Ship Computer Aided Design MR 422

Geometry of Curves Introduction Mathematical Curve Definitions Analytic Properties of Curves Fairness of Curves. Spline Curves. Interpolating Splines Approximating Splines and Smoothness B- spline Curves NURBS Curves Re-parameterization of Parametric Curves Continuity of Curves Projections and Intersections. Relational Curves Points Embedded in Curves

10. Re-parameterization of Parametric Curves A curve is a one-dimensional point set embedded in a 2-D or 3-D space. If it is either explicit or parametric, a curve has a “natural” parameter distribution implied by its construction. However, if the curve is to be used in some further construction, e.g., of a surface, it may be desirable to have its parameter distributed in a different way. Re-parameterization does not change the shape of a curve, but it may have important modeling effects on the curve’s descendants.

11. Continuity of Curves When two curves join or are assembled into a single composite curve, the smoothness of the connection between them can be characterized by different degrees of continuity. Degrees of continuity between the intersecting curves G0 : Two curves that join end-to-end with an arbitrary angle at the junction are said to have G0 continuity, or “geometric continuity of zero order”. G1 : Curves join with zero angle at the junction and have the same tangent direction (1st order geometric continuity or slope continuity or tangent continuity). G2 : curves join with zero angle, and have the same curvature at the junction (2nd order geometric continuity or curvature continuity.)

11. Continuity of Curves Degrees of parametric continuity C0: Two curves that share a common endpoint. They may join with G1 or G2 continuity, but if their parametric velocities are different at the junction, they are only C0. C1: Two curves that are G1 and have the same parametric velocity at the junction. C2: Two curves that are G2 and have the same parametric velocity and acceleration at the junction.

12. Projections and Intersections Curves can arise from various operations on other curves and surfaces. Projection A normal projection of a curve onto a plane which each point of the original curve along a straight line normal to the plane result in a corresponding point on the plane The Projected Curve : the locus of all such projected points The original curve is called the basis curve.

12. Projections and Intersections Intersection (of surfaces with planes or other surfaces) There is no direct formula for finding points on an intersection of a parametric surface; instead, each point located requires the iterative numerical solution of a system of one or more (usually nonlinear) equations. A surface and a plane may not intersect at all, or may intersect in more than one place.

13. Relational Curves In relational geometry, most curves are constructed through defined relationships to point entities or to other curves. Examples Line: is a straight line defined by reference to two control points X1,X2. An arc: is a circular arc defined by reference to three control points X1,X2,X3.

The Arc entity types: A BCurve is a uniform B-spline curve which depends on two or more control points. SubCurve is the portion of any curve between two beads re-parameterized to the range [0,1]. ProjCurve is the projected curve described in preceding section Advantages of relational structure: -The curve can be automatically update if any of its supporting entities changes. -The curves can be durably joined at their endpoints by referencing a given point entity in common.

14. Points Embedded in Curves A curve: consists of 1-D continuous point set embedded in 3-D space. A bead : point embedded in a curve Ways to construct such points: Absolute bead : specified by curve and (t) parameter . Relative bead: specified by parameter offset (Δt) from other bead . Arc length bead: specified by arc length distance from another bead or from one end of a curve. Intersection bead: located at the intersection Of the Curve with a (plane, surface, or another curve).

The uses of beads include: assign a location on the curve to compute a tangent or location of a fitting. Endpoints of a sub curve, a portion of the host curve between two beads. End points