Given that the radius of the cylinder is R and the radius of the pipe is r, the projection of the saddle-shaped weld in the xOy plane is shown as in Figure 2, d being the angle parameter that can uniquely determine the pose of the point on the saddle-shaped

space curve.

Normal curves in the Minkowski space-time are defined in [5] as the

space curves whose position vector (with respect to some chosen origin) always lies in its normal space [T.sup.[perpendicular to]], which represents the orthogonal complement of the tangent vector field of the curve.

For a general parameter t of a

space curve [??], we can calculate the curvature and the torsion as follows:

The new design methodology BladeCAD was motivated by the need to define and modify blade sections within general surfaces of revolution while it provides intuitive interactivity Direct definition and modification of a three-dimensional

space curve, which characterizes both the shape of the blade section and the stream surface, would be extremely difficult.

Automatic Parameterization of Rational Curves and Surfaces IV: Algebraic

Space CurvesIf [alpha] is a null

space curve with a spacelike principal normal [??], then the following Frenet formulas hold

For the first, connect two points on the

space curve by a secant line, and adjoin line segments parallel to the coordinate axis.

[4] Yucesan, A., Coken, A.C., Ayyildiz, N., On the Darboux Rotation Axis of Lorentz

Space Curve, Appl.

In general, two quadric surfaces intersect in a nonsingular quartic

space curve. Under special circumstances, however, this intersection may "degenerate" into a quartic with a double point, or a composite of lines, conics, and twisted cubics whose degrees, counted over the complex projective domain, sum to four.

Miura, "A generalized log aesthetic

space curve," in Proceedings of the Joint International Conference on Human-Centered Computer Environments (HCCE '12), pp.

Integration constants [C.sub.1] and [C.sub.2] from relations (5) and (6) are determined by enforcing condition as for x = [x.sub.i] and x = [x.sub.i+1] the

space curve [s.sub.i] (x) to go through points [P.sub.i] and [P.sub.i+1], it means that, the z coordinate is to take values [z.sub.i] and [z.sub.i+1]:

On singularites, "perestroikas" and differential geometry of

space curve. Ens.