The Study For Hilbert Curve On Moving Frame

The Hilbert curve is a discrete fractal curve which can fill the cube without intersection.Through the transformation from one-dimensional space to n-dimensional space,Hilbert curve maintains the connection of spatial data to some extent;moreover,the Hilbert curve has been proved to be a space-filling curve that can best maintain the local adjacency of spatial points.Therefore,Hilbert curve has been used widely in the field of spatial data indexing,and how to characterised it has received more and more attention.Using the method of moving frame,the invariants of Hilbert curve under rigid body motion is obtained,that is discrete curvature and torsion.Based on the definition of discrete curvature and torsion,the nodes of Hilbert curve are recode,furthermore one obtain the function between the order of curve and the sum of inflection points;find the iterative characteristic of curve's curvature and torsion sequence,which show the mode of generation of Hilbert curve which looks like irregular from a new angle.We establish a map between the discrete curvature and torsion of inflection point and the inflection point location number,and based on that,an algorithm of describing edge tangent vector of inflection points of Hilbert curve is developed.For any real number n,the edge tangent vector of the inflection point can be output which location number is n and draw the relative position.Taking constructing moving frame as main research method,discrete curvature and torsion as tool,we have studied Hilbert curve deeply,developed an algorithm for describing three dimensional Hilbert curves moreover.Compared to the algorithm Hilbert3(n)based on MATLAB to generate Hilbert curve,this algorithm is not limited to the order of the curve and does not depend on the iteration between the coordinates of neighboring order curves,instead,it depicts Hilbert curve in units of each inflection point.Finally,experimental comparison shows that the algorithm is more efficient.