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The curve of the chains of a suspension bridge is always an intermediate curve between a parabola and a catenary, but in practice the curve is generally nearer to a parabola due to the weight of the load (i.e. the road) being much larger than the cables themselves, and in calculations the second-degree polynomial formula of a parabola is used.
A parabolic segment is the region bounded by a parabola and line. To find the area of a parabolic segment, Archimedes considers a certain inscribed triangle. The base of this triangle is the given chord of the parabola, and the third vertex is the point on the parabola such that the tangent to the parabola at that point is parallel to the chord.
A convex curve (black) forms a connected subset of the boundary of a convex set (blue), and has a supporting line (red) through each of its points. A parabola, a convex curve that is the graph of the convex function () = In geometry, a convex curve is a plane curve that has a supporting line through each of its points.
This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate the area of a circle, the surface area and volume of a sphere, area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a ...
In general, parabolic points form a curve on the surface called the parabolic line. The first fundamental form is positive definite , hence its determinant EG − F 2 is positive everywhere. Therefore, the sign of K coincides with the sign of LN − M 2 , the determinant of the second fundamental.
Let φ 1 = 0, φ 2 = 2π; then the area of the black region (see diagram) is A 0 = a 2 π 2, which is half of the area of the circle K 0 with radius r(2π). The regions between neighboring curves (white, blue, yellow) have the same area A = 2a 2 π 2. Hence: The area between two arcs of the spiral after a full turn equals the area of the circle ...
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The scale factors for the parabolic coordinates (,) are equal = = + Hence, the infinitesimal element of area is = (+) and the Laplacian equals = + (+) Other differential operators such as and can be expressed in the coordinates (,) by substituting the scale factors into the general formulae found in orthogonal coordinates.