Parametric continuity conditions , Computer Graphics

Parametric Continuity Conditions   To ensure a smooth transitions from one section of a piecewise parametric curve to the next, we can impose various continuity conditions at the connection points. If each section of a spline is described with a set of parametric coordinate functions of the form we set parametric continuity by matching the parametric derivatives of adjoining curve sections at their common boundary. Zero - order parametric continuity, described as continuity means simply that the curves meet. That is the values of x,y, and z evaluated at for the first curve section are equal respectively to the values of x, y, and z evaluated at for the next curve section, means that the first parametric derivatives of the coordinate function is equation for two successive curve sections are equal at their joining point. Second- order parametric continuity, of c2 continuity means that both the first and second parametric derivatives of the two curve sections are the same at the intersection. Higher - order parametric continuity conditions are defined similarly. Figure shows examples of c0, c1, c2 continuity.  With second - order continuity the rates of change o9f the tangent vectors for connecting sections are equal at their intersection. Thus the tangent line transitions smoothly from one section of the curve to the next. But with first order continuity the rates of change of the tangent vectors for the two sections can be quite different so that the general shapes of the two adjacent sections can change abruptly. First- order continuity is often sufficient for digitizing drawings and some design applications while second order continuity is useful for setting up animation paths for camera motion and for many precision CAD requirements. With equal steps in parameter u would experience an abrupt change in acceleration at the boundary of the two sections, producing a discontinuity in the motion sequence. But if the camera were traveling along the path in Fig (c) the frame sequence for the motion would smoothly transition across the boundary.        

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