Abstract
This paper presents a new method for constructing a third degree parametric spline curve of C1 continuity. Like the Bèzier curve, the proposed curve is constructed and operated by control points. The peculiarity of the proposed algorithm is the assignment of some unknown values of the spline in the control points abscissas, which are based on the conditions of the first derivative continuity of the curve at these points. The position of the touch points, as well as the control points, can be set interactively. Changing of these points positions leads to a change in the curve shape. This allows the user to flexibly adjust the shape of the curve. Systems of algebraic equations with tridiagonal matrix for calculating the coefficients of a spline curve are constructed. Conditions for the existence and uniqueness of such a curve are presented. Examples of the use of the proposed curve, in particular, for monotone data sets, approximation the ellipse and constructing the letter "S" are given.
Highlights
Bèzier method is a powerful tool for constructing curves and surfaces with desired properties in the computer geometric design systems
The combination of Bèzier's curves with the technique of the spline constructing provided significant opportunities for the development of spline curves, which are actively used in computer-aided design systems and computer graphics packages [1, 2, 3]
The proposed algorithm for constructing of a spline curve and the resulting function have the following useful properties: ─ the curve has a continuity of C1 for any set of control points and their arbitrary location; ─ the obtained curve allows to retain the monotony of the original data set; ─ a strategy for selecting and manipulating control points is intuitively understood; ─ manipulations with parameters that allow you to locally adjust the curve shape can be done interactively; ─ the algorithm for curve constructing is well suited for implementation on a computer, since it requires O(N ) arithmetic operations, where N is the number of control points
Summary
Bèzier method is a powerful tool for constructing curves and surfaces with desired properties in the computer geometric design systems. To obtain additional smoothness of the curves, an algorithm for constructing control points, the position of which ensures the C 2 continuity for compound cubic Bèzier curves is proposed in [19]. The proposed algorithm for constructing of a spline curve and the resulting function have the following useful properties: ─ the curve has a continuity of C1 for any set of control points and their arbitrary location; ─ the obtained curve allows to retain the monotony of the original data set; ─ a strategy for selecting and manipulating control points is intuitively understood; ─ manipulations with parameters that allow you to locally adjust the curve shape can be done interactively; ─ the algorithm for curve constructing is well suited for implementation on a computer, since it requires O(N ) arithmetic operations, where N is the number of control points.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
