HTML and CSS Reference
In-Depth Information
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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
:
p0
The starting point of the curve. We will refer to these
x
and
y
values as
x0
and
y0
.
p3
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
