By Nickolas S. Sapidis
This cutting-edge examine of the ideas used for designing curves and surfaces for computer-aided layout functions makes a speciality of the primary that reasonable shapes are continuously freed from unessential good points and are basic in layout. The authors outline equity mathematically, display how newly built curve and floor schemes warrantly equity, and help the consumer in making a choice on and elimination form aberrations in a floor version with no destroying the relevant form features of the version. Aesthetic facets of geometric modeling are of important value in business layout and modeling, relatively within the car and aerospace industries. Any engineer operating in computer-aided layout, computer-aided production, or computer-aided engineering may want to upload this quantity to his or her library. Researchers who've a familiarity with simple innovations in computer-aided picture layout and a few wisdom of differential geometry will locate this ebook a beneficial reference.
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Additional info for Designing Fair Curves and Surfaces: Shape Quality in Geometric Modeling and Computer-Aided Design (Geometric Design Publication)
Example text
5. 2 can be recast as follows. e.. spline degree and knot set. (2) Define geometrical constraints as linear equality constraints on the spline coefficients. (3) Define an initial guess curve satisfying the mathematical constraints and as many of the geometrical constraints as can easily be done. (4) Iterate as needed. , regions where curvature must be positive (negative), or monotone increasing (decreasing). , regions in which the curvature should qualitatively resemble a user-provided target profile, expressed as a pair (p, q) of target curvature numerator and denominator splines.
31), The actual computation uses the numerical curvatures K-, in a more indirect way to guarantee that segments are not too long. e.. yi). ,The original data, which arc the only ones to contain any information regarding the faired curve, are known as (x-ni^j-j.. j/i n j(,;)), i = 1 , . . ,n o r i s-. with iiij(-i) the entries of a suitable integer array that arc1 computed at the time of fill-in. Approximate tangent vectors and normal vectors, displacements. Approximate tangent vectors t,;, and orthogonal to these, approximate normal vectors n, can also be estimated from the, raw data.
Lacking such a "complete theory," mathematical smoothing may attempt to Approximation with Aesthetic Constraints 27 simulate manual smoothing, which can be done by fitting a curve through the "cloud" of data by means of French curves. It is here that aesthetics enters into mathematical smoothing. Our experience suggests that for automobile lines a "complete theory" of curve fairness can be provided by the principle of simplest shape and an analysis of curvature properties like the ones we have studied here.