Usually the conic gnomon is placed on a horizontal dial, I think this is the most natural approach to understand how it works. On this web-site the matter about the conic gnomon is shown on the peaclock pages but here I wish to underline the possibility to use it on any oriented dial and particular on an equinoctial dial. In reality there are two cones, they are opposite to the vertexes; on the horizontal dial the second cone is under the dial so that it is useless but in the other cases both the cones may work; when the dial orientation gets a style altitude lower then the latitude, there is a portion of both the cones on the dial, otherwise the dial shares the cones, one per side; this is the case of the equinoctial dial.

Cause the conic gnomon has a polar axis and the polar axis is orthogonal to the equinoctial dial, we may first observe: at any latitude and with any Sun declination, the shadow of the cone turns on the dial keeping the same amplitude. This feature depends on the Sun altitude which is constant on the equinoctial dial for all day; in reality this is an approximation due to the fact that we assume the Sun declination is costant during all day long.

When the Sun declination is none, the Sun ray which reaches the vertex of the cone lies in the plane of the equinoctiol dial, grazing against the dial; in this case the shadow of the cone has the minimum amplitude equivalent to a line.
Cause the Sun rays are parallel to themselves and they light the cone, some of these rays lie on two planes tangent to the cone. The intersection of these tangent planes is a line which reaches the vertex of the cone and has the same direction of the Sun ray.
The two tangent planes touch the cone on two lines starting from the vertex up to the base of the cone, here they identify two points whose distance is an arc alpha. On the equinoxes alpha is exactly 180° (see picture number 2).

When the Sun declination is greater then zero, a ray reaches the vertex of the cone with an angle delta with the equinoctial dial. Also in this case some of the Sun rays, parallel to the first one, are tangent to the cone and they lie in the planes which bound the shadow of the cone.
Also in this case the two tangent planes intersect themselves on a line which reaches the vertex of the cone: this is just the same as the first ray we considered. When the ray forms an angle delta with the equinoctial dial, the two tangent planes touch the cone on two lines starting from the vertex up to two points on the base, the points form an arc alpha greater then the equinoctial case.

The angle alpha is the day-light arc and it depends on delta and on the latitude, that is the vertex angle of the cone.

The two tangent planes intersect the equinoctial dial on the shadow edge of the cone which forms an angle corresponding to a surplus of 12 hours of the day-light arc.

Assuming the Sun declination is constant during the day, the Sun turns around the cone axis with the same altitude as on the equinoctial dial and casts a shadow with a costant amplitude. This means the shadow of the cone always indicates the day-light arc, showing an angle which is the surplus of 12 hours. On the Equinoxes the shadow is theoretically a line, it means it has no amplitude, so that the day-light arc is properly 12 hours.

On the lower side of the dial the geometry of the tangent planes is analogous to the upper side case but insteed of alpha we have to consider the opposite angle while the shadow amplitude shows the lack of the day-light arc to 12 hours.

Then we may build a conic gnomon sundial with an equinoctial orientation simply to show the day-light arc. A scale on dial with signs every 15° helps to estimate the shadow amplitude at the first glance. The signs may be the same as to indicate co-italic and babilonic hours, the two edges of the shadow will show these hours.
Seen from the constructive aspect it may be difficult to keep the cone balanced on an inclinated plane, the equinoctial plane; the cone can't have an ideal dot-like vertex; we should need to install a style through the cone but it would change the indication on the Equinoxes: in this case the shadow shouldn't be a line but it should be as thick as the style. We also have to consider that on Equinoxes the threadlike shadow is a theorical case because it is very dim so we can install a style but it must be very thin.
The same style may have a function inside the cone, it may work as a polar gnomon to indicate the modern hours.

I suggest a picture with a sundial on Mart, I choose this habitat for two reasons: on Mart the gravity force is weak, so the style may be more thin than on the Earth, moreover it might be appreciated by next colonizers. On a planet where there are no clouds because there is almost no atmoshere, the lighting is continuous although weaker than on the Earth; this kind of sundial might be useful for those who are outdoor and wish to know the day-light arc or the working time left. This might be an important information cause probably there will be no artificial light outdoor to continue working after sunset. The sundial would work on Mart as well as it works on the Earth although it would show the martian hours, that are about 2,5% longer than the Earth's ones.
Here, on the Earth, we have not yet agreed upon unit of length, in fact we lost a satellite, just straight to Mart, for a misunderstanding about length if measured by feet or meters. After solving this problem the martian colonizers will probably face another problem about the time. The Martians will have an Earth's culture, so they will use the Earth's hour and will divide the day into 24 hours helped by the fact that the martian hour is very similar to the terrestrial hour. A longer stay on this planet, probably some months, might lead them to evaluate the habitat with a 'local' point of view. After all a terrestrial is an alien up there! An hour 2,5% (approx) longer needs even a longer second: what is the sense to use terrestrial hours on Mart? They would change their watches with martian seconds, in this way they would adjust local time but would shake up all knowledge about measurement as we use on the Earth; should we aspect a new conflict about unit of measure? Sundials are just ready, they are in condition to adapt to the local martian time without any changes.