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Geometric Curves

On One-Dimensional Geometric Objects: Curves and Lines

In geometry, curve is a general term for one-dimensional objects. These may be curved, but do not have to be. Sharp bends or kinks are also possible. This differs from the colloquial use of the word curve or its usage in contexts such as road construction.
Curves have length but possess neither area nor volume, though they may have other quantifiable properties. While curves are one-dimensional, they can extend across two or three dimensions. A point moving along such a curve can only travel in two directions, forward or backward. However, viewed from the outside, the point may undergo various changes in direction depending on the curve's shape.
Curves can be open or closed. Closed curves enclose areas and are therefore treated alongside two-dimensional shapes, the circle being a prime example. The boundary of a two-dimensional shape is a one-dimensional curve. Curves can also intersect themselves, thereby enclosing one or more areas. Finally, a curve may be composed of several individually defined curves.

These geometry calculators begin with one-dimensional objects. In fact, geometry also includes zero-dimensional objects known as points. However, there are no geometric calculations to be performed on points. Consequently, the first and most fundamental shape presented here is the straight line. The circular arc follows as the first curved line. The helix is ​​a one-dimensional curve that winds through three-dimensional space. The Koch curve is a special case. Constructed in multiple steps, it grows with each iteration and ultimately reaches infinite length in the limit. In this final form, it transcends the first Hausdorff dimension, achieving a value of 1.26.

The following 5 one-dimensional shapes can be calculated on these pages:

Last updated on 08/12/2026. Author: Jürgen Kummer

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