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Having a constant diameter, measured at varying angles around the shape, is often considered to be a simple measurement of roundness.This is misleading. [3]Although constant diameter is a necessary condition for roundness, it is not a sufficient condition for roundness: shapes exist that have constant diameter but are far from round.
The degree of circularity of an ellipse is quantified by eccentricity, with values between 0 to 1, where 0 is a perfect circle (waist circumference same as height) and 1 is a vertical line. [1] To accommodate human shape data in a greater range, Thomas and colleagues mapped eccentricity in a range of 1 to 20 by using the equation: [ 1 ]
The transformation sends the circle to an ellipse by stretching or shrinking the horizontal and vertical diameters to the major and minor axes of the ellipse. The square gets sent to a rectangle circumscribing the ellipse. The ratio of the area of the circle to the square is π /4, which means the ratio of the ellipse to the rectangle is also π /4
A circle bounds a region of the plane called a disc. The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern
More formulas of this nature can be given, as explained by Ramanujan's theory of elliptic functions to alternative bases. Perhaps the most notable hypergeometric inversions are the following two examples, involving the Ramanujan tau function τ {\displaystyle \tau } and the Fourier coefficients j {\displaystyle \mathrm {j} } of the J-invariant ...
The perimeter of a stadium is calculated by the formula = (+) where a is the length of the straight sides and r is the radius of the semicircles. With the same parameters, the area of the stadium is A = π r 2 + 2 r a = r ( π r + 2 a ) {\displaystyle A=\pi r^{2}+2ra=r(\pi r+2a)} .
is equal to one. This parametrization gives the same value for the curvature, as it amounts to division by r 3 in both the numerator and the denominator in the preceding formula. The same circle can also be defined by the implicit equation F(x, y) = 0 with F(x, y) = x 2 + y 2 – r 2. Then, the formula for the curvature in this case gives
where c ∈ ℝ n is the center of the circle (irrelevant since it disappears in the derivatives), a,b ∈ ℝ n are perpendicular vectors of length ρ (that is, a · a = b · b = ρ 2 and a · b = 0), and h : ℝ → ℝ is an arbitrary function which is twice differentiable at t. The relevant derivatives of g work out to be