You are confusing two separate parameters in the determination of the orientation of the orbit. There exists three elements that determine the orientation of the orbital plane - The first is inclination (i), the second is the Right Ascension of the Ascending Node [RAAN] [in your diagram it is referred to as longitude of ascending node] (capital Omega), and the third is the argument of the periapsis (lower Omega or w). These parameters are all independent from one another and are used to essentially constrain each position in the orbit to a fixed position in space - alleviating the conundrum you were alluding to by stating that Kerbin spins. Now, the only element that requires the Reference Direction as a parameter in its calculation is the RAAN (longitude of ascending node). That's it. The back story of the Reference Direction is that it is an arbitrarily chosen convention vector in space used as an external positional reference - like the 12 on a standard clock - Why start at 12? Why not zero? Convention. (Well there's more to it than that but we'll leave it there for another time.) The calculation of the inclination of the orbit is just as you had defined in that equation, which clearly you already know. What seems to be confusing you is the determination of the sign of the resultant of that equation - due to the mathematical behavior of the arccos function it will never be a negative value - which begs your question: "Where do the (-) signs come from?" The answer: Convention and relative positioning. Let us say an object is traveling in an orbit around Kerbin. Since the object is in orbit around Kerbin we define our reference plane to be the equator of Kerbin. Any point above this plane is in the positive domain, and any point below this plane is in the negative domain. It is to this plane we will measure our inclination angles. Now let us observe the object again - this time we notice that it is traveling with some inclination. The value of this inclination has a magnitude found from that equation (0o < i < 180o), but ends up having TWO signs - both positive and negative (Take a close look on either side of the orbit, one is positive and one is negative of the same number). The orbit is divided into two halves, the ascending plane, and the descending plane separated by two nodes, the Ascending Node and Descending Node - which can be seen from the diagram in your original post. One half lies above and one half dips below the reference plane (equator) into the southern hemisphere - it is on this lower side, after the Descending Node, that you receive the negative sign. All this sign means is that your orbit has descended into the southern hemisphere or below the reference plane. Now, you might think that adding a second object would bungle all of this math and such. This is not so; the angle of inclination between two space craft is just the difference between each of their angles of inclinations with respect to the equatorial reference plane. iab = ia - ib = arccos(VaZ/mag(Va)) - arccos(VbZ/mag(Vb)) Written this way it is clear to see how this number can easily be negative now, but note - the same number appears at the opposite side of the orbit only as a positive number. Where each appears depends on the relative positioning of the two orbits. In summation - the polarity of the inclination angle, when written, is typically referred only as positive (0o < i < 180o) - because mathematically speaking it can only be positive (unless you want an imaginary number). Convention and usefulness, however, dictate that there be a distinction between whether the object is in the ascending plane or descending plane, i.e. above or below the plane of reference. This distinction becomes readily apparent in what 5thHorseman stated - it is extremely useful for inclination changes, such as the question: Thrust in which direction is going to be needed to match another object's orbital plane. If you see a negative number for an inclination - that means you are looking at the Descending Node (DN in game) and if your ship is near that node it means you are about to pass below the plane of reference. In the case of a targeted second object you are about to pass below the intersection of the two object's orbital planes. It is all about defining what position you are currently at in the orbit, using relative positioning. I hope this answers most of your question because this subject is notorious for causing confusion; It all ends up being relatively simple trig and vector algebra - mixed with some strange conventional usage. - Nef
