The indicatrix a hyperbola; surfaces with opposite curvatures.
The assymplotes of the indicatrix have a contact of the second order with the surface, and of the first order with the of the surface by its tangent plane.
A ruled surface has contrary curvatures at every point. The second assymplotes of the indicatrices of all the points of the same generatrix form a hyperboloid, if the surface has not directer-plane,—a paraboloid, if it have one.
Curvature of developable surfaces.
There exists upon every surface two systems of orthogonal lines, such that every straight line subject to move by gliding over either of them, and remaining normal to the surface, will engender a developable surface. These lines are called lines of curvature.
The two lines of curvature which cross at a point, are tangents to the principal s of the surface at that point.
Remarks upon the lines of curvature of developable surfaces, and surfaces of revolution.
Determination of the radii of curvature, and assymplotes of the indicatrix at a point of a surface of revolution.
Lessons 29–30. Division of Curves of Apparent Contour, and of Separation of Light and Shadow into Real and Virtual Parts.
When a cone is circumscribed about a surface, at any point whatever of the curve of contact, the tangent to this curve and the generatrix of the cone are parallel to two conjugate diameters of the indicatrix.