Reuleaux Triangles

Geometric figures may be analyzed according to their component parts. Perhaps the most basic components of the Reuleaux triangle consist of those points which comprise it. The position of a point in a plane can be given by means of two numbers. For example, x, y can be the distances of a point, P, from two given perpendicular lines. Given this information, the position of P can be determined when the values of both x and y are known. These numbers then become the cartesian coordinates of P.

Points may also exist in three-dimensional space. For these, distances, x, y, z may be given. Furthermore, a straight line in three-dimensional Euclidean space is determined by any two distinct points that lie on it. For example, points A(x1, y1, z1) and B(x2, y2, z2) might be two points on line AB. Additionally, line AB could be further defined as consisting of the points P(x, y, z) for which a value of the ratio (/( can be found such that

x = (x1 + (x2, y = (y1 + (y2, z = (z1 + (z2.

Every value of (/( determines a single point on the line AB. In addition, ( = 0 determines point B and ( = 0 determines point A. Further

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eal number pairs (u1, u2). Hence, "a plane can be 'parametrized' by a Cartesian coordinate system or by a polar coordinate system." Thus, a surface is given by the following equation: ? = ?(ui) = ?(u1, u2) = {x1(u1, u2), x2(u1, u2), x3(u1, u2)}. Here, the real number pairs (ui) range across a region G of the "Cartesian (u1, u2)-plane." It should also be noted that the functions xi are of class C3 (i.e., all of the function's derivatives consist orders less than k both exist and all are continuous). In addition, a given curve, ? = ?(t), may be considered to lie on the surface, ? = ?(ui), if the following equation holds true: ?(t) = ?(ui(t)). The functions ui(t) are also of class C3 and also satisfy the condition that ªu?1ª + ªu?2ª is not equal to zero for all t in the domain of the curve ?(t). Triangles, curves, and surfaces may be additionally be considered in their total extension, or "in the large." Although, for the most part, global geometry requires individual methods of proof for each problem, one powerful tool developed by Herglotz involves integral formulas. This technique essentially consists of demonstrating that a "sufficient condition for some global property is the identical vanishing of a certain functi
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