Explain why a cubic Bezier curve is completely determined by \({\mathop{\rm x}\nolimits} \left( 0 \right)\), \(x'\left( 0 \right)\), \({\mathop{\rm x}\nolimits} \left( 1 \right)\), and \(x'\left( 1 \right)\).

The cubic Bezier curve is determined by four control points.

Let\({\mathop{\rm x}\nolimits} \left( t \right)\)be the cubic Bezier curve determined by the control points\({{\mathop{\rm p}\nolimits} _0},{{\mathop{\rm p}\nolimits} _1},{{\mathop{\rm p}\nolimits} _2},\)and\({{\mathop{\rm p}\nolimits} _3}\).

The midpoint of the original curve\({\mathop{\rm x}\nolimits} \left( t \right)\)occurs at\({\mathop{\rm x}\nolimits} \left( {.5} \right)\)when\({\mathop{\rm x}\nolimits} \left( t \right)\)has the standard parameterization

\({\mathop{\rm x}\nolimits} \left( t \right) = \left( {1 - 3t + 3{t^2} - {t^3}} \right){{\mathop{\rm p}\nolimits} _0} + \left( {3t - 6{t^2} + 3{t^3}} \right){{\mathop{\rm p}\nolimits} _1} + \left( {3{t^2} - 3{t^3}} \right){{\mathop{\rm p}\nolimits} _2} + {t^3}{{\mathop{\rm p}\nolimits} _3}\)

Differentiation of\({\mathop{\rm x}\nolimits} \left( t \right)\)as shown below:

\(\begin{aligned}{}x'\left( t \right) &= \left( { - 3 + 6t - 3{t^2}} \right){{\mathop{\rm p}\nolimits} _0} + \left( {3 - 12t + 9{t^2}} \right){{\mathop{\rm p}\nolimits} _1} + \left( {6t - 9{t^2}} \right){{\mathop{\rm p}\nolimits} _2} + 3{t^2}{{\mathop{\rm p}\nolimits} _3}\\x'\left( 0 \right) & = - 3{{\mathop{\rm p}\nolimits} _0} + 3{{\mathop{\rm p}\nolimits} _1} & = 3\left( {{{\mathop{\rm p}\nolimits} _1} - {{\mathop{\rm p}\nolimits} _0}} \right)\\x'\left( 1 \right) & = - 3{{\mathop{\rm p}\nolimits} _2} + 3{{\mathop{\rm p}\nolimits} _3} & = 3\left( {{{\mathop{\rm p}\nolimits} _3} - {{\mathop{\rm p}\nolimits} _2}} \right)\end{aligned}\)

This demonstrates that the tangent vector\(x'\left( 0 \right)\)points in the direction from\({{\mathop{\rm p}\nolimits} _0}\)to\({{\mathop{\rm p}\nolimits} _1}\)and is 3 times the length of\({{\mathop{\rm p}\nolimits} _1} - {{\mathop{\rm p}\nolimits} _0}\).

Similarly,\(x'\left( 1 \right)\)points in the direction from\({{\mathop{\rm p}\nolimits} _2}\)to\({{\mathop{\rm p}\nolimits} _3}\)and is three times the length of\({{\mathop{\rm p}\nolimits} _3} - {{\mathop{\rm p}\nolimits} _2}\).

Thus, the cubic Bezier curve is determined by \({\mathop{\rm x}\nolimits} \left( 0 \right),x'\left( 0 \right),{\mathop{\rm x}\nolimits} \left( 1 \right),\)and\(x'\left( 1 \right)\).