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There are continuous curves on which every arc (other than a single-point arc) has infinite length. An example of such a curve is the Koch curve. Another example of a curve with infinite length is the graph of the function defined by f(x) = x sin(1/x) for any open set with 0 as one of its delimiters and f(0) = 0.
However, if both ends of the curve (P 1 and P 2) are at the same level (y 1 = y 2), it can be shown that [58] = where L is the total length of the curve between P 1 and P 2 and h is the sag (vertical distance between P 1, P 2 and the vertex of the curve).
Therefore an intrinsic equation defines the shape of the curve without specifying its position relative to an arbitrarily defined coordinate system. The intrinsic quantities used most often are arc length s {\displaystyle s} , tangential angle θ {\displaystyle \theta } , curvature κ {\displaystyle \kappa } or radius of curvature , and, for 3 ...
The chord function is defined geometrically as shown in the picture. The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle. The angle θ is taken in the positive sense and must lie in the interval 0 < θ ≤ π (radian measure).
3.1 Formula from arc length. 3.2 Formula from chord length. 3.3 Formula from ... a curve with an arc length of 600 units that has an overall sweep of 6 degrees is a 1 ...
In the following equations, denotes the sagitta (the depth or height of the arc), equals the radius of the circle, and the length of the chord spanning the base of the arc. As 1 2 l {\displaystyle {\tfrac {1}{2}}l} and r − s {\displaystyle r-s} are two sides of a right triangle with r {\displaystyle r} as the hypotenuse , the Pythagorean ...
The curve can be parameterised by the equation = (), = / (). [ 4 ] Due to the geometrical way it was defined, the tractrix has the property that the segment of its tangent , between the asymptote and the point of tangency, has constant length a .
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.