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ALGEBRAIC 3D CURVE

An algebraic 3-dimensional curve is the intersection of two algebraic surfaces. Its degree is the product of the degrees of the two surfaces; it is also the number (with multiplicity) of (complex and projective) intersection points between the curve and any given plane.

A curve which is the intersection of two algebraic surfaces can be decomposed into the reunion of curves with smaller degrees the sum of which is equal to the degree of the whole curve.

The algebraic 3D curves of degrees 1 and 2 are the lines and the conics.

The 3D *cubics*, of degree 3, are the intersections
between two ruled quadrics that share a common line.

Examples: the skew
parabola, the horopter.

The 3D *quartics*, of degree 4, can be divided into
two groups:

- first kind: intersection between quadrics that
do not share a common line (called biquadratics
: 4 = 2 . 2)

- second kind: intersection between a cubic surface
and a ruled quadric that have two common lines ().

Example: the striction line of a one-sheeted
hyperboloid.

The names are the same as for plane
curves (which are special cases).

Example of biquartic: the Archytas
curve.

Examples of families of algebraic 3D curves with
any degree:

- the 3D Lissajous
curves (including the cylindrical sine waves)

- the clelias.

Example of section between two algebraic surfaces, showing
that the right framework for all this is complex projective geometry:

Intersection between Plücker's
conoid (degree 3):
and the degree 2 cylinder: ;
in homogeneous coordinates .
Therefore, the intersection is composed of the ellipse
(degree 2), the double line
and the two imaginary lines :
one can verify that 6 = 2 + 2 + 1 + 1.

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© Robert FERRÉOL
2018