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The radius of the curvature of the stressed structure is related to stress tensor in the structure, and can be described by modified Stoney formula. [6] The topography of the stressed structure including radii of curvature can be measured using optical scanner methods.
The curvature is the reciprocal of radius of curvature. That is, the curvature is =, where R is the radius of curvature [5] (the whole circle has this curvature, it can be read as turn 2π over the length 2π R). This definition is difficult to manipulate and to express in formulas.
The formula simplifies: = ′ (). ... The reciprocal of the curvature is called the radius of curvature. A circle with radius r has a constant ...
Where degree of curvature is based on 100 units of arc length, the conversion between degree of curvature and radius is Dr = 18000/π ≈ 5729.57795, where D is degree and r is radius. Since rail routes have very large radii, they are laid out in chords, as the difference to the arc is inconsequential; this made work easier before electronic ...
The Frenet–Serret formulas are: =, = +, =, where is the derivative with respect to arclength, κ is the curvature, and τ is the torsion of the space curve. (Intuitively, curvature measures the failure of a curve to be a straight line, while torsion measures the failure of a curve to be planar.)
The center and radius of the osculating circle at a given point are called center of curvature and radius of curvature of the curve at that point. A geometric construction was described by Isaac Newton in his Principia:
The area formula is intuitive: start with a circle of radius (so its area is ) and stretch it by a factor / to make an ellipse. This scales the area by the same factor: π b 2 ( a / b ) = π a b . {\displaystyle \pi b^{2}(a/b)=\pi ab.} [ 18 ] However, using the same approach for the circumference would be fallacious – compare the integrals
The intrinsic quantities used most often are arc length, tangential angle, curvature or radius of curvature, and, for 3-dimensional curves, torsion. Specifically: The natural equation is the curve given by its curvature and torsion.