2-D Coordinate Systems - Maple Help

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2-D Coordinate Systems

Main Concept

The Cartesian coordinate system is the default 2-D coordinate system used by Maple.

Additionally, Maple supports the following 2-D coordinate systems:

 $\mathrm{bipolar}$ $\mathrm{cardioid}$ $\mathrm{cassinian}$ $\mathrm{elliptic}$ $\mathrm{hyperbolic}$ $\mathrm{invcassinian}$ $\mathrm{invelliptic}$ $\mathrm{logarithmic}$ $\mathrm{logcosh}$ $\mathrm{maxwell}$ $\mathrm{parabolic}$ polar $\mathrm{rose}$ $\mathrm{tangent}$

Conversions

The conversions from the various coordinate systems to cartesian (rectangular) coordinates in 2-space

$\left(u,v\right)\to \left(x,y\right)$

 are given by:

bipolar (Spiegel)

 $x=\frac{\mathrm{sinh}\left(v\right)}{\mathrm{cosh}\left(v\right)-\mathrm{cos}\left(u\right)}$
 $y=\frac{\mathrm{sin}\left(u\right)}{\mathrm{cosh}\left(v\right)-\mathrm{cos}\left(u\right)}$

cardioid

 $x=\frac{{u}^{2}-{v}^{2}}{2{\left({u}^{2}+{v}^{2}\right)}^{2}}$
 $y=\frac{uv}{{\left({u}^{2}+{v}^{2}\right)}^{2}}$

cartesian

 $x=u$
 $y=v$

cassinian (Cassinian-oval)

 $x=\frac{a\sqrt{2}\sqrt{\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}+{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}{2}$
 $y=\frac{a\sqrt{2}\sqrt{\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}-{ⅇ}^{u}\mathrm{cos}\left(v\right)-1}}{2}$

elliptic

 $x=\mathrm{cosh}\left(u\right)\mathrm{cos}\left(v\right)$
 $y=\mathrm{sinh}\left(u\right)\mathrm{sin}\left(v\right)$

hyperbolic

 $x=\sqrt{\sqrt{{u}^{2}+{v}^{2}}+u}$
 $y=\sqrt{\sqrt{{u}^{2}+{v}^{2}}-u}$

invcassinian (inverse Cassinian-oval)

 $x=\frac{a\sqrt{2}\sqrt{\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}+{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}{2\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}$
 $y=\frac{a\sqrt{2}\sqrt{\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}-{ⅇ}^{u}\mathrm{cos}\left(v\right)-1}}{2\sqrt{{ⅇ}^{2u}+2{ⅇ}^{u}\mathrm{cos}\left(v\right)+1}}$

invelliptic (inverse elliptic)

 $x=\frac{a\mathrm{cosh}\left(u\right)\mathrm{cos}\left(v\right)}{{\mathrm{cosh}\left(u\right)}^{2}-{\mathrm{sin}\left(v\right)}^{2}}$
 $y=\frac{a\mathrm{sinh}\left(u\right)\mathrm{sin}\left(v\right)}{{\mathrm{cosh}\left(u\right)}^{2}-{\mathrm{sin}\left(v\right)}^{2}}$

logarithmic

 $x=\frac{a\mathrm{ln}\left({u}^{2}+{v}^{2}\right)}{\mathrm{\pi }}$
 $y=\frac{2a\mathrm{arctan}\left(\frac{v}{u}\right)}{\mathrm{\pi }}$

logcosh (ln cosh)

 $x=\frac{a\mathrm{ln}\left({\mathrm{cosh}\left(u\right)}^{2}-{\mathrm{sin}\left(v\right)}^{2}\right)}{\mathrm{\pi }}$
 $y=\frac{2a\mathrm{arctan}\left(\mathrm{tanh}\left(u\right)\mathrm{tan}\left(v\right)\right)}{\mathrm{\pi }}$

maxwell

 $x=\frac{a\left(u+1+{ⅇ}^{u}\mathrm{cos}\left(v\right)\right)}{\mathrm{\pi }}$
 $y=\frac{a\left(v+{ⅇ}^{u}\mathrm{sin}\left(v\right)\right)}{\mathrm{\pi }}$

parabolic

 $x=\frac{{u}^{2}}{2}-\frac{{v}^{2}}{2}$
 $y=uv$

polar

 $x=u\mathrm{cos}\left(v\right)$
 $y=u\mathrm{sin}\left(v\right)$

rose

 $x=\frac{\sqrt{\sqrt{{u}^{2}+{v}^{2}}+u}}{\sqrt{{u}^{2}+{v}^{2}}}$
 $y=\frac{\sqrt{\sqrt{{u}^{2}+{v}^{2}}-u}}{\sqrt{{u}^{2}+{v}^{2}}}$

tangent

 $x=\frac{u}{{u}^{2}+{v}^{2}}$
 $y=\frac{v}{{u}^{2}+{v}^{2}}$

Explore by choosing from the different functions and coordinate systems. Adjust the sliders to change parameters such as the domain and the linear factor of the selected function.

 Function: sin(x)cos(x)csc(x)sec(x)tan(x)x^2 - 4ln(x)exp(x) Coordinate System: bipolarcartesiancardioidcassinianelliptichyperbolicinvcassinianinvellipticlogarithmiclogcoshmaxwellparabolicpolarrosetangent Lower limit of Domain, $\mathrm{x1}$  Upper limit of Domain, $\mathrm{x2}$ Linear Factor, $a$



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