We have learned about cubic spline interpolation. Here you will work with Bézier curves, another  kind of spline which are essential for computer aided-design, manufacturing, and digital animation.

Bézier curves are splines that allow the user to control the slopes at the knots.  In return for the extra freedom, the smoothness of the first and second derivatives across the knot, which were main features of the cubic splines we derived in class, are no longer guaranteed.  Bézier splines are appropriate for cases where corners (discontinuous first derivatives) and abrupt changes in curvature (discontinuous second derivatives) are occasionally needed.

Pierre Bézier developed the idea during his work for the Renault automobile company. The same idea was discovered independently by Paul de Casteljau, working for Citroen, a rival automobile company.  It was considered an industrial secret by both companies, and the fact that both had developed the idea came to light only after Bézier published his research.

Theory:  Equations for Bézier Curves

Each piece of a planar Bézier spline is determined by endpoints of the spline curve (x1 , y1) , (x4 , y4 ) and control points (x2 , y2) , (x3 , y3 ) as shown in the  Figure below.   The  curve  leaves (x1 , y1 ) along the tangent direction (x2  − x1, y2  −  y1 )  and  ends  at (x4 , y4)  along the  tangent direction (x4  − x3 , y4  −  y3 ).  The equations that accomplish this are expressed as a parametric curve (x(t), y(t)) for 0 ≤ t  ≤ 1 .

We could show that these equations lead to x(0) = x1 , x(1) = x4 , x′ (1)  =  3(x4  − x3 ), and that analogous facts hold for y(t).

Your Programming Assignment:  Drawing Bézier Curves

Here you will explore how to use Bézier curves to draw letters and related shapes.  They can be implemented by writing MATLAB code that creates its own PostScript file. You may work with a partner for this project.

PostScript fonts are built directly from Bézier curves.   Here  is a complete PostScript file that illustrates the curve shown in Figure 1.  You can also find this file inside the Project 3 .zip file as beziersample.ps.

%!PS

newpath

100 100 moveto

100 300 300 300 200 200 curveto

stroke

The first line identifies the file as a PostScript file.  The newpath command initiates the curve. PostScript uses a stackand postfixnotation.  The moveto command sets the current plot point to be the (x, y) point specified by the last two numbers on the stack—in this case, the point (100,100).   The  curveto command  accepts  three  (x, y)  points  from  the  current  stack  and constructs the Bézierspline starting at the current plot point, treating the three (x, y) pairs as the two control points and the endpoint, respectively.  The stroke command draws the curve.  If the preceding file is sent to a PostScript printer (for example, double-clicking on the file on my Mac opens the file with Adobe Distiller), the Bézier curve of Figure 1 will be printed.  The coordinates have been multiplied by 100 to match the conventions of PostScript, which are 72 units to the inch. A sheet of letter-sized paper is 612 PostScript units wide and 792 high.

PostScript fonts are drawn on computer screens and printers, using the Bézier curves of the curveto command. The fill command will fill in the outline if the figure is closed.

For example, the upper case T character in the Times Roman font is constructed out of the following 16 Bézier curves (each line consists of the numbers x1 y1 x2 y2 x3 y3 x4 y4 that define one piece of the Bézier spline. This data is also available as a text file , proj2_exercise2_letterT.txt):

To write a PostScript file that writes the letter T, one needs to moveto the initial endpoint (237,620), add the next two control points and endpoint (237, 120), and follow with a curveto command. After that, the current position will be (237, 120), and the second curve is drawn by the command

237  35   226  24   143  19 curveto

To test your code, reproduce the letter T from Figure 1 above by reading proj2_exercise2_letterT.txt. Then run your code on the two files, proj2_exercise2mysteryOne.txt and proj2_exercise2mysteryTwo.txt. What symbol does each of these ‘mystery’ files create? Include the answer in your live script.

2.   Now write a MATLAB function that reads in a text file whose format matches the proj2*.txt files, but now prints the Bézier curves in a MATLAB figure. (That is, your MATLAB figure should look analogous to what you get from the .ps files from Exercise #1.) Use the parametric Bézier curve equations defined above.  The axis square command may be useful here, to give your figure a nice aspect ratio.

3.   Consider the shield from F&M Diplomats logo below.  Use your Bézier curve plotting function to create a MATLAB figure which reproduces the outline of the shield and the ‘shiny reflection ’ as accurately as you can.  Estimate the endpoints and control points of each Bézier curve by hand; be ready to revise them later as needed to improve your result. You can either keep the shield as two separate  pieces, or connect them as in the example  shown  Appendix 1. Gridlines have been added to the logo below for reference.  Submit the endpoints and control points as a single .txt file where each line includes the endpoints and control points of a Bézier curve, consistent with the proj2*.txt files above. Also submit a marked-up printout/screenshot of the Diplomats logo, which shows evidence of your work.