Преобразование координат. Coords - системы координат, поддерживаемые MAPLE V

Coords - системы координат, поддерживаемые MAPLE V.

В настоящее время MAPLE поддерживает следующие системы координат:

В трехмерном пространстве - bipolarcylindrical, bispherical, cardioidal, cardioidcylindrical, casscylindrical, confocalellip, confocalparab, conical, cylindrical, ellcylindrical, ellipsoidal, hypercylindrical, invcasscylindrical, invellcylindrical, invoblspheroidal, invprospheroidal, logcoshcylindrical, logcylindrical, maxwellcylindrical, oblatespheroidal, paraboloidal, paraboloidal2, paracylindrical, prolatespheroidal, rectangular, rosecylindrical, sixsphere, spherical, tangentcylindrical, tangentsphere, and toroidal.

В двухмерном пространстве - bipolar, cardioid, cassinian, cartesian, elliptic, hyperbolic, invcassinian, invelliptic, logarithmic, logcosh, maxwell, parabolic, polar, rose, and tangent.

ОБРАТИТЕ ВНИМАНИЕ, что только положительные числа используются для следующих преобразований: casscylindrical, confocalellip, confocalparab, conical, ellipsoidal, hypercylindrical, invcasscylindrical, paraboloidal2, rosecylindrical (в трехмерном пространстве);

cassinian, hyperbolic, invcassinian, and rose (в двухмерном пространстве).

Формулы перехода к декартовым координатам в 3-мерном пространстве:

(u, v, w) - > (x, y, z)

bipolarcylindrical:

x = a*sinh(v)/(cosh(v)-cos(u))

y = a*sin(u)/(cosh(v)-cos(u))

z = w

bispherical:

x = sin(u)*cos(w)/d

y = sin(u)*sin(w)/d

z = sinh(v)/d (где d= cosh(v) - cos(u))

cardioidal:

x = u*v*cos(w)/(u^2+v^2)^2

y = u*v*sin(w)/(u^2+v^2)^2

z = (u^2-v^2)/2/(u^2+v^2)^2

cardioidcylindrical:

x = (u^2-v^2)/2/(u^2+v^2)^2

y = u*v/(u^2+v^2)^2

z = w

casscylindrical:

x = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2)+exp(u)*cos(v)+1)^(1/2)

y = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2)-exp(u)*cos(v)-1)^(1/2)

z = w

confocalellip: (confocal elliptic)

x = ((a^2-u)*(a^2-v)*(a^2-w)/(a^2-b^2)/(a^2-c^2))^(1/2)

y = ((b^2-u)*(b^2-v)*(b^2-w)/(b^2-a^2)/(b^2-c^2))^(1/2)

z = ((c^2-u)*(c^2-v)*(c^2-w)/(c^2-a^2)/(c^2-b^2))^(1/2)

confocalparab: (confocal parabolic)

x = ((a^2-u)*(a^2-v)*(a^2-w)/(b^2-a^2))^(1/2)

y = ((b^2-u)*(b^2-v)*(b^2-w)/(b^2-a^2))^(1/2)

z = (a^2+b^2-u-v-w)/2

conical:

x = u*v*w/(a*b)

y = u/b*((v^2 - b^2)*(b^2-w^2)/(a^2-b^2))^(1/2)

z = u/a*((a^2 - v^2)*(a^2 - w^2)/(a^2-b^2))^(1/2)

cylindrical:

x = u*cos(y)

y = u*sin(y)

z = w

ellcylindrical: (elliptic cylindrical)

x = a*cosh(u)*cos(v)

y = a*sinh(u)*sin(v)

z = w

ellipsoidal:

x = u*v*w/a/b

y = ((u^2-b^2)*(v^2-b^2)*(b^2-w^2)/(a^2-b^2))^(1/2)/b

z = ((u^2-a^2)*(a^2-v^2)*(a^2-w^2)/(a^2-b^2))^(1/2)/a

hypercylindrical: (hyperbolic cylinder)

x = ((u^2+v^2)^(1/2)+u)^(1/2)

y = ((u^2+v^2)^(1/2)-u)^(1/2)

z = w

invcasscylindrical: (inverse Cassinian-oval cylinder)

x = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) +

exp(u)*cos(v)+1)^(1/2)/(exp(2*u)+2*exp(u)*cos(v)+1)^(1/2)

y = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) -

exp(u)*cos(v)-1)^(1/2)/(exp(2*u)+2*exp(u)*cos(v)+1)^(1/2)

z = w

invellcylindrical: (inverse elliptic cylinder)

x = a*cosh(u)*cos(v)/(cosh(u)^2-sin(v)^2)

y = a*sinh(u)*sin(v)/(cosh(u)^2-sin(v)^2)

z = w

invoblspheroidal: (inverse oblate spheroidal)

x = a*cosh(u)*sin(v)*cos(w)/(cosh(u)^2-cos(v)^2)

y = a*cosh(u)*sin(v)*sin(w)/(cosh(u)^2-cos(v)^2)

z = a*sinh(u)*cos(v)/(cosh(u)^2-cos(v)^2)

