This app is a port of a Bézier curve Java applet I wrote in 1996 (!), and was published in the App Store in October 2009. Version 1.01 is the same app, updated for recent versions of iOS. New features will be added in subsequent releases.
In this version, you can create up to 4 cubic Bézier curve segments (13 control points, because the last point of each segment is the first point of the next).
Touch the screen to add control points, move them by dragging.
Touch the 
to display the convex hull of each set of 4 control points.
You can now move the points around and see that a
Bézier curve always stays within the convex hull of its control points.
Touch the 
to see a line going through the
third and fourth point of each segment. The second point of the next
segment should be at the other end of that line in order for the curve
to be C1 continuous at the connecting point.
Touch the 
to send an e-mail with the current curve as an SVG (Scalable Vector Graphics) file
(which you can import into and edit in programs like
Inkscape and
Illustrator), and as a PNG image.
Finally, touch the 
to delete the last control point, and
the 
for a usage summary.