Write a program to compute a Bezier curve in two segments such that the curve is continuously differentiable between the segments. Use the program to draw the letter "S" as two letter "C"s.
Solution:
I draw the letter "S" as two letter "C"s in the domain [0,1]*[0,2]. For the upper "C", I chose the control points at (0,2) and (0,1); and for the lower "C", I chose the control points at (1,1) and (1,0). The two segments meet at (0.5,1). Since the endpoints for each segments and the chosen control points satisfy that $x_i-x_{i-1/3}=-x_i+x_{i+1/3}$. The curve is shown in Figure 1.
% Write a program to compute a Bezier curve in two segments % such that the curve is continuously differentiable between % the segments. Use the program to draw the letter "S" as two letter "C"s. clear all; close all; M = [-1 3 -3 1; 3 -6 3 0; -3 3 0 0; 1 0 0 0]; t = linspace(0,1,1000); x1 = [1,0,0,0.5; 2,2,1,1]; x2 = [0.5,1,1,0; 1,1,0,0]; T = [t.^3;t.^2;t;ones(size(t))]; B1 = x1*M*T; B2 = x2*M*T; set(gca,'XTick',0:.1:1); plot(B1(1,:),B1(2,:),'ro-');hold on; plot(B2(1,:),B2(2,:),'bo-')
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