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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 curve (red) is what the algorithm is trying to draw. The raster points (black) are used as starting points to find the closest points on the curve (red circles). The spacing between each raster point is exaggerated to show the individual curve points; to more accurately trace the curve, more raster points would be used. [4]
Free-space diagram of the red and the blue curve. In contrast to the definition in the text, which uses the parameter interval [0,1] for both curves, the curves are parameterized by arc length in this example. An important tool for calculating the Fréchet distance of two curves is the free-space diagram, which was introduced by Alt and Godau. [4]
The equation of a line is given by = +. The equation of the normal of that line which passes through the point P is given = +. The point at which these two lines intersect is the closest point on the original line to the point P. Hence:
The distance between any two points on the real line is the absolute value of the numerical difference of their coordinates, their absolute difference.Thus if and are two points on the real line, then the distance between them is given by: [1]
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.
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 ...
Given the two red points, the blue line is the linear interpolant between the points, and the value y at x may be found by linear interpolation.. In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.