HTML and CSS Reference
InDepth Information
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
smoothcurvesfordrawing,buttheycanalsobeusedinanimationtodefineapathformotion.
A cubic Bezier curve is created using four distinct pointsâ€”
p0
,
p1
,
p2
, and
p3
:
pp0
The starting point of the curve. We will refer to these
x
and
y
values as
x0
and
y0
.
pp3
The ending point of the curve. We will refer to these
x
and
y
values as
x3
and
y3
.
pp1
and
pp2
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
.
NOTE
The usage of the
p1
and
p2
points is the biggest stumbling block for understanding Bezier curves.
The easiest way to understand the relationship between these points and the curve is to draw them on
a bitmapped canvas, which we will do several times in this chapter.
Afteryouhavethefourpoints,youneedtocalculatesixcoefficientvaluesthatyouwilluseto
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