By Fujio Yamaguchi
This booklet comprises a number of different types of mathematical descriptions of curves and surfaces, equivalent to Ferguson, Coons, Spline, Bézier and B-spline curves and surfaces. The fabrics are categorized and organized in a unified method in order that novices can simply comprehend the full spectrum of parametric curves and surfaces. This publication may be invaluable to many researchers, designers, academics, and scholars who're engaged on curves and surfaces. The publication can be utilized as a textbook in laptop aided layout periods.
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Additional info for Curves and Surfaces in Computer Aided Geometric Design
43) is completely determined by setting the parameter equal to 0, the position vector to Qo and the tangent vector to Qo at one end, and the parameter to 1, the posItion vector to Ql and the 40 1. BasIc Theory of Curves and Surfaces tangent vector to This gives: QI at the other end (for further details refer to Sect. 1). 23. pe~ 0f p"r"ml'tric cubic curves. (a) case with no abnormalities, (b) cusp; (c) loop, (d) one inflection pomt; (e) two inflectIOn pomts 41 1 2 Curve Theory then, if the magnitudes of the tangent vectors at these end points are different, the shapes will be different.
7) s ds _. '. 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.
BasIc Theory of Curves and Surfaces P"S=~(~)~= () dt tJPl ds ptJPl . (2P p) 1 p-p~. 19) where : B= P- P (P tJPl tJPl· p") =IBln. 18) gives the relation between the curvature vector and the derivative vectors P(t) and p(t) with respect to the parameter t. Taking the absolute value of Eq. 21) By using Eqs. 21), the center of curvature can be determined graphically as shown in Fig. 17. If we use the relation: p B o (a ) Fig. 17. How to find the center of curvature graphIcally o (b) 27 I 2 Curve Theory P ..
Curves and Surfaces in Computer Aided Geometric Design by Fujio Yamaguchi