Geometric Constraints On Quadratic Bezier Curves Using Minimal Length And Energy

This paper derives expressions for the arc length and the bending energy of quadratic Bezier curves. The formulas are in terms of the control point coordinates. For fixed start and end points of the Bezier curve, the locus of the middle control point is analyzed for curves of fixed arc length or bending energy. In the case of arc length this locus is convex. For bending energy it is not. Given a line or a circle and fixed end points, the locus of the middle control point is determined for those curves that are tangent to a given line or circle. For line tangency, this locus is a parallel line. In the case of the circle, the locus can be classified into one of six major types. In some of these cases, the locus contains circular arcs. These results are then used to implement fast algorithms that construct quadratic Bezier curves tangent to a given line or circle, with given end points, that minimize bending energy or arc length.

Young Joon Ahn, Christoph Hoffmann, and Paul Rosen. Geometric Constraints On Quadratic Bezier Curves Using Minimal Length And Energy. Journal of Computational and Applied Mathematics, 2014.

@article{ahn2014geometric,
  title = {Geometric Constraints on Quadratic B{'e}zier Curves Using Minimal Length and
    Energy},
  author = {Ahn, Young Joon and Hoffmann, Christoph and Rosen, Paul},
  journal = {Journal of Computational and Applied Mathematics},
  volume = {255},
  pages = {887--897},
  year = {2014},
  abstract = {This paper derives expressions for the arc length and the bending energy of
    quadratic Bezier curves. The formulas are in terms of the control point coordinates. For
    fixed start and end points of the Bezier curve, the locus of the middle control point is
    analyzed for curves of fixed arc length or bending energy. In the case of arc length
    this locus is convex. For bending energy it is not. Given a line or a circle and fixed
    end points, the locus of the middle control point is determined for those curves that
    are tangent to a given line or circle. For line tangency, this locus is a parallel line.
    In the case of the circle, the locus can be classified into one of six major types. In
    some of these cases, the locus contains circular arcs. These results are then used to
    implement fast algorithms that construct quadratic Bezier curves tangent to a given line
    or circle, with given end points, that minimize bending energy or arc length.}
}