The formula for a conic section, relative to the origin:

     z = c*y² / ( 1 + sqrt( 1 - q*(c*y)² ) )

or:

     z = ( 1 - sqrt( 1 - q*(c*y)² ) ) / (q*c)

where:

     c is the curvature of the surface at the origin
     q is a measure of the eccentricity:
          < 0 hyperbola
          0 parabola
          0 .. 1 ellipse (narrow axis of ellipse)
          1 circle
          > 1 oblate (wide axis of ellipse)

Another form of this equation is:

     y² = 2*R*z - q*z²



where R is the radius of curvature at the origin or 1/c.

The slope, tan U, at the zone is:

     tan n = c*y/sqrt( 1 - q*(c*y)² )

The normal to the surface thus intersects the axis of symmetry at a distance: z + y / tan n from the node. The curvature at the zone is:

     c / sqrt³( 1 + (1-q)*(c*y)² )

Going 3D, the surface of revolution of a conic scoops out a volume of:

     pi*( R*s² - q*s³/3 )

where s is the full sagitta. This volume is scooped out of the cylinder with volume:

     pi*y²*s = pi*( 2*R*s² - q*s³ )

This implies the volume not removed from this cylinder is:

     pi*( R*s² - 2*q*s³/3 )

The volume of aspherization is that amount of volume removed from the center which converts the sphere with the calculate curvature at the zone into the given surface. This volume measure tacitly assumes scooping from the center for a concave prolate ellipsoid, paraboloid, or hyperboloid. It will be negative for a concave oblate ellipsoid.

The center of gravity, relative to the node, of this remaining piece is:

     s/3*((1 - 3*q*s*c/4)/(1 - 2*q*s*c/3))


The center of gravity, relative to the node, of the scooped out chunk is:

     s/3*((1 - 3*q*s*c/4)/(1 - q*s*c/3))

CONIC CALCULATOR

To use this calculator, either the radius of curvature or the curvature and either the Schwartzschild constant or conic constant of the secondary need to be known. These values are entered into the appropriate box and the button under it is clicked, selecting that option. A convex surface can be modelled using a negative value for the radius of curvature or curvature, but care should be taken regarding the sign of the results. Clicking the selection automaticly calculates the non-selected option value.

After the above are provided, the diameter needs to be supplied. After the diameter is entered with the clicking of the button under it, the remaining values, described above, are calculated.