the number of coordinates required to allow this extension is one more ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola. Hyperbolas arise in many ways: * as the curve representing the reciprocal function y(x) = 1/x in the Cartesian plane, * as the path followed by the shadow of the tip of a sundial, * as the shape of an open orbit (as distinct from a closed elliptical orbit), such as the orbit of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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