Hi Shutao,

Many image editing an animation programs have a standard interface for editing cubic B-spline curves that pass through a set of points. Typically the user clicks to place down the interpolation points, and then the program generates two "handle" points associated with each interpolation point. So the middle red part of the curve below is a spline generated from the two red interpolation points and the two red handle points that stick out of them.

From the following animations you can see that adjacent parts of the overall curve will be smooth if an interpolation point and its two handle points are all co-linear.

So a reasonable way to automatically place the handle points to generate a simple curve is shown below. Given interpolation points in red, we can generate the points for the B-spline segment between red points 2 and 3 in blue:

The first point is just red point 2 and the fourth is just red point 3:

{#2, ..., ..., #3}&

The second point is a location that starts at red point 2, and extends a short distance parallel to the line connecting red point 1 and 3.

{#2, #2+(#3-#)/4, ..., #3}&

Now when we are generating the third point for the segment connecting interpolation points 1 and 2, it will also start at point 2 and extend in the opposite direction parallel to the line connecting points 1 and 3. This will make the handles of point 2 co-linear, so the segment connecting points 1 and 2, and 2 and 3 will be smooth.

Dear @Michael Thanks sincerely! . I can understand your method by the following code and your detailed explonation?)

pts = {{0, 0}, {1, 0}, {2, .5}, {1, 1}, {0, 1}, {2, 2}};
Graphics[{Hue[RandomReal[]], #}] & /@
    (BSplineCurve@{#2, #2 + (#3 - #)/4, #3 - (#4 - #2)/4, #3} & @@@ 
    Partition[pts, 4, 1, {2, 2}]) // Show

So this solution is a method that based on the cubic B-spline. Lastly, I would like to know which sofeware/tools did you use to generate the following animation.