Arc length

Arc length is the distance measured along a curve between two points. For a circular arc, it is the corresponding fraction of the circumference.

Arc length formula: An integral formula that computes the length of a differentiable curve from its coordinates or parametrization. It adds infinitesimal pieces of the curve to obtain the total distance.
Euclidean distance: The straight-line distance between two points in Euclidean space. Each short segment in a curve-length approximation uses this distance.
Circular arc: A portion of a circle's circumference between two points. For a circle, length follows directly from radius and central angle.
Chord length: The straight-line distance between two points on a circle. A chord cuts across the circle, while arc length follows its circumference.
Line integral: An integral that accumulates a scalar or vector quantity along a curve. Arc length is a line integral of the constant function one.
Derivative: A quantity describing the instantaneous rate of change of a function. Coordinate derivatives give the curve's instantaneous velocity.
Radian: An angle unit equal to the angle subtended by an arc whose length equals the circle's radius. Radians make the circular arc-length relation simply length equals radius times angle.
Manhattan distance: A distance metric that sums coordinate differences along axis-aligned directions. It measures constrained grid travel rather than distance along a specified curve.
Parametric curve: A curve described by coordinate functions of a parameter. Its coordinate derivatives determine the speed used in the arc-length integral.
Speed: The magnitude of an object's velocity or a path's rate of traversal. Integrating a parametrized curve's speed gives its length.

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