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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.
Special cases
The names are the same as for plane curves (which are special cases).
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.
Examples of biquartic (degree 8):
- the Archytas curve
- the Archimedean (or isocylindrical) projection of a
circle—termed the "Archimedean window"—is defined as follows:
Given a sphere of radius a and a right circular
cylinder with axis (D) that circumscribes the sphere, the Archimedean
projection maps a point M0 on the
sphere (not located on the axis) to a point M defined as the intersection
of the line (HM0) with the cylinder,
where H is the orthogonal projection of M0
onto the axis. This projection possesses a property
noted by Archimedes: it is area-preserving.
The curve in question is the Archimedean projection of
a great circle of the sphere situated at a distance b from the axis;
it is, therefore, the intersection of a right
conoid—having axis (D) and a circle as its directrix (a conocuneus)—with
the cylinder..
Consequently, the cylindrical region defined by the Archimedean
window has the same area as the spherical
cap bounded by the circle, namely 
.
After development, the Archimedean window can be described
as the image, under the Lambert
projection, of a great circle that lies parallel to the polar axis.
|
System of Cartesian equations:  ,
where  .
System of cylindrical equations:  .
Cartesian parametrization:  ( ). |
Case b = a/2 ; in black, the conocuneus. |
 |
 equation  . |
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
2026