Show that if C is a spherical helix then the projection of the helix on a plane orthogonal to its axis is an arc of an epicycloid.

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This is what I have solved so far.

$fracd(sigmarhoprime)ds+(rho/sigma)=0$(Condition for a space curve lying on sphere) s$rightarrow$arc length of the helix

For helix, $rho$/$sigma$=$eta$(constant)

$sigmarhoprime=-eta s+c$(integration constant)

$rhorhoprime/eta=-eta s+c$

$-dk/k^3=(-eta^2 s+c)ds$

$kappa^-2=eta^2 s^2-pcs-2beta$(integration constant)

$s_1rightarrow$arc length of $C_1$

$fracds_1ds=sinalpha$, $alpharightarrow$angle between axis of the helix and its tangent

$s_1=(sinalpha) s+gamma$(integration constant)$~~Eq.(1)$

$kappa=kappa_1 sin^2alpha, ~kappa_1rightarrow$curvature of the projected plane curve

$kappa_1=fraccosec^2alphasqrteta^2 s^2-pcs-2beta$

Use Eq.(1)

$kappa_1=fraccosec^2alphasqrteta^2 cosec^2alpha(s_1-gamma)^2-pc(cosecalpha)(s_1-gamma)-2beta$

I cannot proceed any further.

asked Aug 20 at 6:26

Asit Srivastava

196

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up vote
0
down vote

favorite

This is what I have solved so far.

$fracd(sigmarhoprime)ds+(rho/sigma)=0$(Condition for a space curve lying on sphere) s$rightarrow$arc length of the helix

For helix, $rho$/$sigma$=$eta$(constant)

$sigmarhoprime=-eta s+c$(integration constant)

$rhorhoprime/eta=-eta s+c$

$-dk/k^3=(-eta^2 s+c)ds$

$kappa^-2=eta^2 s^2-pcs-2beta$(integration constant)

$s_1rightarrow$arc length of $C_1$

$fracds_1ds=sinalpha$, $alpharightarrow$angle between axis of the helix and its tangent

$s_1=(sinalpha) s+gamma$(integration constant)$~~Eq.(1)$

$kappa=kappa_1 sin^2alpha, ~kappa_1rightarrow$curvature of the projected plane curve

$kappa_1=fraccosec^2alphasqrteta^2 s^2-pcs-2beta$

Use Eq.(1)

$kappa_1=fraccosec^2alphasqrteta^2 cosec^2alpha(s_1-gamma)^2-pc(cosecalpha)(s_1-gamma)-2beta$

I cannot proceed any further.

asked Aug 20 at 6:26

Asit Srivastava

196

add a commentÂ |Â

up vote
0
down vote

favorite

This is what I have solved so far.

$fracd(sigmarhoprime)ds+(rho/sigma)=0$(Condition for a space curve lying on sphere) s$rightarrow$arc length of the helix

For helix, $rho$/$sigma$=$eta$(constant)

$sigmarhoprime=-eta s+c$(integration constant)

$rhorhoprime/eta=-eta s+c$

$-dk/k^3=(-eta^2 s+c)ds$

$kappa^-2=eta^2 s^2-pcs-2beta$(integration constant)

$s_1rightarrow$arc length of $C_1$

$fracds_1ds=sinalpha$, $alpharightarrow$angle between axis of the helix and its tangent

$s_1=(sinalpha) s+gamma$(integration constant)$~~Eq.(1)$

$kappa=kappa_1 sin^2alpha, ~kappa_1rightarrow$curvature of the projected plane curve

$kappa_1=fraccosec^2alphasqrteta^2 s^2-pcs-2beta$

Use Eq.(1)

$kappa_1=fraccosec^2alphasqrteta^2 cosec^2alpha(s_1-gamma)^2-pc(cosecalpha)(s_1-gamma)-2beta$

I cannot proceed any further.

asked Aug 20 at 6:26

Asit Srivastava

196

This is what I have solved so far.

$fracd(sigmarhoprime)ds+(rho/sigma)=0$(Condition for a space curve lying on sphere) s$rightarrow$arc length of the helix

For helix, $rho$/$sigma$=$eta$(constant)

$sigmarhoprime=-eta s+c$(integration constant)

$rhorhoprime/eta=-eta s+c$

$-dk/k^3=(-eta^2 s+c)ds$

$kappa^-2=eta^2 s^2-pcs-2beta$(integration constant)

$s_1rightarrow$arc length of $C_1$

$fracds_1ds=sinalpha$, $alpharightarrow$angle between axis of the helix and its tangent

$s_1=(sinalpha) s+gamma$(integration constant)$~~Eq.(1)$

$kappa=kappa_1 sin^2alpha, ~kappa_1rightarrow$curvature of the projected plane curve

$kappa_1=fraccosec^2alphasqrteta^2 s^2-pcs-2beta$

Use Eq.(1)

$kappa_1=fraccosec^2alphasqrteta^2 cosec^2alpha(s_1-gamma)^2-pc(cosecalpha)(s_1-gamma)-2beta$

I cannot proceed any further.

asked Aug 20 at 6:26

Asit Srivastava

196

asked Aug 20 at 6:26

Asit Srivastava

196

asked Aug 20 at 6:26

Asit Srivastava

196

asked Aug 20 at 6:26

Asit Srivastava

196

add a commentÂ |Â

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