Create a curve that pass through 3 points

geometry

Suppose I have 3 points in space, $A$, $B$ and $C$.

Suppose there is a segment between $A$ and $B$ and in the middle of this segment, there is a perpendicular line where C is, just like that:

enter image description here

What I want is basically to create a function $curve(t)$ that represents the green curve, that is:

  • $curve(0) = A$
  • $curve(0.5) = C$
  • $curve(1) = B$

and the other $t$ values represent points of the curve in the space.

How can I build such function?

If $A$ and $B$ were at the same height, I was using $curve(t) = ( A.x + t * (B.x – A.x),\; A.x + h * sin\left(t * \dfrac{\pi}{2}\right),\; A.z + t * (B.z – A.z))$

But this constraint doesn't exist so I don't know how to do yet.

Best Answer

Let consider for $t\in[0,1]$, then

  • $A+t(B-A)$ is the line from $A$ to $B$

the mid point $M$ between $A$ and $B$ is

  • $M= \frac{A+B}2$

finally we need to add a term to reach $C$, that is for example

$$\operatorname{curve}(t)=A+t(B-A)+4t(t-1)(C-M)$$

with

  • $\operatorname{curve}(0)=A$
  • $\operatorname{curve}(1/2)=C$
  • $\operatorname{curve}(1)=B$
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