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In this version, every time we detect a number of taps, instead of calling the corresponding method immediately, we use performSelector:withObject:afterDelay: to call it four-tenths of a second in the future, and we cancel the perform request done by our method when the previous tap count was received. So, when we receive one tap, we call the singleTap method four-tenths of a second in the future. When we receive notification of a double-tap, we cancel the call to singleTap and call doubleTap four-tenths of a second in the future. We do the same thing with triple-taps and quadruple-taps so that only one of the four methods gets called for any particular tap sequence. Compile and run this version, and when you double-, triple-, or quadruple-tap, you should only see one label displayed.

So all we need to do is drop in our angle (don t forget to convert from degrees to radians if you are using C math library functions), and we have our vector well, almost. The second problem still needs to be solved. The other problem is that our spaceship s starting orientation is different from the unit circle. Notice that when we start our game, for angle = 0, the ship is pointing toward the top of the screen (along the y axis). But the unit circle for angle = 0 points along the x axis (refer to Figure 11 6). This means our spaceship is turned, or phase-shifted, by 90 degrees compared to the unit circle. So if we just plug into the above equations, our results will be wrong. If you look at the x and y values around the unit circle in Figure 11 6, you ll notice that their magnitudes repeat around the circle, and it is only the positive and negative signs that change. I point this out to also remind you that cosine and sine are periodic (repeating) functions. (You can think of waveforms, too.) In fact, you can express sine in terms of cosine or vice versa, because they are essentially the same function, just shifted (based on the angle parameter). This means that we can just look up a table of trigonometric identities (or derive it, if you really want to) to figure out what it means to phase-shift cosine and sine by 90 degrees to handle the fact that our spaceship starting orientation points along the y axis. This reveals the following useful identities:

Another common gesture is the two-finger pinch. It s used in a number of applications, including Mobile Safari, Mail, and Photos to let you zoom in (if you pinch apart) or zoom out (if you pinch together). Detecting pinches is pretty easy. First, when the gesture begins, we check to make sure there are two touches, because pinches are two-finger gestures. If there are two, we store the distance between them. Then, as the gesture progresses, we keep checking the distance between the user s fingers, and if the distance increases or decreases more than a certain amount, we know there s been a pinch. Create a new project in Xcode, again using the view-based application template, and call this one PinchMe. In this project and the next one, we re going to need to do some fairly standard analytic geometry to calculate such things as the distance between two points (in this project) and later the angle between two lines. Don t worry if you don t remember much geometry, we ve provided you with functions that will do the calculations for you. Look in the 13 PinchMe folder for two files, named CGPointUtils.h and CGPointUtils.c. Drag both of these to the Classes folder of your project. Feel free to use these utility functions in your own applications. The PinchMe application is only going to need a single outlet for a label, but it also needs an instance variable to hold the starting distance between the fingers and, as with the previous