Normal plane

to a curve in space at a point $M$

The plane passing through $M$ and perpendicular to the tangent at $M$. The normal plane contains all normals (cf. Normal) to the curve passing through $M$. If the curve is given in rectangular coordinates by the equations

$$x=f(t),\quad y=g(t),\quad z=h(t),$$

then the equation of the normal plane at the point $M(x_0,y_0,z_0)$ corresponding to the value $t_0$ of the parameter $t$ can be written in the form

$$(x-x_0)\frac{df(t_0)}{dt}+(y-y_0)\frac{dg(t_0)}{dt}+(z-z_0)\frac{dh(t_0)}{dt}=0.$$

If the equation of the curve has the form $\mathbf r=\mathbf r(t)$, then the equation of the normal plane is

$$(\mathbf R-\mathbf r)\frac{d\mathbf r}{dt}=0.$$

References

[a1]	M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 142