Mathematically a conformal map has the property that the Jacobian of the transformation is a scaled version of a rotation matrix. The transformation involved in a spherical Lambert conformal conic map projection doesn't have that property. In what sense is the LCC conformal?
Map projection in pseudocode (
x,y are the map coordinates); I'm using the notation as described on pg. 104 of Map Projections - a Working Manual
lam refer to latitude, longitude respectively):
J = (drho/dphi)*sin(theta) rho*cos(theta)*(dtheta/dlam) -(drho/dhpi)*cos(theta) rho*sin(theta)*(dtheta/dlam)
This would be a scaled rotation matrix if
(drho/dphi)==rho*(dtheta/dlam) however, since
rho=R*F*cot( phi/2+pi/4)**n theta=n*(lam-lam0)
R,F,n, lam0 are constants derived from the parameters defining the projection.) we have
drho/dphi=-n*R*F*cot( phi/2+pi/4)**(n-1)*csc( phi/2+pi/4)*0.5 =-n*rho*csc(phi/2+pi/4)**2*tan(phi/2+pi/4)*0.5 rho*(dtheta/dlam)=n*rho
Thus, the Jacobian is not a scaled version of a rotation matrix, therefore the LCC is not a conformal map.