# Hermite Curves

These are curves defined by four control points and a cubic polynomial defined in terms of a parameter $$t$$. The control points $$q_0$$ and $$q_1$$ define the position of the curve at $$t=0$$ and $$t=1$$ respectively, and $$q_0'$$ and $$q_1'$$ its derivative. Drag the control points with the mouse to see how this affects the shape of the curve.

If we consider just the $$x$$ coordinate of points along the curve, then our cubic polynomial takes the form:

$$x(t)=a_3t^3+a_2t^2+a_1t+a_0$$

Then we can find the values of $$a_3,a_2,a_1,a_0$$ that correspond to a given set of control points based on the constraints that the curve passes through $$q_0$$ with derivative $$q_0'$$ and $$q_1$$ with derivative $$q_1'$$, see the lecture slides. Note that to find the $$x$$ coordinate of the curve we can simply use the $$x$$ components of each $$q$$, here written $$x_0$$, $$x_0'$$, $$x_1$$ and $$x_1'$$, and we can do the same for the $$y$$ coordinate. Then we find that we can rearrange our polynomial to the form

$$x(t)=(2t^3-3t^2+1)x_0+(t^3-2t^2+t)x_0'+(-2t^3+3t^2)x_1+(t^3-t^2)x_1'$$

This gives us a set of basis functions described in terms of $$t$$ that are used to blend the four control points to decide the final coordinate of a point on the curve (these are the curves plotted in the figure above). At $$t=0$$, only $$x_0$$ contributes to the position, whilst at $$t=1$$, only $$x_1$$ contributes.