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As indicated in item 2, if the Fj is a velocity constraint, then the Ei,j in the equation containing the Fj will be rst derivatives If the Fj is an acceleration, then the Ei,j will be second derivatives, and so on Written in matrix notation the system of equations above becomes

As you can see, y is always reinitialized to 1 each time the inner for loop is entered Even though it is subsequently assigned the value 100, this value is lost There is one quirk to C# s scope rules that may surprise you: Although blocks can be nested, no variable declared within an inner scope can have the same name as a variable declared by an enclosing scope For example, the following program, which tries to declare two separate variables with the same name, will not compile

For clarity, we let (x) = ln x, (x) = 1/x Thus the (inde nite) integral becomes 1/3 (x) dx = 3 4 (x) (x) Thus

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/* This program attempts to declare a variable in an inner scope with the same name as one defined in an outer scope *** This program will not compile *** */ using System; class NestVar { static void Main() { int count; for(count = 0; count < 10; count = count+1) { ConsoleWriteLine("This is count: " + count); int count; // illegal!!! for(count = 0; count < 2; count++) ConsoleWriteLine("This program is in error!"); } } }

[ E ][ A] = [ F ]

If you come from a C/C++ background, then you know that there is no restriction on the names you give variables declared in an inner scope Thus, in C/C++ the declaration of count within the block of the outer for loop is completely valid However, in C/C++, such a declaration hides the outer variable The designers of C# felt that this type of name hiding could easily lead to programming errors and disallowed it

3:

(56)

In programming, it is common to assign one type of variable to another For example, you might want to assign an int value to a float variable, as shown here:

Thus the area under the curve and above the x-axis is 1/(3 ln3 2)

When compatible types are mixed in an assignment, the value of the right side is automatically converted to the type of the left side Thus, in the preceding fragment, the value in i is converted into a float and then assigned to f However, because of C# s strict type-checking, not all types are compatible, and thus, not all type conversions are implicitly allowed For example, bool and int are not compatible Fortunately, it is still possible to obtain a conversion between incompatible types by using a cast A cast performs an explicit type conversion Both automatic type conversion and casting are examined here

4 Solve the system of equations above for the coef cients Aj 5 Using the methods below, evaluate the B-splines or B-spline derivatives as needed to determine displacement, velocity, acceleration, jerk, and other elements between constraints Evaluation of B-Splines and Their Derivatives B-splines can be evaluated by means of a well-known recurrence relation (Cox, 1972; deBoor, 1972) This relationship for the case here, in terms of the cam angle, x, and given the knot sequence, T, is N j ,k ( x ) = x - Tj Tj + k -1 - Tj N j ,k -1 ( x ) + Tj + k - x Tj + k - Tj +1 N j +1,k -1 ( x ) (57)

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