EXPANSION OF BEZIER CURVE EXPRESSION

Some interesting features are observed when the equation for a Bezier curve is expanded algebraically.

Note that each term is a cubic, although lower order terms disappear as the control point subscript increases. Also, as seen before,

The terms can be collected to write the expression as a cubic polynomial in u.

Again, note the obvious effect of u = 0 and the evaluation for u = 1.

What happens when all four control points of a cubic Bezier curve are the same? (Does this make any sense?)

What if all the control points are in a straight line? The result should be a linear expression in u.

A simple example ( in 2D ) is to set the x coordinate of each the same and allow the y values to vary. In this case the x value for each point on the curve would be a constant ( as in the case where all four points are the same.)

What would happen to the y value? Note that this question generalizes to answer the above more general question.

The analysis in the general case is complicated. However, if a simple case is examined, the results are reasonable.

Although the above equations are linear in u, this is NOT the case in general – the expression will be cubic.