In the --plane a spiral with parametric representation
a third coordinate can be added such that the space curve lies on the cone with equation :
Such curves are called conical spirals.[2] They were known to Pappos.
Parameter is the slope of the cone's lines with respect to the --plane.
A conical spiral can instead be seen as the orthogonal projection of the floor plan spiral onto the cone.
Examplesedit
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 .
Propertiesedit
The following investigation deals with conical spirals of the form and , respectively.
Slopeedit
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 is 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.
Arclengthedit
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:
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:
Tangent traceedit
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.
conical, spiral, mathematics, conical, spiral, also, known, conical, helix, space, curve, right, circular, cone, whose, floor, projection, plane, spiral, floor, projection, logarithmic, spiral, called, conchospiral, from, conch, with, archimedean, spiral, floo. In mathematics a conical spiral also known as a conical helix 1 is a space curve on a right circular cone whose floor projection is a plane spiral If the floor projection is a logarithmic spiral it is called conchospiral from conch Conical spiral with an archimedean spiral as floor projection Floor projection Fermat s spiral Floor projection logarithmic spiral Floor projection hyperbolic spiral Contents 1 Parametric representation 1 1 Examples 2 Properties 2 1 Slope 2 2 Arclength 2 3 Development 2 4 Tangent trace 3 References 4 External linksParametric representation editIn the x displaystyle x nbsp y displaystyle y nbsp plane a spiral with parametric representation x r f cos f y r f sin f displaystyle x r varphi cos varphi qquad y r varphi sin varphi nbsp a third coordinate z f displaystyle z varphi nbsp can be added such that the space curve lies on the cone with equation m 2 x 2 y 2 z z 0 2 m gt 0 displaystyle m 2 x 2 y 2 z z 0 2 m gt 0 nbsp x r f cos f y r f sin f z z 0 m r f displaystyle x r varphi cos varphi qquad y r varphi sin varphi qquad color red z z 0 mr varphi nbsp Such curves are called conical spirals 2 They were known to Pappos Parameter m displaystyle m nbsp is the slope of the cone s lines with respect to the x displaystyle x nbsp y displaystyle y nbsp plane A conical spiral can instead be seen as the orthogonal projection of the floor plan spiral onto the cone Examples edit 1 Starting with an archimedean spiral r f a f displaystyle r varphi a varphi nbsp gives the conical spiral see diagram x a f cos f y a f sin f z z 0 m a f f 0 displaystyle x a varphi cos varphi qquad y a varphi sin varphi qquad z z 0 ma varphi quad varphi geq 0 nbsp 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 r f a f displaystyle r varphi pm a sqrt varphi nbsp as floor plan 3 The third example has a logarithmic spiral r f a e k f displaystyle r varphi ae k varphi nbsp as floor plan Its special feature is its constant slope see below Introducing the abbreviation K e k displaystyle K e k nbsp gives the description r f a K f displaystyle r varphi aK varphi nbsp 4 Example 4 is based on a hyperbolic spiral r f a f displaystyle r varphi a varphi nbsp Such a spiral has an asymptote black line which is the floor plan of a hyperbola purple The conical spiral approaches the hyperbola for f 0 displaystyle varphi to 0 nbsp Properties editThe following investigation deals with conical spirals of the form r a f n displaystyle r a varphi n nbsp and r a e k f displaystyle r ae k varphi nbsp respectively Slope edit nbsp Slope angle at a point of a conical spiral The slope at a point of a conical spiral is the slope of this point s tangent with respect to the x displaystyle x nbsp y displaystyle y nbsp plane The corresponding angle is its slope angle see diagram tan b z x 2 y 2 m r r 2 r 2 displaystyle tan beta frac z sqrt x 2 y 2 frac mr sqrt r 2 r 2 nbsp A spiral with r a f n displaystyle r a varphi n nbsp gives tan b m n n 2 f 2 displaystyle tan beta frac mn sqrt n 2 varphi 2 nbsp For an archimedean spiral is n 1 displaystyle n 1 nbsp and hence its slope is tan b m 1 f 2 displaystyle tan beta tfrac m sqrt 1 varphi 2 nbsp For a logarithmic spiral with r a e k f displaystyle r ae k varphi nbsp the slope is tan b m k 1 k 2 displaystyle tan beta tfrac mk sqrt 1 