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The relationships between x and x and y and y are linear (Rogers, 2001), namely, Ax = 6Cx, Ay = 6Cy (729)

Info for t1: Triangle is right Width and height are 8 and 12 Color is Blue Area is 48 Info for t2: Triangle is isosceles Width and height are 2 and 2 Color is Red Area is 2

Because of inheritance, ColorTriangle can make use of the previously defined classes of Triangle and TwoDShape, adding only the extra information it needs for its own, specific application This is part of the value of inheritance; it allows the reuse of code This example illustrates one other important point: base always refers to the constructor in the closest base class The base in ColorTriangle calls the constructor in Triangle The base in Triangle calls the constructor in TwoDShape In a class hierarchy, if a base class constructor requires parameters, then all derived classes must pass those parameters up the line This is true whether or not a derived class needs parameters of its own

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Note that the A and C matrices appearing above are themselves functions of the coordinates of the SP In fact the n n matrices A and C are de ned as 2a 1,n a 1 0 A= M 0 a n and - b1,n b 1 0 C= M 0 bn where b1 - b1,2 b2 M 0 0 0 b2 - b 2 ,3 O L 0 0 0 b3 O b n L L L L O - b n ,n b n bn 0 0 M b n - b n ,n a1 2a 1,2 a2 M 0 0 0 0 a2 0 2a 2 ,3 a 3 O O L a n 0 L L L L O 2a n ,n a n a n 0 0 M a n 2a n ,n

In the foregoing discussion of inheritance and class hierarchies, an important question may have occurred to you: When a derived class object is created, whose constructor is executed first The one in the derived class or the one defined by the base class For example, given a derived class called B and a base class called A, is A s constructor called before B s, or vice versa The answer is that in a class hierarchy, constructors are called in order of derivation, from base class to derived class Furthermore, this order is the same whether or not base is used If base is not used, then the default (parameterless) constructor of each base class will be executed The following program illustrates the order of constructor execution:

Math Note: In mathematics, we commonly write log x to mean the natural logarithm Thus you will sometimes encounter ln x and sometimes encounter log x (without any subscript); they are both understood to mean loge x, the natural logarithm The exponential function exp x is de ned to be the inverse function to ln x Figure 159 shows the graph of y = exp x In fact we will see later that exp x = ex More generally, the function a x is the inverse function to loga x The exponential has these properties: (a) a b+c = a b a c ;

// Demonstrate when constructors are called using System; // Create a base class class A { public A() { ConsoleWriteLine("Constructing A"); } } // Create a class derived from A class B : A { public B() {

a k = Dpk , a i , j = a i + a j , b k = 1 a k , bi , j = bi + b j for i, j, k = 1, , n , and n = n - 1, n = n - 2, n = n - 3 On the other hand, the relations between x and x , and between y and y can be written as x = (F - GC)x, y = (F - GC)y where the n n matrices F and G are given by: -a 1 0 0 F= M a n a1 -a 2 0 M 0 0 L a2 0 L -a 3 a 3 L O 0 0 0 O O M a n -a n L L L O 0 0 0 M b n 2b n (730)

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