TY - GEN
T1 - Fast conversion of dynamic B spline curves into a set of power form polynomial curves
AU - Kim, Deok Soo
AU - Ryu, Joonghyun
AU - Lee, Hyunchan
AU - Shin, Hayong
AU - Park, Joonyoung
AU - Jang, Taeboom
N1 - Publisher Copyright:
© 2000 IEEE.
PY - 2000
Y1 - 2000
N2 - Computation of the characteristic points such as inflection points or cusp on a curve is often necessary in CAGD applications. When a curve is represented in a B-spline form, such computations can be made easier once it is transformed in a set of polynomial curves in a power form. Once a curve is represented in a power form, a point evaluation can be also made faster due to Horner's rule even though some issues of stability remains. In addition, the implicitization process of a parametric curve using a resultant usually requires the geometry represented in a power form. Usual practice of the transformation of a B-spline curve into a set of piecewise polynomial curves in a power form is done by either a knot refinement followed by basis conversions, or applying a Taylor expansion on the B-spline curve for each knot span. Presented in this paper is a new algorithm, called direct expansion, for the problem. The algorithm first locates the coefficients of all the linear terms that make up the basis functions in a knot span, and then the algorithm directly obtains the power form representation of basis functions by expanding the summation of products of appropriate linear terms. Then, a polynomial segment of a knot span can be easily obtained by the summation of products of the basis functions within the knot span with corresponding control points. Repeating this operation for each knot span, all of the polynomials of the B-spline curve can be transformed into a power form.
AB - Computation of the characteristic points such as inflection points or cusp on a curve is often necessary in CAGD applications. When a curve is represented in a B-spline form, such computations can be made easier once it is transformed in a set of polynomial curves in a power form. Once a curve is represented in a power form, a point evaluation can be also made faster due to Horner's rule even though some issues of stability remains. In addition, the implicitization process of a parametric curve using a resultant usually requires the geometry represented in a power form. Usual practice of the transformation of a B-spline curve into a set of piecewise polynomial curves in a power form is done by either a knot refinement followed by basis conversions, or applying a Taylor expansion on the B-spline curve for each knot span. Presented in this paper is a new algorithm, called direct expansion, for the problem. The algorithm first locates the coefficients of all the linear terms that make up the basis functions in a knot span, and then the algorithm directly obtains the power form representation of basis functions by expanding the summation of products of appropriate linear terms. Then, a polynomial segment of a knot span can be easily obtained by the summation of products of the basis functions within the knot span with corresponding control points. Repeating this operation for each knot span, all of the polynomials of the B-spline curve can be transformed into a power form.
UR - http://www.scopus.com/inward/record.url?scp=84969567805&partnerID=8YFLogxK
U2 - 10.1109/GMAP.2000.838265
DO - 10.1109/GMAP.2000.838265
M3 - Conference contribution
AN - SCOPUS:84969567805
T3 - Proceedings - Geometric Modeling and Processing 2000: Theory and Applications
SP - 337
EP - 343
BT - Proceedings - Geometric Modeling and Processing 2000
A2 - Martin, Ralph
PB - Institute of Electrical and Electronics Engineers Inc.
T2 - Geometric Modeling and Processing 2000, GMP 2000
Y2 - 11 April 2000 through 12 April 2000
ER -