Normal plane (geometry)
In geometry, the normal plane of a curve at a point P is the plane that is perpendicular to the tangent vector of the curve at P. In other words, it is the plane that contains all lines that are normal to the curve at P.
More formally, let r(t) be a parametric representation of a curve in three-dimensional space, where t is a parameter. The tangent vector to the curve at the point corresponding to t is given by the derivative r'(t). The normal plane at that point is then the plane passing through the point r(t) and having a normal vector given by r'(t).
The equation of the normal plane at a point P(x0, y0, z0) on the curve, where the tangent vector at P is (a, b, c), can be expressed as:
a(x - x0) + b(y - y0) + c(z - z0) = 0
It is important to note that the normal plane is defined only at points where the tangent vector is non-zero. Also, the normal plane should not be confused with the osculating plane or the rectifying plane, which are other planes associated with a curve in space. The normal plane is orthogonal to the tangent vector, while the osculating plane contains both the tangent and principal normal vectors, and the rectifying plane contains the tangent and binormal vectors.