I'm trying to model a smooth path between several control points in three dimensions, the problem is that there doesn't appear to be an explanation on how to use splines to achieve this. Are splines a subset of other types of curves such as Bezier curve or the Hermite curve?
I have successfully found cubic splines in 2 dimensions, but I'm not sure how to extend it into 3 dimensions and why there is no explanation about this.
Is there a better and more documented type of curve I could use to achieve this? My goal is to move an object along the smooth curve going through the control points. Please help.
$\endgroup$2 Answers
$\begingroup$First, let's understand parametric splines.
Let's assume we already know how to find $y$ as a (spline) function of $x$. And suppose we have a sequence of 2D points $P_i = (x_i,y_i)$. First, we assign a parameter value $t_i$ to each point $P_i$. The usual way to do this is to use chord-lengths -- you choose the $t_i$ values such that $t_i - t_{i-1} = \|P_i - P_{i-1}\|$. Then you compute $x$ as a function of $t$. The calculation is the one you already know, but it's just $x=f(t)$ instead of $y=f(x)$. Now do the same thing with $y$ and $t$. So, now you have both $x$ and $y$ as functions of $t$. In other words, you have a 2D point $(x,y)$ as function of $t$, which means you have a 2D parametric curve.
Now, the extension to 3D is straightforward. We just make $z$ a function of $t$, also. So, now we have $(x,y,z)$ as a function of $t$, so we have a parametric space curve.
To do 3D spline interpolation using Matlab functions, see here.
A better reference is this web site.
Bezier curves are also easy to extend to 3D. As you probably know, the equation of a cubic Bezier curve is$$ \mathbf{C}(t) = (1-t)^3\mathbf{P}_0 + 3t(1-t)^2\mathbf{P}_1 + 3t^2(1-t)\mathbf{P}_2 + t^3\mathbf{P}_3 $$In this equation it doesn't matter whether the control points $\mathbf{P}_0$, $\mathbf{P}_1$, $\mathbf{P}_2$, $\mathbf{P}_3$ are 2D or 3D.
$\endgroup$ 3 $\begingroup$This seems relevant:
There is a MATLAB toolkit with this functionality (spline toolkit).
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