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Thus the length of a curve is a non-negative real number. Usually no curves are considered which are partly spacelike and partly timelike. In theory of relativity, arc length of timelike curves (world lines) is the proper time elapsed along the world line, and arc length of a spacelike curve the proper distance along the curve.
The equation defining a plane curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining r as a function of φ. The resulting curve then consists of points of the form (r(φ), φ) and can be regarded as the graph of the polar function r.
The curve must be continuous (no jump) between the two ends. The sinuosity value is really significant when the line is continuously differentiable (no angular point). The distance between both ends can also be evaluated by a plurality of segments according to a broken line passing through the successive inflection points (sinuosity of order 2).
Radius of curvature and center of curvature. In differential geometry, the radius of curvature, R, is the reciprocal of the curvature.For a curve, it equals the radius of the circular arc which best approximates the curve at that point.
Note that changing F into –F would not change the curve defined by F(x, y) = 0, but it would change the sign of the numerator if the absolute value were omitted in the preceding formula. A point of the curve where F x = F y = 0 is a singular point, which means that the curve is not differentiable at this point, and thus that the curvature is ...
Ptolemy used a circle of diameter 120, and gave chord lengths accurate to two sexagesimal (base sixty) digits after the integer part. [2] 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.
If the horizontal cross-section moves up or down, toward or away from the apex of the cone, D and E move along the parabola, always maintaining the relationship between x and y shown in the equation. The parabolic curve is therefore the locus of points where the equation is satisfied, which makes it a Cartesian graph of the quadratic function ...
The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. Formulas are known for computing distances between different types of objects, such as the distance from a point to a line .