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5.5.2.1 Implicitization

Implicitization is possible for RPP surfaces but it is computationally expensive for high degree surfaces and with the necessary use of exact rational arithmetic for robustness. For a rational polynomial surface with maximum degrees in and equal to and , i.e. , the implicit equation is of the form where the highest total degree of the polynomial is [374]. Therefore, for , and for , . The implicitization method is useful for special surfaces such as cylindrical and conical ruled surfaces, surfaces of revolution, etc:
  1. Implicitization of a surface of revolution

    Figure 5.9: Surface of revolution

    Let us consider a planar profile curve to be a rational polynomial of degree

        (5.34)

    as illustrated in Fig. 5.9. By simple implicitization of , we obtain
        (5.35)

    where is a polynomial in and of total degree . Also,
        (5.36)

    Next we eliminate from (5.35) and (5.36) by implicitization. The resultant of the two polynomial equations is
        (5.37)

    and is a polynomial in , and of total degree . For example, a torus results in a degree four algebraic surface.

  2. Implicitization of a cylindrical ruled surface
    Figure 5.10: Cylindrical ruled surface

    Let

        (5.38)

    be a planar base curve or directrix of a ruled surface as shown in Fig. 5.10. The directrix is a rational polynomial curve of degree in the plane. The resulting implicit equation of the curve
        (5.39)

    is a polynomial in and of total degree . Let
        (5.40)

    be a constant direction unit vector which gives the direction of the ruling at each point on the directrix, then the three equations
        (5.41)
        (5.42)
        (5.43)

    describe a cylindrical ruled surface. If we assume , we can eliminate by substituting into the first two equations. Then solving for and the implicit cylindrical ruled surface equation becomes:
        (5.44)

    This equation can be expanded to a standard form using a symbolic manipulation program such as MATHEMATICA [446], MAPLE [51] etc.

  3. Implicitization of a conical ruled surface
    Figure 5.11: Conical ruled surface
    Let a conical ruled surface be defined by its apex
        (5.45)

    and its planar base curve
        (5.46)

    which is a degree planar rational polynomial curve on the plane. Its implicit equation
        (5.47)

    is a total degree polynomial. The equation of the resulting conical ruled surface is
        (5.48)
        (5.49)
        (5.50)

    Eliminating by substituting into (5.48) and (5.49), where we assume , and solving for yields:
        (5.51)

    This equation can be expanded to the standard form using a symbolic manipulation program such as MATHEMATICA [446], MAPLE [51] etc.


Next: 5.5.2.2 Newton's method Up: 5.5.2 Point/rational polynomial parametric Previous: 5.5.2 Point/rational polynomial parametric   Contents   Index
December 2009