Bezier curves

The original idea of Bezier curves came from Pierre Bezier in 1970.

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Bezier Curves

Bezier curves commonly refers to cubic curves and are useful in vector oriented applications.

The idea behind bezier curves is to write the constant function 1 on the interval [0, 1] in a special way:

$1 = (1 - t + t)^3 = (1 - t)^3 + 3(1 - t)^2 t + 3(1 - t) t^2 + t^2, \;\;\; \textrm{for} \; t\; \textrm{in}\; [0, 1].$

Supose you want to draw a curve in space starting at the point $\mathbf{p}_1$ , with a initial velocity $\mathbf{v}_1$ and a final point $\mathbf{p}_2$ , with final velocity $\mathbf{v}_2$ . A solution to this problem is: take each term in the above formula and multiply it by a vector and get

$\mathbf{c}(t) = (1 - t)^3\,\mathbf{q}_1 + 3(1 - t)^2 t\,\mathbf{q}_2 + 3(1 - t) t^2\,\mathbf{q}_3 + t^3\,\mathbf{q}_4$ ,

Now we have

$\mathbf{c}(0) = \mathbf{q}_1 = \mathbf{p}_1$

and

$\mathbf{c}(1) = \mathbf{q}_4 = \mathbf{p}_2.$

In this way

$\mathbf{c}(t) = (1 - t)^3\,\mathbf{p}_1 + 3\,(1 - t)^2 t\,\mathbf{q}_2 + 3\,(1 - t) t^2\mathbf{q}_3 + t^3\mathbf{p}_2.$

To take into account the initial and final velocities, take the t derivative of the above formula and evaluate it at $t = 0$ and $t = 1$ .

$\mathbf{c}'(t) = -3\,(1 - t)^2\mathbf{p}_1 + 3\,(-2\,(1 - t) t + (1 - t)^2)\,\mathbf{q}_2 + 3\,( 2\,(1 - t) t - t^2)\,\mathbf{q}_3 + 3\,t^2\,\mathbf{p}_2.$