I have a spline generation system using a cubic Bézier algorithm.

I created a tool that splits the spline and adds a new point where the user clicked on the spline, but it causes the spline to deform from its current shape.

How can I calculate Bézier control point positions in 3D space to avoid deformation, as software like 3ds Max and Blender do?


1 Answer 1


What you're looking for is called "de Casteljau's algorithm"

First, you need to find the segment and the parameter value \$t\$ at the point that was clicked. Search "closest point on cubic Bézier spline" for algorithms to find that.

Now you have a segment with control points \$\{ P_1, P_2, P_3, P_4 \}\$ and a parameter value \$0 < t < 1\$ (if \$t\$ is exactly zero or one, then control point \$P_1\$ or \$P_4\$ respectively already split the spline at the clicked point), at which the Bézier segment passes through the clicked point \$P_t\$.

We can now form new points by interpolating between the existing control points by a factor of \$t\$:

$$ P_5 = (1 - t) P_1 + t P_2\\ P_6 = (1 - t) P_2 + t P_3\\ P_7 = (1 - t) P_3 + t P_4\\ $$

and again:

$$ P_8 = (1 - t) P_5 + t P_6\\ P_9 = (1 - t) P_6 + t P_7\\ $$

(If we wanted, we could also find \$P_t = (1 - t) P_8 + t P_9\$ this way, but by this stage you probably already know it)

Your two new Bézier segments are:

$$ \{P_1, P_5, P_8, P_t\}\\ \{P_t, P_9, P_7, P_4\} $$

  • \$\begingroup\$ is t calculated over single segment or entire spline? \$\endgroup\$
    – Cyclone
    Sep 25, 2022 at 5:51
  • 1
    \$\begingroup\$ "a value 0<t<1 ... at which the Bézier segment passes through the clicked point Pt" Note that if t = 0, the split point is P1, and if t= 1, the split point is P4. This only makes sense if t is the parameter for the segment. \$\endgroup\$
    – DMGregory
    Sep 25, 2022 at 10:58

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