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Cubic Bezier Curve Movement
Cubic Bezier curves can be used to define a movement path for an object. Pierre Bezier
first popularized these curves in the 1960s. They are widely used in 2D vector graphics
to define smooth curves for drawing, but they can also be used in animation to define
a path for motion.
A cubic Bezier curve is created using four distinct points— p0 , p1 , p2 , and p3 :
The starting point of the curve. We will refer to these x and y values as x0 and y0 .
The ending point of the curve. We will refer to these x and y values as x3 and y3 .
p1 and p2
The control points for the curve. The curve does not pass through these points;
instead, the equation uses these points to determine the arc of the curve. We will
refer to these x and y values as x0 , x1 , x2 , x3 , y0 , y1 , y2 , and y3 .
The usage of the p1 and p2 points is the biggest stumbling block for
understanding Bezier curves. The easiest way to understand the rela-
tionship between these points and the curve is to draw them on a
bitmapped canvas, which we will do several times in this chapter.
After you have the four points, you need to calculate six coefficient values that you will
use to find the x and y locations as you move an object on the curve. These coefficients
are known as ax , bx , cx , ay , by , and cy . They are calculated as follows:
cx = 3 (x1 - x0)
bx = 3 (x2 - x1) - cx
ax = x3 - x0 - cx - bx
cy = 3 (y1 - y0)
by = 3 (y2 - y1) - cy
ay = y3 - y0 - cy - by
After you've calculated the six coefficients, you can find the x and y locations based on
the changing t value using the following equations. The t value represents movement
over time:
x(t) = axt3 + bxt2 + cxt + x0
y(t) = ayt3 + byt2 + cyt + y0
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