hi, hope this helps but this is my interpretation of the contours (on page 18 of the ch14 of the course notes):
*each of the contours are representative of 2 variables, U1 and U2
the numbers on the lines (on the contour) indicate the probability of the outcome, and any point on the "perspective" surface is a cumulative distribution value, e.g. any point on the top-most surface can be associated with a high cdf value.
Counter-monotonicity
as the name suggests, the 2 events move in opposite directions. e.g. a game of tennis, with U1 being player1 winning and U2 being player2 winning. we can't have both players winning - this is probability 0. Hence the flat surface for the "triangle" - because you can't have both events occurring in the same direction. As it is monotonic - you would expect the surface of the cdf for anything that interprets as "success of only one" to be like a "flat" slope. (can't think of a better way to explain this one)
Independence
the 2 variables are independent to one another. The probabilities associated with the materialisation of the 2 events are simply the multiplication of the 2 events. So, in the event that both events are likely to happen (i.e. close to 1), e.g. P(U1) = 0.9 and P(U2) = 0.9, then this lies in the top most surface of the cdf (indicating a high level of probability).
Co-monotonicity
The 2 variables move in the same direction, for example U2, the price of a burger that moves linearly with U1, the cost of its inputs (if input costs increases, the price of the burger will increase). Even if there was a probability of 0.2 that burger prices will go up, we would say that there is a 0.1 chance that its prices increases if there is a 0.1 probability that the input cost increases (taking the minimum), regardless of how much the increase is (does this make sense?).
with Counter and Co-monotonicity, the "surfaces" are usually "flat" slopes because they move in linear fashion (i.e. either move in the same direction or opposite directions).
does this help? or is it still unclear?