Intersections of the following types of surfaces -

There are two types of algebraic space curves of the 4th order (type I and type II). The curves of the type I are intersection curves of two quadric surfaces. The curves of the type II are curves in the decomposition of curves of the 6th order that are intersections of a ruled surface of 3rd order and a ruled surface of 2nd order that have one common line and that line is a double line of the surface of the 3rd order.

We will only work with space curves of type I that are intersections of cones, cylinders and spheres.

Those curves can have at most

There are

There are

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one-branch |
two-branch |
with a double point |

curve of the 3rd order and a line |
two curves of the 2nd order |
curve of the 2nd order and two lines |
four lines\(^*\) |

Every plane in the space intersects a space curve of the \(n\)th order in \(n\) points, therefore, any line in the projection plane intersects the projection of the curve in \(n\) points. In other words, the projection of a space curve is a plane curve of the \(n\)th order.

Since a plane curve can have more double points than a space curve to remain proper (can you remember that number?), the projection of a space curve is a plane curve with possibly more double points.

\(^*\) The point which is the intersection of the four lines is a fourfold point of the degenerated intersection curve of the 4th order. This point counts as 6 double points. There are different types of degeneration of a space curve of order 4 into 4 lines but we will only encounter this type of degeneration. In some examples, this fourfold point will be a point at infinity.

**INTERSECTION OF TWO CONES**

**INTERSECTION OF A CONE AND A CYLINDER **

**INTERSECTION OF TWO CYLINDERS **

**INTERSECTION WITH A SPHERE **

Created by Sonja Gorjanc, translated by Helena Halas and Iva Kodrnja - 3DGeomTeh - Developing project of the University of Zagreb