• Problem #1, pg. 705:
  
  
  • Start by defining the points P_i, i=0,1,2,3, by typing:
    
    > x0 := 4;
    > y0 := 1;
    
    etc...
  
  
  • Define x(t) and y(t):
    
    > x := t-> x0*(1-t)^3 + 3*x1*t*(1-t)^2 + ...;
    > y := t-> y0*(1-t)^3 + 3*y1*t*(1-t)^2 + ...;
  
  
  • To plot, use:
    
    > with(plots):
    > plot([x(t), y(t), t=0..1], options);
    
    replacing options acording to what you want the graph to look like (e.g., color, axis, etc.).
    
    You may want to try different ranges of t (including negative values) to see what the curve looks like, but for the final answer, you only need t=0..1.
  
  
  • To plot the line segments P_iP_i+1, you can define parametric equations for these by writing:
    
    > lsx0 := t -> (1-t)*x0 + t*x1;
    > lsy0 := t -> (1-t)*y0 + t*y1;
    
    for the first line segment, and similar expressions for the other two. They describe the desired line segments in the interval t=0..1 (this is something that, coincidentally, we'll cover tomorrow in class).
    
    Use the plot command with multiple arguments to generate a graph with the Bézier Curve and the four line segments "bordering" it:
    
    > plot({[x(t), y(t), t=0..1],[lsx0(t), lsy0(t), t=0..1],[lsx1(t), lsy1(t), t=0..1],[lsx2(t), lsy2(t), t=0..1]}, options);
    
    You don't need to specify any options to get the plot, but using thickness=2 may help when printing.
  
  
• Problem #4, pg. 705:
  
  The graph from problem #1 should give you a clue as to how to "draw the letter C": Choose four points in the plane that describe a box similar to the one in problem #1, but open to the right (i.e., choose the point P₀ as the lower right point and then move clockwise to choose the other three). Try the Bézier Curve defined earlier with the new control points.