By Fujio Yamaguchi
This ebook includes a variety of sorts of mathematical descriptions of curves and surfaces, comparable to Ferguson, Coons, Spline, Bézier and B-spline curves and surfaces. The fabrics are labeled and organized in a unified approach in order that novices can simply comprehend the total spectrum of parametric curves and surfaces. This e-book might be worthwhile to many researchers, designers, lecturers, and scholars who're engaged on curves and surfaces. The ebook can be utilized as a textbook in desktop aided layout sessions.
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Extra resources for Curves and Surfaces in Computer Aided Geometric Design
IP'I= 1. That is, P' is in the same direction as P, and its magnitude is 1. P' is called the unit tangent vector (refer to Fig. 10(b)). It can be seen from Eq. 8) that s is the magnitude of the tangent vector. , and the unit tangent vector by t. 11) P'=t. 13) Here the parameter u is the distance from the tangent point. At a point where a curve IS regular, the tangent line is umque. However, at a singular point there occur various anomalous cases. Examples of singular POIllts are shown in Figs. 12.
104) To find the extreme value of Kn , take the partial derivatives with respect to and I] and set OKn /O~ =O and OKn/Ol]=O to obtain: (L -KnE) ~+(M -KnF)I] = 0 }. 106) This quadratic equation always has two real solutions Knmax and K nmlO . Knmax and KnmlO are the maximum and minimum values of the normal curvature, called the principal curvatures. From the relation between the roots and the coeffIcients we have: LN-M2 K=:=KnmaxKnmlO= EG-F 2 (e . Puu ) (e . Pww) - (e . 108) K is called the total curvature or the Gauss curvature, while H IS called the mean curvature.
Normally, in the curved surface representation encountered in CAD, it often happens that Puw = Pwu' In such a case, G is a symmetrical matrix. Let a space curve u=u(t) be given on the curved surface P(u, w) and let C be the curve formed by the intersection of the curved surface with the plane which includes the tangent vector j> = itA at a point P on the curve and the unit normal vector e (refer to Fig. 27). The curvature of the curve C is called the normal curvature relative to the direction itA at point P.