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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 of a line can be given in vector form: = + Here a is the position of a point on the line, and n is a unit vector in the direction of the line. Then as scalar t varies, x gives the locus of the line. The distance of an arbitrary point p to this line is given by
A ray with a terminus at A, with two points B and C on the right. Given a line and any point A on it, we may consider A as decomposing this line into two parts. Each such part is called a ray and the point A is called its initial point. It is also known as half-line, a one-dimensional half-space. The point A is considered to be a member of the ray.
A line through three-dimensional space between points of interest on a spherical Earth is the chord of the great circle between the points. The central angle between the two points can be determined from the chord length. The great circle distance is proportional to the central angle.
The coordinate-independent definition of the square of the line element ds in an n-dimensional Riemannian or Pseudo Riemannian manifold (in physics usually a Lorentzian manifold) is the "square of the length" of an infinitesimal displacement [2] (in pseudo Riemannian manifolds possibly negative) whose square root should be used for computing curve length: = = (,) where g is the metric tensor ...
Initially it is given all the points between the first and last point. It automatically marks the first and last point to be kept. It then finds the point that is farthest from the line segment with the first and last points as end points; this point is always farthest on the curve from the approximating line segment between the end points.
The two squared formulas inside the square root give the areas of squares on the horizontal and vertical sides, and the outer square root converts the area of the square on the hypotenuse into the length of the hypotenuse. [3] It is also possible to compute the distance for points given by polar coordinates.
The arc length of one branch between x = x 1 and x = x 2 is a ln y 1 / y 2 . The area between the tractrix and its asymptote is π a 2 / 2 , which can be found using integration or Mamikon's theorem. The envelope of the normals of the tractrix (that is, the evolute of the tractrix) is the catenary (or chain curve) given by y = a ...
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