Bézier curves are very common in computer graphics and also touch on mathematical aspects. In the following article we offer a short introduction to working with Bézier curves and describe them first visually and then mathematically.
There are different forms of Bézier curves; linear, square, cubic etc. Linear Bézier curves are just straight lines. The most common type of Bézier curve in drawing programs is the cubic one and the one and the one we will describe below.
A cubic is defined by four points: two points determine where the curve begins and ends and two other points determine the shape. Assume that the points are identified by P0, P1, P2 and P3. The curve starts at P0 and initially moves towards P1. It ends at P3 in the direction of a line connecting P2 and P3. If you move P1 further away from P0, the curve becomes flatter and continues in direction P1 before turning. Similar remarks apply when moving P2 away from P3.
Let us now move on to the equations. The Bézier cubic curve is given by
The equation shows that B(0) = P(0) and B(1) = P3. A small calculation shows that the derivatives meet the requirements of
and
Further extension of points P1 and P2 increases the derivatives and allows the curve to continue towards these points before bending.
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