Normal plane
From Encyclopedia of Mathematics
to a curve in space at a point
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.
Comments
References
[a1] | M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 142 |
How to Cite This Entry:
Normal plane. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_plane&oldid=32595
Normal plane. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_plane&oldid=32595
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article