A curve in the plane can be approximated by connecting a finite number of points on the curve using (straight) line segments to create a polygonal path. Since it is straightforward to … See more

If a planar curve in $${\displaystyle \mathbb {R} ^{2}}$$ is defined by the equation $${\displaystyle y=f(x),}$$ where $${\displaystyle f}$$ is continuously differentiable, … See more

Antiquity
For much of the history of mathematics, even the greatest thinkers considered it impossible to … See more

• Arc (geometry)
• Circumference
• Crofton formula
• Elliptic integral
• Geodesics
• Intrinsic equation See more

Let $${\displaystyle f\colon [a,b]\to \mathbb {R} ^{n}}$$ be an injective and continuously differentiable (i.e., the derivative is a continuous function) function. The length of the curve defined by $${\displaystyle f}$$ can be defined as the See more

Arcs of circles
Arc lengths are denoted by s, since the Latin word for length (or size) is spatium.
In the following lines, See more

As mentioned above, some curves are non-rectifiable. That is, there is no upper bound on the lengths of polygonal approximations; the … See more