How do bezier handles work?

Question

On Wikipedia I found information about bezier curves and made a function to generate the inbetween points for a bezier polygon. I noticed that Expression Design uses bezier handles. This allows a circle to be made with 4 points each with a bezier handle.

I'm just not sure mathematically how this works in relation with the formula for bezier point at time T. How do these handle vectors work to modify the shape? Basically what's there relation to the bezier formula?

Thanks

Answer 1

Basically, the 4 points used in the cubic bezier formula are the 2 points the curve is between, plus the two points of the handles on that "side" of the first two points (1 handle from each of the first points). If there are double handles on each point, the handles on the "opposite" side of the points from the curve currently being calculated are ignored (they're used for generating the curve that comes out of the opposite side).

The actual generation method used for cubic bezier curves is outlined on the Wikipedia page you linked in your question.

Answer 2

The 4 points of a Bezier segment are the two endpoints of the segment and two handles, one per endpoint. The handles determine the initial direction of the line as it leaves the endpoint. The distance from the handle to the endpoint determines the amount of "pull" that the handle exerts on the path.

Often you will find multiple Beziers connected end to end, with the endpoint of one being shared as the starting point of the next. This guarantees an unbroken curve. If the handles on either side of a point are directly across from each other, the angle at the joint will match up; if the handles are also the same distance from the point, the angle will be completely smooth and there will not be a visible discontinuity at the point.

An interesting property of Bezier segments is that the curve will fit entirely within the parallelogram defined by the 4 points.

What I have been describing is the most common form of Bezier, the cubic. There is also a quadratic which only has a single handle between the two endpoints; the most common application is TrueType fonts.