Conical spiral
In mathematics, a conical spiral, also known as a conical helix, is a space curve on a right circular cone, whose floor plan is a plane spiral. If the floor plan is a logarithmic spiral, it is called "conchospiral" (from conch). Parametric representation. In the formula_1-formula_2-plane a spiral with parametric representation a third coordinate formula_4 can be added such that the space curve lies on the cone with equation formula_5 : Such curves are called conical spirals. They were known to Pappos. Parameter formula_7 is the slope of the cone's lines with respect to the formula_1 -formula_2-plane. A conical spiral can instead be seen as the orthogonal projection of the floor plan spiral onto the cone. Properties. The following investigation deals with conical spirals of the form formula_18 and formula_19, respectively. Slope. The "slope" at a point of a conical spiral is the slope of this point's tangent with respect to the formula_1-formula_2-plane. The corresponding angle is its "slope angle" (see diagram): A spiral with formula_18 gives: For an "archimedean" spiral is formula_25 and hence its slope isformula_26 Because of this property a conchospiral is called an "equiangular" conical spiral. Arclength. The  length of an arc of a conical spiral can be determined by For an "archimedean" spiral the integral can be solved with help of a table of integrals, analogously to the planar case: For a "logarithmic" spiral the integral can be solved easily: In other cases elliptical integrals occur. Development. For the development of a conical spiral the distance formula_33 of a curve point formula_34 to the cone's apex formula_35and the relation between the angle formula_36 and the corresponding angle formula_37 of the development have to be determined: Hence the polar representation of the developed conical spiral is: In  case of formula_18 the polar representation of the developed curve is which describes a spiral of the same type. In case of a "logarithmic" spiral formula_19 the development is a logarithmic spiral: Tangent trace. The collection of intersection points of the tangents of a conical spiral with the formula_1-formula_2-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 formula_1-formula_2-plane has parameter formula_55 and the intersection point is formula_18 gives formula_58 and the tangent trace is a spiral. In  the case formula_43 (hyperbolic spiral) the tangent trace degenerates to a "circle" with radius formula_60 (see diagram). For formula_61 one has formula_62 and the tangent trace is a logarithmic spiral, which is congruent to the floor plan, because of the self-similarity of a logarithmic spiral.