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You can have a pointer point to another pointer that points to the target value This situation is called multiple indirection, or pointers to pointers Pointers to pointers can be confusing Figure 20-1 helps clarify the concept of multiple indirection As you can see, the value of a normal pointer is the address of the variable that contains the value desired In the case of a pointer to a pointer, the first pointer contains the address of the second pointer, which points to the variable that contains the value desired Multiple indirection can be carried on to whatever extent desired, but more than a pointer to a pointer is rarely needed In fact, excessive indirection is difficult to follow and prone to conceptual errors A variable that is a pointer to a pointer must be declared as such You do this by placing an additional asterisk after the type name For example, the following declaration tells the compiler that q is a pointer to a pointer of type int:

int** q;

Occasionally, the follower requires a particular velocity and acceleration at some critical points in the machine motion This can be accomplished by skewing the acceleration pro le as seen in Fig 37 Neklutin (1969) has treated the modi ed trapezoidal curve with unequal periods of acceleration, positive and negative Ragsdell and Gilkey (1969) have related the skewed acceleration to a correspondingly symmetrical one Before skewing is considered, the follower rise h and angle b have been determined and will be considered as constants Let b1 and b2 be the periods during positive and negative acceleration, respectively, b and let p = 1 be the skew ratio (Fig 37) Thus b2 b1 + b 2 = b h1 + h2 = h The velocity match at the transition point that

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SOLUTION Notice that x 2 + while e3x 0 So the limit is indeterminate of the form 0 We rewrite the limit as

You should understand that q is not a pointer to an integer, but rather a pointer to an int pointer

20:

( ymax )1 = ( ymax )2

U n s a f e C o d e , P o i n t e r s , N u l l a b l e Ty p e s , a n d M i s c e l l a n e o u s To p i c s

Now both numerator and denominator tend to in nity and we may apply l H pital s Rule The result is that the limit equals 2x x 3e 3x lim

To access the target value indirectly pointed to by a pointer to a pointer, you must apply the asterisk operator twice, as in this example:

(310)

using System; class MultipleIndirect { unsafe static void Main() { int x; // holds an int value int* p; // holds an int pointer int** q; // holds a pointer to an int pointer x = 10; p = &x; // put address of x into p q = &p; // put address of p into q ConsoleWriteLine(**q); // display the value of x } }

The output is the value of x, which is 10 In the program, p is declared as a pointer to an int and q as a pointer to an int pointer One last point: Do not confuse multiple indirection with high-level data structures, such as linked lists These are two fundamentally different concepts

Transition point h/2 b b1 b /2 b 1 b2 b /2

Again the numerator and denominator both tend to in nity so we apply l H pital s Rule to obtain:

Pointers can be arrayed like any other data type The declaration for an int pointer array of size 3 is

int * [] ptrs = new int * [3];

(311)

To assign the address of an int variable called var to the third element of the pointer array, write

x

ptrs[2] = &var;

The relationship between the skewed and corresponding symmetrical acceleration, since the maximum velocity is the same in both skewed and symmetric cases, is: 2b1a1 = asym b = Vmax \ a1 = asym Thus using Eq (310) a1 = Similarly 1+ p a2 = a 2 sym (313) 1+ p a 2 p sym (312) b 2b1

*ptrs[2]

When working in an unsafe context, you might occasionally find it useful to know the size, in bytes, of one of C# s value types To obtain this information, use the sizeof operator It has this general form: sizeof(type) Here, type is the type whose size is being obtained In general, sizeof is intended primarily for special-case situations, especially when working with a blend of managed and unmanaged code

Figure 38a shows the normalized displacement plot and Fig 38b shows the normalized velocity plot where h = 1, b = 1

Part I:

x

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