. 1.2 In the 3d space Coordinates x, y, z.

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Geodesics on the sphere 1 Coordinates

1.1 On the sphere

Geographical coordinates (Latitude) and (Longitude).

Sometimes we use:

= ₂

. 1.2 In the 3d space Coordinates x, y, z.

N

x

y z

In space

We have:

x=cos

( )

^cos

( )

y=cos

( )

^sin

( )

z=sin

( )

1.3 In the plane

1.3.1 Cartesian coordinates

Coordinates u, v. The first coordinate u in the vertical direction.

u

v

Cartesian coordinates

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1.3.2 Polar coordinates

Coordinates r (Distance) and t (Angle). Zero-direction for t vertical.

r t

Polar coordinates

2 Geodesics

The idea is to rotate the equator about the x-axes about the angle . The plane of this great circle has the equation:

z= ytan

( ) From

x=cos

( )

^cos

( )

y=cos

( )

^sin

( )

z=sin

( ) we get

sin

( )

⁼^cos

( )

^sin

( )

^tan

( )

and

( )

⁼^{arctan sin}

( ( )

^tan

( )

)

Note: This is not the general case of geodesics, since all these geodesics are passing

through the points ( )

, ⁼

( )

^{0, 0}

^and ( )

^, ⁼

( )

^0,

^.

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3 Programs

The following programs are written in MuPAD. This software is very similar to Maple.

To every example you see the corresponding figure.

3.1 Sphere

Sphere x:=(phi,lambda)-> cos(phi)*cos(lambda):

y:=(phi,lambda)-> cos(phi)*sin(lambda):

z:=(phi,lambda)-> sin(phi):

phig:=(lambda, alpha)->arctan(sin(lambda)*tan(alpha)):

sphere:=plot::Surface([x(phi,lambda), y(phi,lambda), z(phi,lambda)],

phi=-PI/2..PI/2, lambda=0..2*PI, Mesh=[13, 25], Submesh=[1,1], FillColor=[1,1,1], FillColor2=[1,1,1], LineWidth=0.5, LineCo- lor=[0,0,0]):

plot(sphere,

Scaling=Constrained, Width=90, Height=90, Axes=None, CameraDirection=[8,6,4]);

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3.2 Sphere with geodesics

Sphere with geodesics x:=(phi,lambda)-> cos(phi)*cos(lambda):

y:=(phi,lambda)-> cos(phi)*sin(lambda):

z:=(phi,lambda)-> sin(phi):

sphere:=plot::Surface([x(phi,lambda), y(phi,lambda), z(phi,lambda)],

phi=-PI/2..PI/2, lambda=0..2*PI, Mesh=[13, 25], Submesh=[1,1], FillColor=[1,1,1], FillColor2=[1,1,1], LineWidth=0.5, LineCo- lor=[0,0,0]):

geod:=alpha->plot::Curve3d([x(phig(lambda, alpha),lambda), y(phig(lambda, alpha),lambda),z(phig(lambda, alpha),lambda)], lambda=-PI..PI, LineColor=[0,0,1], LineWidth=0.8):

plot(sphere, geod(al*PI/180)$al=15..75 step 30, Scaling=Constrained, Width=90, Height=90,

Axes=None, CameraDirection=[8,6,4]);

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3.3 Cylindrical equidistant projection u

( )

, ⁼ v

( )

, ⁼

Cylindrical equidistant projection u:=(phi,lambda)->phi:

v:=(phi,lambda)->lambda:

phiLine:=lambda->plot::Curve2d([v(phi,lambda), u(phi,lambda)], phi=-PI/2..PI/2, LineColor=[0,0,0], LineWidth=0.5):

lambdaLine:=phi->plot::Curve2d([v(phi,lambda), u(phi,lambda)], lambda=-PI..PI, LineColor=[0,0,0], LineWidth=0.5):

geod:=alpha->plot::Curve2d([v(phig(lambda, alpha),lambda), u(phig(lambda, alpha),lambda)],

lambda=-PI..PI, LineColor=[0,0,1], LineWidth=0.8):

plot(phiLine(la*PI/180)$la=-180..180 step 15, lambdaLine(ph*PI/180)$ph=-90..90 step 15, geod(al*PI/180)$al=15..75 step 30,

Scaling=Constrained, Axes=None, Width=120, Height=60);

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3.4 Azimuthal equidistant projection

r

( )

