Parallelkurven

Parallelkurven, also known as offset curves, are curves that maintain a constant distance from a given original curve. For a planar curve, the parallel curve at a distance 'd' consists of all points that are exactly 'd' units away from the closest point on the original curve.

The concept of parallel curves is fundamental in various fields, including computer-aided design (CAD), computer graphics, and differential geometry. In CAD, parallel curves are essential for tasks such as creating toolpaths for manufacturing, generating offset surfaces, and designing parallel piping or wiring.

For a simple closed curve like a circle, the parallel curve at any distance 'd' is another concentric circle. However, for more complex curves, such as a parabola or an ellipse, the parallel curves can exhibit more intricate shapes. A notable characteristic of parallel curves is that they can self-intersect or become degenerate under certain conditions, particularly when the distance 'd' is large enough or when the original curve has points of high curvature.

The mathematical definition of a parallel curve involves the concept of the normal vector to the original curve. For a point P on the original curve, the corresponding point on the parallel curve is found by moving along the normal vector at P by a distance 'd'. The normal vector indicates the direction perpendicular to the curve at that point.

The study of parallel curves also leads to interesting geometric properties, such as relationships between their arc lengths and curvatures. For instance, the rate of change of the distance between a point on the original curve and its corresponding point on the parallel curve is related to the curvature of the original curve.