2-D Coordinate Systems

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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:

$bipolar$	$cardioid$	$cassinian$	$elliptic$
$hyperbolic$	$invcassinian$	$invelliptic$	$logarithmic$
$logcosh$	$maxwell$	$parabolic$	polar
$rose$	$tangent$

Conversions

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

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

are given by:

bipolar (Spiegel)

$x = \frac{\sinh (v)}{\cosh (v) - \cos (u)}$

$y = \frac{\sin (u)}{\cosh (v) - \cos (u)}$

cardioid

$x = \frac{u^{2} - v^{2}}{2 {(u^{2} + v^{2})}^{2}}$

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

cartesian

$x = u$

$y = v$

cassinian (Cassinian-oval)

$x = \frac{a \sqrt{2} \sqrt{\sqrt{{&ExponentialE;}^{2 u} + 2 {&ExponentialE;}^{u} \cos (v) + 1} + {&ExponentialE;}^{u} \cos (v) + 1}}{2}$

$y = \frac{a \sqrt{2} \sqrt{\sqrt{{&ExponentialE;}^{2 u} + 2 {&ExponentialE;}^{u} \cos (v) + 1} - {&ExponentialE;}^{u} \cos (v) - 1}}{2}$

elliptic

$x = \cosh (u) \cos (v)$

$y = \sinh (u) \sin (v)$

hyperbolic

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

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

invcassinian (inverse Cassinian-oval)

$x = \frac{a \sqrt{2} \sqrt{\sqrt{{&ExponentialE;}^{2 u} + 2 {&ExponentialE;}^{u} \cos (v) + 1} + {&ExponentialE;}^{u} \cos (v) + 1}}{2 \sqrt{{&ExponentialE;}^{2 u} + 2 {&ExponentialE;}^{u} \cos (v) + 1}}$

$y = \frac{a \sqrt{2} \sqrt{\sqrt{{&ExponentialE;}^{2 u} + 2 {&ExponentialE;}^{u} \cos (v) + 1} - {&ExponentialE;}^{u} \cos (v) - 1}}{2 \sqrt{{&ExponentialE;}^{2 u} + 2 {&ExponentialE;}^{u} \cos (v) + 1}}$

invelliptic (inverse elliptic)

$x = \frac{a \cosh (u) \cos (v)}{{\cosh (u)}^{2} - {\sin (v)}^{2}}$

$y = \frac{a \sinh (u) \sin (v)}{{\cosh (u)}^{2} - {\sin (v)}^{2}}$

logarithmic

$x = \frac{a \ln (u^{2} + v^{2})}{π}$

$y = \frac{2 a \arctan (\frac{v}{u})}{π}$

logcosh (ln cosh)

$x = \frac{a \ln ({\cosh (u)}^{2} - {\sin (v)}^{2})}{π}$

$y = \frac{2 a \arctan (\tanh (u) \tan (v))}{π}$

maxwell

$x = \frac{a (u + 1 + {&ExponentialE;}^{u} \cos (v))}{π}$

$y = \frac{a (v + {&ExponentialE;}^{u} \sin (v))}{π}$

parabolic

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

$y = u v$

polar

$x = u \cos (v)$

$y = u \sin (v)$

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.