Display of Algebraic Hypersurfaces

Implicit Algebraic curves.

Parametric Algebraic curves.

Implicit Algebraic surfaces.

Parametric Algebraic surfaces:

This display of parametric equations has to address the following issues:

1. Infinite parametric range  ( the parameter value may have to be infinity to account for some points )

                   For Eg: 

                   The unit sphere:

                   imlpicit form:                f ( x , y , z ) = x² + y² + z² -1 = 0, 

                   parametric form:          x = 2s / ( 1 + s² + t² )

                                                      y =  2t / ( 1 + s² + t² )

                                                      z = 1 - s² - t² / ( 1 + s² + t² )

                   The point ( 0 , 0 , -1 )  can only be reached when both s and t tend to infinity.

2. Complex parameter values ( we may need complex values to get real points )

                   For Eg: 

                   Consider a rational algebraic curve:

                   imlpicit form:                f ( x , y ) = x³ + x² + y² = 0, 

                   parametric form:          x ( s ) = - s² + 1

                                                      y (s ) =  - s ( s² + 1 )

                   The origin can only be reached with s = sqrt ( - 1 ) or i.

3. Poles ( the function f₄ ( s , t ) may be 0, yielding a pole curve ) 

                   For Eg: 

                   Consider a hyperboloid of 2 sheets:

                   imlpicit form:                f ( x , y , z ) = z² + yz + xz - y² -xy -x² -1 = 0

                   parametric form:          x ( s , t ) = 4s / ( 5t² +6st + 5s² -1 )

                                                      y (s , t ) =  4t / ( 5t² +6st + 5s² -1 )

                                                      z ( s , t ) = ( 5t² + 6st -2t + 5s² -2s + 1)  / ( 5t² +6st + 5s² -1 ) 

                   Then problems arise because of the pole curve described by 5t² + 6st -2t + 5s² -2s + 1 = 0 in the parametric domain.

4. Base points ( all the functions may equal 0 for some values of s and t, thus causing curves ( seam curves ) to be missing from the parametric surface ) 

                For Eg: 

                Consider a cubic parametric surface with seam curves,

                parametric form:          x ( s , t ) = t³ - t + s³ - s² + 1 / ( t³ + s³ + 1 )

                                                   y (s , t ) = 2t³ - t² -s²t + 2s³ + 2 / ( t³ + s³ + 1 ) 

                                                   z ( s , t ) = - st - s³ / ( t³ + s³ + 1 )

                Hyperboloid of 1 sheet with seam curve gaps caused by base points.

Other examples of parametric surfaces:

triangulation of a parametric surface with a point singularity

( equation of surface : [ x ( s , t ) = X ( s , t ) / W ( s , t ) , y ( s , t ) = Y ( s , t ) / W ( s , t ) , z ( s , t ) = Z ( s , t ) / W ( s , t ) ]

    X ( s , t ) = s³ + st² - 3s

    Y ( s , t ) =  ( s² + t² )² - 3 ( s² + t² )

    Z ( s , t ) = s²t + t³ -3t

    W ( s , t ) = ( s² + t² )² + 2 ( s² + t² ) + 1

)

triangulation of a parametric surface with a point and line singularity ( Steiner surface )

Rational parametric surface with nine sheets

Rational parametric surface with two sheets

Rational parametric surface with two sheets self-intersecting