A cubic Bezier curve is defined by four points. Two are
*endpoints*. (*x _{0},y_{0}*) is the

Two equations define the points on the curve. Both are
evaluated for an arbitrary number of values of *t* between 0 and 1.
One equation yields values for *x*, the other yields values for *y*.
As increasing values for *t* are supplied to the equations, the point
defined by *x(t),y(t)* moves from the origin to the destination. This
is how the equations are defined in Adobe's PostScript references.

x(t) = a_{x}t^{3}+ b_{x}t^{2}+ c_{x}t + x_{0}

x_{1}= x_{0}+ c_{x}/ 3

x_{2}= x_{1}+ (c_{x}+ b_{x}) / 3

x_{3}= x_{0}+ c_{x}+ b_{x}+ a_{x}

y(t) = a_{y}t^{3}+ b_{y}t^{2}+ c_{y}t + y_{0}

y_{1}= y_{0}+ c_{y}/ 3

y_{2}= y_{1}+ (c_{y}+ b_{y}) / 3

y_{3}= y_{0}+ c_{y}+ b_{y}+ a_{y}

This method of definition can be reverse-engineered so that it'll give up the coefficient values based on the points described above:

c_{x}= 3 (x_{1}- x_{0})

b_{x}= 3 (x_{2}- x_{1}) - c_{x}

a_{x}= x_{3}- x_{0}- c_{x}- b_{x}

c_{y}= 3 (y_{1}- y_{0})

b_{y}= 3 (y_{2}- y_{1}) - c_{y}

a_{y}= y_{3}- y_{0}- c_{y}- b_{y}

Now, simply by knowing coördinates for any four points, you can create the equations for a simple Bézier curve.

Shockwave Bézier demo | The Bézier curve slider demo

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