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First review the properties of inverse functions that we learned in Subsection 185 The graph of exp(x) is obtained by re ecting the graph of ln x in the line y = x We exhibit the graph of y = exp(x) in Fig 65

A derived class can call a constructor defined in its base class by using an expanded form of the derived class constructor declaration and the base keyword The general form of this expanded declaration is shown here: derived-constructor(parameter-list) : base(arg-list) { // body of constructor } Here, arg-list specifies any arguments needed by the constructor in the base class Notice the placement of the colon To see how base is used, consider the version of TwoDShape in the following program It defines a constructor that initializes the Width and Height properties This constructor is then called by the Triangle constructor

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In Eq (513), if all the Wj = 1 ( j = 1, , n), Rj,k(x) = Nj,k(x), then, Eqs (511) and (512) become the same as Eqs (51) and (54), respectively Adjustments to these weight values constitute the additional degrees of freedom mentioned earlier and provide additional design exibility as well Note that, in particular, a change in Wj affects the rational B-splines only in the interval of [Tj, Tj+k] As a result, the designer can exercise local control of motion characteristics by adjusting a particular Wj Hence, for a general case, the rational B-spline procedure permits the designer to re ne the synthesized motion by adjusting the order of B-splines and the knot sequence and to exercise local control through the weight sequence, all without violating any of the motion constraints Moreover, motion constraints can also be added or adjusted to tune motion programs When required for the equations above, derivatives of rational B-splines must be evaluated They can be obtained by differentiating Eq (513) For example, the rst and the second derivatives of rational B-splines required for motion constraints of velocity and acceleration become:

// Add constructor to TwoDShape using System; // A class for two-dimensional objects class TwoDShape { double pri_width; double pri_height;

11:

) ) R(j1k ( x ) = Wj N (j1k ( x ) , ,

public int a; public X(int x) { a = x; } } class Test : ICloneable { public X o; public int b; public Test(int x, int y) { o = new X(x); b = y; } public void Show(string name) { ConsoleWrite(name + " values are "); ConsoleWriteLine("oa: {0}, b: {1}", oa, b); } // Make a deep copy of the invoking object public object Clone() { Test temp = new Test(oa, b); return temp; } } class CloneDemo { static void Main() { Test ob1 = new Test(10, 20); ob1Show("ob1"); ConsoleWriteLine("Make ob2 a clone of ob1"); Test ob2 = (Test) ob1Clone(); ob2Show("ob2"); ConsoleWriteLine("Changing ob1oa to 99 and ob1b to 88"); ob1oa = 99; ob1b = 88; ob1Show("ob1"); ob2Show("ob2"); } }

We see, from inspection of this gure, that exp(x) is increasing and is concave up Since ln(1) = 0 we may conclude that exp(0) = 1 Next we turn to some of the algebraic properties of the exponential function

The output is shown here:

Wj N j ,k ( x ) - Wj N j ,k ( x ) Wj N (j1,k) ( x )

ob1 values are oa: Make ob2 a clone of ob2 values are oa: Changing ob1oa to ob1 values are oa: ob2 values are oa: 10, b: ob1 10, b: 99 and 99, b: 10, b: 20 20 ob1b to 88 88 20

Part II:

For all real numbers a and b we have (a) exp(a + b) = [exp(a)] [exp(b)]

(515)

As the output shows, ob2 is a clone of ob1, but ob1 and ob2 are completely separate objects Changing one does not affect the other This is accomplished by constructing a new Test object, which allocates a new X object for the copy The new X instance is given the same value as the X object in the original To implement a shallow copy, simply have Clone( ) call MemberwiseClone( ) defined by Object For example, try changing Clone( ) in the preceding program as shown here:

// Make a shallow copy of the invoking object public object Clone() { Test temp = (Test) MemberwiseClone(); return temp; }

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