June D Object Representation Shmuel Wimer Bar Ilan Univ., School of Engineering

June Spline Representation Spline is a flexible strip used to produce a smooth curve through a set of points. Splines are used to design curve and surface shapes like automobile bodies, aircrafts, home appliances and more.

June At each of n-1 internal point we require that two successive curve sections pass through that point and also the equality of their 1 st and 2 nd derivatives, thus imposing 4(n-1) equations of 4n polynomial coefficients. Two more equations are obtained from passing through the first and last points. The last two equations can be obtained by requiring the 2nd derivative at first and last points to be zero. A major drawback is that a change in the position of one control point needs recalculation of all coefficients. This is solved by specifying the tangent at each control point. Then, a change in position of one point affects only two sections of the curve. This is called Hermite interpolation.

June 20104

5

6 Other Splines The requirements of curve derivative values control point may be a problem. Instead, the derivatives can naturally be defined based on the control points. In Cardinal splines a curve section is completely defined by the position of four consecutive points.

June t < 0: looser curve t > 0: tighter curve Cardinal matrix and blending functions are derived similarly as in Hermite splines. Setting s=(1-t)/2

June 20108

9 There are few more splines with more parameters than tension, designed to model animation paths with abrupt changes in motion.

June Bezier Spline Curves Developed by Pierre Bezier at Renault corporation for the design of automobile bodies. Bezier splines are very useful for design of curves and surfaces. Bezier curve is a polynomial of degree one less the number of control points. Coefficients and blending functions satisfy recursive relations as follows:

June

June Properties of Bezier Curves

June Cubic Bezier Curves Bezier curves with many points are expensive to compute due to high degree of polynomials. It is common to “patch” curves of four points and construct it from piecewise cubic curves. The following illustrates the four blending functions and how they relate to various curves.

June

June Complex curves can be constructed by patching cubic Bezier splines. Given two control points, in order to obtain C 1 continuity of the patch a new point is introduced and its position is defined by equation the expression of first derivative for u=1 with the next one for u=0. If C 2 continuity is desired, a second point is introduces and the expressions of the second derivatives are also equated.

June Displaying Spline Curves and Surfaces We must determine positions of points on curves and surfaces for displaying those. Parametric polynomial splines must be calculated by steps increments over range of parameter, e.g. [0,1]. This is very time consuming and efficient computations methods are in order. A method called Forward-Difference is very efficient and requires mostly additions.

June

June