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Curves and Surfaces in Computer Aided Geometric Design by Fujio Yamaguchi

By Fujio Yamaguchi

This booklet comprises numerous kinds of mathematical descriptions of curves and surfaces, resembling Ferguson, Coons, Spline, Bézier and B-spline curves and surfaces. The fabrics are labeled and organized in a unified manner in order that newbies can simply comprehend the total spectrum of parametric curves and surfaces. This publication might be valuable to many researchers, designers, academics, and scholars who're engaged on curves and surfaces. The e-book can be utilized as a textbook in desktop aided layout periods.

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51) Next, let us find the condition for the centers of curvature of the two curve segments at the connectIOn point to vary continuously. 17) for the curvature is: P"=Kn. In this equation K is the magnitude of the curvature and n is the unit principal normal vector, pointing toward the center of curvature. From the above equation, for the center of curvature of the curve to vary smoothly at the connection point, it is necessary for both K and n to be continuous. If the binormal b = t x nand t are continuous then n is also continuous.

80) IS equivalent to saying that the vectors Pu and Pw are hnearly independent at each point on the curved surface. 80) IS also equivalent to Pu x Pw =f 0*). A curve III an arbitrary direction on a curved surface can be expressed in terms of a parameter t as P{u(t), w(t)) (a ~ t ~ b). 81 ) 46 1 BasIc Theory of Curves and Surfaces Note that uo=u(t o) and wo=w(t o). The above formula expresses the fact that a tangent vector in an arbitrary direction on the curved surface lies on the plane formed by Pu and Pw ' This plane is called the tangent plane at the point P(u o, wo).

I1m - - = - =K. d s {! Here {! is the radIUs of curvature and in terms of them as: K is the curvature. pI! can be expressed 1 P = - n=Kn. II (! 17) Here n is the unit vector pointing toward the center of curvature. (s) is a vector that has a magnitude equal to the curvature at point s and points toward the center of curvature, it is sometimes called the curvature vector. Curvature is the rate of turning of the unit tangent vector t with respect to the length of the curve s, in other words, a quantity that indicates how rapidly or slowly the curve is turning.

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