Numerical methods for ordinary differential equations pdf
Email Address. Sign In. Access provided by: anon Sign Out. Discrete approximation methods for linear fractional-order systems: A comparative study Abstract: This paper deals with approximation methods for fractional operators. Fractional calculus is a new concept for engineers and we need tools to analyze and utilize the same. The defining points have support over the entire length of the spline, meaning moving one affects the entire length of the curve. That doesn't seem to be the thrust of you question, but it plays a major roll in the history of surface representation.
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By contrast, with nurbs and subdivision surfaces, control points exercise local control, beyond which the surface is unaffected. When designing a surface, it is often very useful to be able to say with certainty that some variation is localized. Imagine wanting to run a series of different ship bow designs through a performance model, all affixed the same aft hull. From Local Support: Wikipedia. In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero.
If the domain of f is a topological space, the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis. From Nurbs: Wikipedia. In many applications the fact that a single control point only influences those intervals where it is active is a highly desirable property, known as local support.
In modeling, it allows the changing of one part of a surface while keeping other parts unchanged. Now for what you did ask about - interpolation vs approximation schemes. These can apply to either case - local or non-local, but the effort required increases as the span of the variation increases ie. There is no reason why you can't define B-splines using only points on the curve. It doesn't require any math beyond high-school algebra. That is a bit disingenuous with respect to the algorithms needed to pull off interpolating schemes.
It requires an optimization step not found in approximation schemes.
These have to be implemented with some care. From Subdivision Surfaces: Wikipedia.
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Subdivision surface refinement schemes can be broadly classified into two categories: interpolating and approximating. Interpolating schemes are required to match the original position of vertices in the original mesh. Approximating schemes are not; they can and will adjust these positions as needed. In general, approximating schemes have greater smoothness, but editing applications that allow users to set exact surface constraints require an optimization step.
After subdivision, the control points of the original mesh and the new generated control points are interpolated on the limit surface. The earliest work was the butterfly scheme by Dyn, Levin and Gregory , who extended the four-point interpolatory subdivision scheme for curves to a subdivision scheme for surface. Kobbelt further generalized the four-point interpolatory subdivision scheme for curves to the tensor product subdivision scheme for surfaces. Deng and Ma further generalized the four-point interpolatory subdivision scheme to arbitrary degree.
For free-form design, the approximation technique is easier to implement. The control points need to be chosen with care by the designer in both cases, but it is easier to understand why you have to exercise care with the approximation techniques. For importing a surface from another format, interpolation techniques may be desirable, at least to form an initial coarse mesh, with an operator putting these in the right place for the job.
Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Why choose approximation over interpolation to implement digital lofting? Ask Question. Asked 3 years, 11 months ago. Active 8 months ago. Viewed times. Glorfindel 1 1 gold badge 1 1 silver badge 9 9 bronze badges. This seems like an issue with your drawing tool, not CADD in general. I know that there are CADD programs that work in both ways.
I'm talking about the history of the development of CAD tools.
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The first digital models to draw complex curves on technical drawings were bezier curves, B-splines and NURBS, all of them controlled by a set of points which not necessarily are included in the set of curve points. B-splines: The math was actually discovered in the 19th century, but efficient computer algorithms date from the s. Nurbs - first computer implementation in , first practical CAD systems in the s.
I'm comparing digital splines vs mechanical splines.