Wolfram alpha is telling me there isn't. I guess this reduces down to finding the inflection point in some way and testing against points A and D.

Here's the equation I'm using.
$$ x=(1-t)^3*A_{x}+3*((1-t)^2)*t*B_{x}+3*(1-t)*(t^2)*C_{x}+(t^3)*D_{x} $$ $$ y=(1-t)^3*A_{y}+3*((1-t)^2)*t*B_{y}+3*(1-t)*(t^2)*C_{y}+(t^3)*D_{y} $$

  • $\begingroup$ I presume first terms in your formulas are $(1-t)^3$, right? $\endgroup$ – Intelligenti pauca Mar 11 at 18:12
  • $\begingroup$ What do you mean by "domain" here? The above functions are defined for any value of $t$. $\endgroup$ – Intelligenti pauca Mar 11 at 18:15
  • $\begingroup$ @Aretino DOH. Also Re Domains: Splines are defined for t=[0:1] $\endgroup$ – Audiomatt Mar 12 at 3:27
  • $\begingroup$ If so, why are you asking for the domain? $\endgroup$ – Intelligenti pauca Mar 12 at 7:03
  • 1
    $\begingroup$ The [0,1] portion of the cubic polynomial will exist inside the rectangle defined by the xmin/xmax and ymin/ymax of the control points. In fact, it will lie inside the convex hull of the control points. $\endgroup$ – fang Mar 12 at 18:11

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