Motivated by a well-known representation of the affine normal vector of a hypersurface in equi-affine differential geometry, the author defines a ‘Laplace normal’ in relative differential geometry. It is obtained (up to a normalization) by applying the Laplace operator with respect to the first fundamental form of relative differential geometry to the position vector. The author shows that the Laplace normal can be interpreted as a relative normal with respect to a suitable gauge surface. The positions of relative and Laplace normals are discussed, especially for the cases of the Euclidean, equi-affine and centro-affine normalizations.