In this photo would you say the near points are normal or reverse?
http://acm.jhu.edu/~sthurmovik/Railpics ... Ladder.jpg
To me looks normal for the train you are riding and reverse for the track at right at the same time?
Normal and reverse are terms only defined for the switch's frame of reference. You can't just look at a switch and know which direction is "Normal" (although it is usually intuitive which is which). In the case of the K ladder at ZOO you first look at the diagram
. You see that the first double slip is controlled by levers 69 and 71 and the second double slip by levers 71 and 73. To route a train straight down the K Ladder you reverse 69, 71 and 73. Trains going straight down the 36th St connector would see those 3 levers set normal. Any move between the lines would see one lever normal and the other reverse.
I was thinking of conventional CTC crossover, say a crossover between two mains, where both points at each end of crossover throw in unison.
A double slip represents two adjacent crossovers, not the same one. On the K ladder the 71 switch is part of two different double slips.
The more I study all the photos it looks like the switch is half thrown for a diverging route at all times when normal?
I wouldn't necessarily call it being "half thrown", but a double slip is two switches in one. To switch a train from one route to the other requires one half of the double slip being normal and the other reverse.
Then it only takes one set of points to throw and not both (meaning points on opposite ends together) to make a diverging route?
Well you need to make sure that the second set of points aren't set against the movement. On the K-ladder if you normal 69 trains will diverge onto the 36th St connector, but if you leave 71 normal a train would run through the other set of points so you need to reverse 71 to have a valid route.
From the model board it would be just two switches with N and R that just happen to be close together?
Yes exactly. You have seen how double slips are diagrammed on model boards and track charts, right?