k 2 nbsp constant displaystyle color red text constant nbsp Because of this property a conchospiral is called an equiangular conical spiral Arclength edit The length of an arc of a conical spiral can be determined by L f 1 f 2 x 2 y 2 z 2 d f f 1 f 2 1 m 2 r 2 r 2 d f displaystyle L int varphi 1 varphi 2 sqrt x 2 y 2 z 2 mathrm d varphi int varphi 1 varphi 2 sqrt 1 m 2 r 2 r 2 mathrm d varphi nbsp For an archimedean spiral the integral can be solved with help of a table of integrals analogously to the planar case L a 2 f 1 m 2 f 2 1 m 2 ln f 1 m 2 f 2 f 1 f 2 displaystyle L frac a 2 left varphi sqrt 1 m 2 varphi 2 1 m 2 ln left varphi sqrt 1 m 2 varphi 2 right right varphi 1 varphi 2 nbsp For a logarithmic spiral the integral can be solved easily L 1 m 2 k 2 1 k r f 2 r f 1 displaystyle L frac sqrt 1 m 2 k 2 1 k r big varphi 2 r varphi 1 big nbsp In other cases elliptical integrals occur Development edit nbsp Development green of a conical spiral red right a side view The plane containing the development is designed by p displaystyle pi nbsp Initially the cone and the plane touch at the purple line For the development of a conical spiral 3 the distance r f displaystyle rho varphi nbsp of a curve point x y z displaystyle x y z nbsp to the cone s apex 0 0 z 0 displaystyle 0 0 z 0 nbsp and the relation between the angle f displaystyle varphi nbsp and the corresponding angle ps displaystyle psi nbsp of the development have to be determined r x 2 y 2 z z 0 2 1 m 2 r displaystyle rho sqrt x 2 y 2 z z 0 2 sqrt 1 m 2 r nbsp f 1 m 2 ps displaystyle varphi sqrt 1 m 2 psi nbsp Hence the polar representation of the developed conical spiral is r ps 1 m 2 r 1 m 2 ps displaystyle rho psi sqrt 1 m 2 r sqrt 1 m 2 psi nbsp In case of r a f n displaystyle r a varphi n nbsp the polar representation of the developed curve is r a 1 m 2 n 1 ps n displaystyle rho a sqrt 1 m 2 n 1 psi n nbsp 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 n 1 displaystyle n 1 nbsp the development is congruent to the floor plan spiral In case of a logarithmic spiral r a e k f displaystyle r ae k varphi nbsp the development is a logarithmic spiral r a 1 m 2 e k 1 m 2 ps displaystyle rho a sqrt 1 m 2 e k sqrt 1 m 2 psi nbsp Tangent trace edit nbsp The trace purple of the tangents of a conical spiral with a hyperbolic spiral as floor plan The black line is the asymptote of the hyperbolic spiral The collection of intersection points of the tangents of a conical spiral with the x displaystyle x nbsp y displaystyle y nbsp plane plane through the cone s apex is called its tangent trace For the conical spiral r cos f r sin f m r displaystyle r cos varphi r sin varphi mr nbsp the tangent vector is r cos f r sin f r sin f r cos f m r T displaystyle r cos varphi r sin varphi r sin varphi r cos varphi mr T nbsp and the tangent x t r cos f t r cos f r sin f displaystyle x t r cos varphi t r cos varphi r sin varphi nbsp y t r sin f t r sin f r cos f displaystyle y t r sin varphi t r sin varphi r cos varphi nbsp z t m r t m r displaystyle z t mr tmr nbsp The intersection point with the x displaystyle x nbsp y displaystyle y nbsp plane has parameter t r r displaystyle t r r nbsp and the intersection point is r 2 r sin f r 2 r cos f 0 displaystyle left frac r 2 r sin varphi frac r 2 r cos varphi 0 right nbsp r a f n displaystyle r a varphi n nbsp gives r 2 r a n f n 1 displaystyle tfrac r 2 r tfrac a n varphi n 1 nbsp and the tangent trace is a spiral In the case n 1 displaystyle n 1 nbsp hyperbolic spiral the tangent trace degenerates to a circle with radius a displaystyle a nbsp see diagram For r a e k f displaystyle r ae k varphi nbsp one has r 2 r r k displaystyle tfrac r 2 r tfrac r k nbsp and the tangent trace is a logarithmic spiral which is congruent to the floor plan because of the self similarity of a logarithmic spiral nbsp Snail shells Neptunea angulata left right Neptunea despectaReferences edit Conical helix MATHCURVE COM Retrieved 2022 03 03 Siegmund Gunther Anton Edler von Braunmuhl Heinrich Wieleitner Geschichte der mathematik G J Goschen 1921 p 92 Theodor Schmid Darstellende Geometrie Band 2 Vereinigung wissenschaftlichen Verleger 1921 p 229 External links editJamnitzer Galerie 3D Spiralen Archived 2021 07 02 at the Wayback Machine Weisstein Eric W Conical Spiral MathWorld Retrieved from https en wikipedia org w index php title Conical spiral amp oldid 1190996322, wikipedia, wiki, book, books, library,