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In a rational Bezier curve, the weighted control points are given as $$\mathbf{P}^w=(wx, wy, wz, w)=(X,Y,Z,W).$$ We can extract the control point $\mathbf{P}$ from $\mathbf{P}^w$ by dividing each of the coordinates in $\mathbf{P}^w$ by the last coordinate $w$. That is, $$\mathbf{P} = (X/W,Y/W,Z/W)=(wx/w,wy/w,wz/w)=(x,y,z).$$

But how do we get $\mathbf{P}$ if $w$ is equal to zero? The NURBS Book says that $\mathbf{P}=\text{direction}(X,Y,Z)$ in this case without explaining what $\text{direction}(X,Y,Z)$ means. Can somebody explain it to me, please?

nonremovable
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  • I presume this is referring to the direction $(X,Y,Z)$ is pointing to - in other words, the normalised $(X,Y,Z)$. – Parcly Taxel May 10 '17 at 12:40
  • That was my initial thought as well, but I couldn't decide if it made sense or not. – nonremovable May 10 '17 at 12:41
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    It does make sense. The weighted control points are given in homogenous coordinates, which reduce to regular coordinates when $w=1$. The advantage of using them is that they allow the encoding of points at infinity when $w=0$, and those points naturally relate to lines through the origin. – Parcly Taxel May 10 '17 at 12:43

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The homogeneous coordinates $(x,y,z,0)$ represent a vector in the direction $(x,y,z)$. Or alternatively, they represent a "point at infinity" in the direction $(x,y,z)$.

This should make sense. Take the point with homogeneous coordinates $(wx,wy,wz,w)$, and let $w$ tend towards zero. As the value of $w$ decreases, the point $(x/w,y/w,z/w)$ moves along a line through the origin in the direction of the vector $(x,y,z)$. As $w$ approaches zero, the point goes to infinity.

bubba
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