invprospheroidal: (inverse prolate spheroidal)

x = a*sinh(u)*sin(v)*cos(w)/(cosh(u)^2-sin(v)^2)

y = a*sinh(u)*sin(v)*sin(w)/(cosh(u)^2-sin(v)^2)

z = a*cosh(u)*cos(v)/(cosh(u)^2-sin(v)^2)

logcoshcylindrical: (ln cosh cylinder)

x = a/Pi*ln(cosh(u)^2-sin(v)^2)

y = 2*a/Pi*arctan(tanh(u)*tan(v))

z = w

maxwellcylindrical:

x = a/Pi*(u+1+exp(u)*cos(v))

y = a/Pi*(v+exp(u)*sin(v))

z = w

oblatespheroidal:

x = a*cosh(u)*sin(v)*cos(w)

y = a*cosh(u)*sin(v)*sin(w)

z = a*sinh(u)*cos(v)

paraboloidal:

x = u*v*cos(w)

y = u*v*sin(w)

z = (u^2 - v^2)/2

paraboloidal2:

x = 2*((u-a)*(a-v)*(a-w)/(a-b))^(1/2)

y = 2*((u-b)*(b-v)*(b-w)/(a-b))^(1/2)

z = u+v+w-a-b

paracylindrical:

x = (u^2 - v^2)/2

y = u*v

z = w

prolatespheroidal:

x = a*sinh(u)*sin(v)*cos(w)

y=a*sinh(u)*sin(v)*sin(w)

z=a*cosh(u)*cos(v)

rectangular:

x = u

y = v

z = w

rosecylindrical:

x = ((u^2+v^2)^(1/2)+u)^(1/2)/(u^2+v^2)^(1/2)

y = ((u^2+v^2)^(1/2)-u)^(1/2)/(u^2+v^2)^(1/2)

z = w

sixsphere: (6-sphere)

x = u/(u^2+v^2+w^2)

y = v/(u^2+v^2+w^2)

z = w/(u^2+v^2+w^2)

spherical:

x = u*cos(v)*sin(w)

y = u*sin(v)*sin(w)

z = u*cos(w)

tangentcylindrical:

x = u/(u^2+v^2)

y = v/(u^2+v^2)

z = w

tangentsphere:

x = u*cos(w)/(u^2+v^2)

y = u*sin(w)/(u^2+v^2)

z = v/(u^2+v^2)

toroidal:

x = a*sinh(v)*cos(w)/d

y = a*sinh(v)*sin(w)/d

z = a*sin(u)/d (где d = cosh(v) - cos(u))

Формулы перехода к декартовым координатам в 2-мерном пространстве:

(u, v) --> (x, y)

bipolar:

x = sinh(v)/(cosh(v)-cos(u))

y = sin(u)/(cosh(v)-cos(u))

cardioid:

x = 1/2*(u^2-v^2)/(u^2+v^2)^2

y = u*v/(u^2+v^2)^2

cartesian:

x = u

y = v

cassinian: (Cassinian-oval)

x = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) +

exp(u)*cos(v)+1)^(1/2)

y = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) - exp(u)*cos(v)-1)^(1/2)]

elliptic:

x = cosh(u)*cos(v)

y = sinh(u)*sin(v)

hyperbolic:

x = ((u^2+v^2)^(1/2)+u)^(1/2)

y = ((u^2+v^2)^(1/2)-u)^(1/2)

invcassinian: (inverse Cassinian-oval)

x = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) +

+ exp(u)*cos(v)+1)^(1/2)/(exp(2*u)+2*exp(u)*cos(v)+1)^(1/2)

y = a*2^(1/2)/2*((exp(2*u)+2*exp(u)*cos(v)+1)^(1/2) -

- exp(u)*cos(v)-1)^(1/2)/(exp(2*u)+2*exp(u)*cos(v)+1)^(1/2)

invelliptic: (inverse elliptic)

x = a*cosh(u)*cos(v)/(cosh(u)^2-sin(v)^2)

y = a*sinh(u)*sin(v)/(cosh(u)^2-sin(v)^2)

logarithmic:

x = a/Pi*ln(u^2+v^2)

y = 2*a/Pi*arctan(v/u)

logcosh: (ln cosh)

x = a/Pi*ln(cosh(u)^2-sin(v)^2)

y = 2*a/Pi*arctan(tanh(u)*tan(v))

maxwell:

x = a/Pi*(u+1+exp(u)*cos(v))

y = a/Pi*(v+exp(u)*sin(v))

parabolic:

x = (u^2-v^2)/2

y = u*v

polar:

x = u*cos(v)

y = u*sin(v)

rose:

x = ((u^2+v^2)^(1/2)+u)^(1/2)/(u^2+v^2)^(1/2)

y = ((u^2+v^2)^(1/2)-u)^(1/2)/(u^2+v^2)^(1/2)

tangent:

x = u/(u^2+v^2)

y = v/(u^2+v^2)

Если a, b, с не заданы, то по умолчанию a = 1, b = 1/2, и с = 1/3.

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