We may develop the idea of principal lines at any point on a curve of (n−1)-triple curvature geometrically in the following way:
Two consecutive points on the curve determine the tangent, three consecutive points the osculating points, four consecutive points the osculating 3-space and so on, at any point on the curve. At the same point we have an (n−1)-space perpendicular to the tangent and we shall call this space the first normal space at the point; the intersection of the first normal space with the osculating plane is a line which we shall name as the first normal at the point. Similarly all lines perpendicular to the osculating plane determine an (n−2)-space, the second normal space at the point, and the inter-section of this space with the osculating 3-space is the second normal at the point. Proceeding thus we have lastly the (n−1)th normal which is perpendicular to the osculating (n−1)-space at the point. We thus see that the rth normal lies in the osculating (r+1)-space and is perpendicular to r consecutive tangents. These n−1 normals with the tangent constitute the n principal lines at the point which are mutually orthogonal.