$$ a$)$ As shown in the Practical Perspective, the self$-$capacitance design does not permit a true multitouch screen$$if the screen is touched at two different points, a total of four touch points are identified, the two actual touch points and two ghost points. If a self$-$capacitance touch screen is touched at the x$,$ y coordinates $(2\mathrm{.}1,4\mathrm{.}3)$ and $(3\mathrm{.}2,2\mathrm{.}5),$ what are the four touch locations that will be identified? $($Assume the touch coordinates are measured in inches from the upper left corner of the screen.$)$

$$ b$)$ A self$-$capacitance touch screen can still function as a multi$-$touch screen for several common gestures. For example, suppose at time t$1$ the two touch points are those identified in part $($a$),$ and at time t$2$ four touch points associated with the x$,$ y coordinates $(1\mathrm{.}8,4\mathrm{.}9)$ and $(3\mathrm{.}9,1\mathrm{.}8)$ are identified. Comparing the four points at t$1$ with the four points at t$2,$ software can recognize a pinch gesture$$should the screen be zoomed in or zoomed out?

$$ c$)$ Repeat part $($b$),$ assuming that at time t$2$ four touch points associated with the x$,$ y coordinates $(2\mathrm{.}8,3\mathrm{.}9)$ and $(3\mathrm{.}0,2\mathrm{.}8)$ are identified.