Algebraic curves and surfaces
Curves
A real implicit algebraic plane curve is a hypersurface of dimension 1 in R2.
f (
x, y ) = 0
A parametric plane curve is an algebraic variety of dimension 1 in R3.It is a rational mapping from R1 into R2.
x =
f1 ( s ) / f3 ( s )
y =
f2 ( s ) / f3 ( s )
Similarly,
An algebraic space curve can be implicitely defined as the intersection of
two surfaces given in polynomial form:
f 1(
x, y, z ) = 0
f 2(
x, y, z ) = 0
The corresponding parametric form is two sets of parametric equations:
( x = f1,1 ( s1,
t1 ) , y = f2,1 ( s1,
t1 ) , z = f3,1 ( s1,
t1 ) )
( x = f1,2 ( s2,
t2 ) , y = f2,2 ( s2,
t2 ) , z = f3,2 ( s2,
t2 ) ) where all f
are rational functions
Rational algebraic space curves can also be represented as:
x =
f1 ( s ), y =
f2 ( s ), z =
f3 ( s ) where f are rational
functions in s
Surfaces
A real implicit algebraic surface is a hypersurface of dimension 2 in R3.
f (
x, y , z ) = 0
A parametric surface is an algebraic variety of dimension 2 in R5.
It is a rational mapping from R2 into R3.
x =
f1 ( s , t ) / f4 ( s , t )
y =
f2 ( s , t ) / f4 ( s , t
)
z = f3
( s , t ) / f4 ( s , t )
algebraic set
The zero set of a collection of polynomial equations
- f1( x1, .... xd
) = 0
- .....
- .....
- .....
- fm( x1, .... xd
) = 0
with coefficients over the reals or comlpexes.
algebraic variety ( irreducible algebraic set )
An algebraic set that cannot be represented as the union of
two other distinct algebraic sets with neither set containing the other.
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