## Intersections of cones, cylinders and spheres

Intersections of the following types of surfaces - cones or cylinders of the 2nd order or spheres are algebraic space curves of the 4th order.

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 one double point if they are proper.

There are 3 types of proper space curves of the 4th order and type I:
• one-branch,
• two-branch and
• space curves of the 4th order with one double point.

There are 4 possible decompositions of space curves of the 4th order:
• curve of the 3rd order and a line,
• 2 curves of the 2nd order,
• curve of the 2nd order and 2 lines,
• 4 lines.

EXAMPLES OF PROPER SPACE CURVES OF THE 4th ORDER

EXAMPLES OF DECOMPOSITIONS OF A SPACE CURVE OF THE 4th ORDER

 Loading sample geometry. curve of the 3rd order and a line Loading sample geometry. two curves of the 2nd order Loading sample geometry. curve of the 2nd order and two lines Loading sample geometry. four lines$$^*$$

Order of a space curve is invariant under projection - the projection of a space curve of the order $$n$$ is a plane curve of the same order. The reason why this is true is as follows: a line in the plane of projection is a projection of a projecting plane.
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.

• Rotate the figures above and observe how the projection of the curves changes. Try to find the position when the projection curve has one, two, three double points or a triple point.

$$^*$$ 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.

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