There are several different definitions of conical coordinates defined by Morse and Feshbach (1953), Byerly (1959), Arfken (1970), and Moon and Spencer (1988). The system defined in Mathematica is

			(1)
			(2)
			(3)

where . Byerly (1959) uses a system which is essentially the same coordinate system as above, but replacing with , with , and with . Moon and Spencer (1988) use instead of .

The above equations give

(4)

(5)

(6)

The scale factors are

			(7)
			(8)
			(9)

The Laplacian is

(10)

The Helmholtz differential equation is separable in conical coordinates.

Arfken, G. "Conical Coordinates (, , )." §2.16 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 118-119, 1970.

Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, p. 263, 1959.

Moon, P. and Spencer, D. E. "Conical Coordinates ." Table 1.09 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 37-40, 1988.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 659, 1953.

Spence, R. D. "Angular Momentum in Sphero-Conal Coordinates." Amer. J. Phys. 27, 329-335, 1959.

CITE THIS AS:

Weisstein, Eric W. "Conical Coordinates." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ConicalCoordinates.html