, ⁼ ₂ ⁼ t

( )

, ⁼

Azimuthal equidistant projection theta:=phi->PI/2-phi:

r:=(phi,lambda)->theta(phi):

t:=(phi,lambda)->lambda:

phiLine:=lambda-

>plot::Curve2d([r(phi,lambda)*sin(t(phi,lambda)), -r(phi,lambda)*cos(t(phi,lambda))],

phi=-PI/2..PI/2, LineColor=[0,0,0], LineWidth=0.5):

lambdaLine:=phi-

geod:=alpha->plot::Curve2d([r(phig(lambda, alpha),lambda)*

sin(t(phig(lambda, alpha),lambda)),

-r(phig(lambda, alpha),lambda)*cos(t(phig(lambda, alpha),lambda))],

plot(phiLine(la*PI/180)$la=-180..180 step 15,

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lambdaLine(ph*PI/180)$ph=-90..75 step 15, geod(al*PI/180)$al=15..75 step 30,

3.5 Cylindrical equal-area projection u

( )

, ⁼^sin

( )

v

( )

, ⁼

Cylindrical equal-area projection u:=(phi,lambda)->sin(phi):

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3.6 Azimuthal equal-area projection

r

( )

, ⁼^{2 sin}

( )

₄ ₂ ⁼^{2 sin}

^{( )}

²

t

( )

, ⁼

Azimuthal equal-area projection theta:=phi->PI/2-phi:

r:=(phi,lambda)->2*sin(theta(phi)/2):

phiLine:=lambda-

lambdaLine:=phi-

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3.7 Mercator's cylindrical conformal projection u

( )

, ⁼^{ln tan}

( ( )

₂ ⁺ ₄

)

v

( )

, ⁼

Mercators cylindrical conformal projection u:=(phi,lambda)->ln(tan(float(phi/2+PI/4))):

phiLine:=lambda->plot::Curve2d([v(phi,lambda), u(phi,lambda)], phi=-5*PI/12..5*PI/12, LineColor=[0,0,0], LineWidth=0.5):

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3.8 Stereographic projection

r

( )

, ⁼^{2 tan}

( )

₄ ₂ ⁼^{2 tan}

^{( )}

²

t

( )

, ⁼ 3.8.1 -75° phi 0°

Stereographic projection theta:=phi->PI/2-phi:

r:=(phi,lambda)->2*tan(theta(phi)/2):

phiLine:=lambda->plot::Curve2d([r(phi,lambda)

*sin(t(phi,lambda)),

-r(phi,lambda)*cos(t(phi,lambda))],

phi=-5*PI/12..PI/2, LineColor=[0,0,0], LineWidth=0.5):

lambdaLine:=phi-

sin(t(phig(lambda, alpha),lambda)), -r(phig(lambda, alpha),lambda)

*cos(t(phig(lambda, alpha),lambda))],

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3.8.2 Northern hemisphere

Northern hemisphere theta:=phi->PI/2-phi:

r:=(phi,lambda)->2*tan(theta(phi)/2):

phiLine:=lambda->plot::Curve2d([r(phi,lambda)

*sin(t(phi,lambda)),

-r(phi,lambda)*cos(t(phi,lambda))],

phi=0..PI/2, LineColor=[0,0,0], LineWidth=0.5):

lambdaLine:=phi-

sin(t(phig(lambda, alpha),lambda)), -r(phig(lambda, alpha),lambda)

*cos(t(phig(lambda, alpha),lambda))],

lambda=0..PI, LineColor=[0,0,1], LineWidth=0.8):

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plot(phiLine(la*PI/180)$la=-180..180 step 15, lambdaLine(ph*PI/180)$ph=0..90 step 15, geod(al*PI/180)$al=15..75 step 30,

3.9 Mercator / Sanson

u

( )

, ⁼

v

( )

, ⁼^cos

( )

Mercator / Sanson u:=(phi,lambda)->phi:

v:=(phi,lambda)->lambda*cos(phi):

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3.10 Stab / Werner

r

( )

, ⁼ ₂ ⁼

t

( )

, ⁼ ^cos

^{( )}

2 = ^cos

( )

Stab / Werner theta:=phi->PI/2-phi:

r:=(phi,lambda)->theta(phi):

t:=(phi,lambda)->lambda*cos(phi)/theta(phi):

phiLine:=lambda-

lambdaLine:=phi-

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