Geometry of the normal ruled surfaces

Abstract

The wide circle of the surfaces formed by the motion of the right line in the normal plain of some base directrix curve is regarded. The generate right line may rotate at some low at the normal plane of the base curve. The vector equation of the surface with any plane or space base curve is received. There are given the formulas of the geometry characteristics of the surfaces, on the base of them there is shown that the coordinate system of the normal ruled surfaces is orthogonal but there is not conjugated in common, that is that the normal ruled surfaces there are not developable surfaces in common way. The condition of the rotation of directrix plane line when the coordinate system of the normal ruled surfaces will be conjugated and the normal ruled surface will be developable is received. The condition that the normal ruled surface with space base curve will be the developable surface there is connected with its curvature of base curve. The developable normal ruled surface with plane base curve is formed by motion of right line at the normal plane of the base curve with the constant angle to the plane of the base curve; the received surface is a surface of constant slope. On the base of the vector equation of the surfaces there are made the figures of the normal ruled surfaces with the help of program complex MathCAD.

Keywords

• geometry of the curves
• geometry of the surfaces
• normal surfaces
• line surfaces
• geometrical characteristics
• surface