1) Starting with an archimedean spiral gives the conical spiral (see diagram)
In this case the conical spiral can be seen as the intersection curve of the cone with a helicoid.
2) The second diagram shows a conical spiral with a Fermat's spiral as floor plan.
3) The third example has a logarithmic spiral as floor plan. Its special feature is its constant slope (see below).
Introducing the abbreviation gives the description: .
4) Example 4 is based on a hyperbolic spiral. Such a spiral has an asymptote (black line), which is the floor plan of a hyperbola (purple). The conical spiral approaches the hyperbola for .
The slope at a point of a conical spiral is the slope of this point's tangent with respect to the --plane. The corresponding angle is its slope angle (see diagram):
A spiral with gives:
For an archimedean spiral, , and hence its slope is
For a logarithmic spiral with the slope is ( ).
Because of this property a conchospiral is called an equiangular conical spiral.
For the development of a conical spiral[3] the distance of a curve point to the cone's apex and the relation between the angle and the corresponding angle of the development have to be determined:
Hence the polar representation of the developed conical spiral is:
In case of the polar representation of the developed curve is
which describes a spiral of the same type.
If the floor plan of a conical spiral is an archimedean spiral than its development is an archimedean spiral.
In case of a hyperbolic spiral () the development is congruent to the floor plan spiral.
In case of a logarithmic spiral the development is a logarithmic spiral:
The collection of intersection points of the tangents of a conical spiral with the --plane (plane through the cone's apex) is called its tangent trace.
For the conical spiral
the tangent vector is
and the tangent:
The intersection point with the --plane has parameter and the intersection point is
gives and the tangent trace is a spiral. In the case (hyperbolic spiral) the tangent trace degenerates to a circle with radius (see diagram). For one has and the tangent trace is a logarithmic spiral, which is congruent to the floor plan, because of the self-similarity of a logarithmic spiral.