iphone - how can i trace the finger movement on touch for drawing smooth curves?

Question

like what i want is if i move my finger fast on the iphone screen , then i want like something that it make a proper curve using quartz 2d or opengl es whatever.

i want to draw a path in curve style...... i had seen that GLPaint(OpenglES) example ,but it will not help me alot , considering if your finger movement is fast.....

something like making a smooth curve..... any one have some kind of example please tell me

thanks

Edit: Moved from answer below:

thanks to all.......

but i had tried the bezier curve algo with two control points but problem is first how to calculate the control points whether there is no predefined points....

as i mentioned my movement of finger is fast...... so most of the time i got straight line instead of curve, due to getting less number of touch points.......

now as mark said piecewise fashion, ihad tried it like considering first four touch points and render them on screen , then remove the first point then again go for next four points ex. step 1: 1,2,3,4 step 2: 2,3,4,5 like that where as in that approach i got an overlap , which is not the issue actually , but didn't get smooth curve........

but for fast movement of finger i have to find something else?????

Answer 1

Depending on the number of sample points you are looking at, there are two approaches that I would recommend:

Simple Interpolation

You can simply sample the finger location at set intervals and then interpolate the sample points using something like a Catmull-Rom spline. This is easier than it sounds since you can easily convert a Catmull-Rom spline into a series of cubic Bezier curves.

Here's how. Say you have four consecutive sample points P0, P1, P2 and P3, the cubic Bezier curve that connects P1 to P2 is defined by the following control points:

B0 = P1
B1 = P1 + (P2 - P0)/6
B3 = P2 + (P1 - P3)/6
B4 = P2

This should work well as long as your sample points aren't too dense and it's super easy. The only problem might be at the beginning and end of your samples since the first and last sample point aren't interpolated in an open curve. One common work-around is to double-up your first and last sample point so that you have enough points for the curve to pass through each of the original samples.

To get an idea of how Catmull-Rom curves look, you can try out this Java applet demonstrating Catmull-Rom splines.

Fit a curve to your samples

A more advance (and more difficult) approach would be to do a Least Squares approximation to your sample points. If you want try this, the procedure looks something like the following:

1. Collect sample points
2. Define a NURBS curve (including its knot vector)
3. Set up a system of linear equations for the samples & curve
4. Solve the system in the Least Squares sense

Assuming you can pick a reasonable NURBS knot vector, this will give you a NURBS curve that closely approximates your sample points, minimizing the squared distance between the samples and your curve. The NURBS curve can even be decomposed into a series of Bezier curves if needed.

If you decide to explore this approach, then the book "Curves and Surfaces for CAGD" by Gerald Farin, or a similar reference, would be very helpful. In the 5th edition of Farin's book, section 9.2 deals specifically with this problem. Section 7.8 shows how to do this with a Bezier curve, but you'd probably need a high-degree curve to get a good fit.