Novel methods for curvature analysis and their application to TA worm

• The computation of the principal curvatures and directions is handled from the angle of the eigenvalue problem of matrix.
• The simplified generalized Euler and Bertrand formulae are established.
• The equivalency of analyzing the curvature through the principal frame and an arbitrary unit orthogonal frame is proved.
• The curvature analysis a modified TA worm surface is performed and a number of basic and important formulae are obtained.
• The points on a TA worm surface are the hyperbolic points and the modification has no influence on the type of the points.
• A TA worm surface is an undevelopable ruled surface and the straight cutting edge forms one of its two asymptotic directions.

The computation of the principal curvatures and directions for a surface is handled from the viewpoint of the generalized characteristic value problem of matrix. Consequently many properties of the principal curvature and direction are proved conveniently. Moreover the formula for the principal direction is established specifically. The reduced generalized Euler and Bertrand formulae are proposed. These formulae are particularly suitable for the case that the calculation of the principal directions is extraordinarily complicated. Additionally, the equivalency of the curvature analysis results, procured by right of the principal frame and an arbitrary unit orthogonal frame, is proved and verified laconically. The curvature analysis for the helicoidal surface of a modified TA worm is taken to be an example to explain the theory and method suggested in the current study. For the helicoid, a number of basic and important formulae are obtained. Both the theoretical analysis and the numerical consequences manifest that, the points on a modified TA worm surface are the hyperbolic points and the modification has no influence on the type of the points. The spiral surface of a TA worm is an undevelopable ruled surface and the straight cutting edge forms one of its two asymptotic directions.