<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" > <head> <title>Contents</title> <link rel="Stylesheet" type="text/css" href="../styles/pagestyle.css") /> </head> <body> <div id="contents"> <h3> SPLINE REPRESENTATIONS </h3> <hr /> In drafting terminology, a spline is a flexible strip used to produce a smooth curve through a designated set of points several small weighs are distributed along the length of the strip to hold it in position on the drafting table as the curve is drawn. We can mathematically describe such a curve with a piecewise cubic <br /> Figure below <br /> <img src="img/pic007.jpg" alt="" width="245" height="200" /> <br /> <br /> Polynomial function whose first and second derivatives are continuous across the various curve sections. In computer graphics the term spline curve now refers to any composite curve formed with polynomial sections satisfying specified continuity conditions at the boundary of the pieces. A spline surface can be described with two sets of orthogonal spline curves. There are several different kinds of spline specifications that are used in graphics applications. Each individual specification simply refers to a particular type of polynomial with certain specified boundary conditions.<br /> Splines are used in graphics applications to design curve and surface shapes, to digitize drawings for computer storage, and to specify animation paths for the objects or the camera in a scene. Typical CAD applications for splines include the design of automobile bodies, aircraft and spacecraft surfaces, and ship hulls.<br /> <br /> <br /> <strong><span style="text-decoration: underline"> Interpolation and Approximation Splines</span><br /> <br /> </strong> <img class="float_right" src="img/pic008.jpg" alt="" width="100" height="222" /> We specify a spline curve by giving a set of coordinate positions, called control points, which indicates the general shape of the curve. These control points are then fitted with piecewise continuous parametric polynomial functions in one of two ways. When polynomial sections are fitted so that the curve passes through each control point, as in fig. below. The resulting curve is said to interpolate the set of control points. On the other hand, when the polynomials are fitted to the general control  point path without necessarily passing through any control point, the resulting curve is said to approximate the set of control points figure below.<br /> <br /> <img class="float_right" src="img/pic009.jpg" alt="" width="244" height="100" /> Interpolation curves are commonly used to digitize drawings or to specify animation paths. Approximation curves are primarily used as design tools to structure object surfaces. Figure below shows an approximation spline surface created for a design application. Straight lines connect the control-point positions above the surface.<br /> <br /> A spline curve is defined, modified and manipulated with operations of the control points. By interactively selecting spatial positions for the control points, a designer can set up an initial curve. After the polynomial fit is displayed for a given set of control points, the designer can then reposition some or all of the control points to restructure the shape of the curve. In addition, to the control points. CAD packages can also insert extra control points to aid a designer in adjusting the curves shapes.<br /> </div> <script src="http://www.google-analytics.com/urchin.js" type="text/javascript"> </script> <script type="text/javascript"> _uacct = "UA-2741197-1"; urchinTracker(); </script> </body> </html>