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For the parabola, the eccentricity e = 1. Referring to Fig. B.7, since e = SA/AZ, by definition it follows that SA = AZ. Let f = SA = AZ. This is known as the focal length. Consider a line LP drawn parallel to the directrix. The path length from S to P is r + PP . But PP = AL f r cos , and hence the path length is r(1 cos ) + AL f. Substituting for r from Eq. (B.4) with e = 1 yields a path length of p + AL f. This shows that the path length of a ray originating from the focus and reflected parallel

terminology Two functions f and g are inverses of each other if f undoes the effect of g, and g undoes the effect of f . In terms of compositions (Section 15.1), this means: f g(x) = x and g f (x) = x

to the axis (ZZ ) is constant. It is this property which results in a parallel beam being radiated when a source is placed at the focus. It will be seen that when = 90 , r = SP = p. But SP = eSZ = 2f (since e = 1), and therefore, p = 2f. Thus, the radius as given by Eq. (B.4) can be written as

This table can also be viewed even after the program terminates by pressing Ctrl+B from within the Editor Screen or by selecting the View Variables Table menu option from the Run menu Although this example is a good introduction to debugging it really won t help us nd the problem discussed earlier This is because the program stops only after the infrared sensors have detected an object meaning that a collision will stop the program before you get a chance to view the sensory data If we move the Debug-statement just before the wend statement instead of after it (as shown in Fig 61), we can obtain the necessary information If you try it, you will see that the program will stop and display the debug window after every move As long as an object has not been encountered, the sensor data will be 0.

(B.33)

It can take some time before the robot reaches an object because you have to press Step each time the robot moves forward one pixel One solution to this problem is to draw the object you want to test very close to the robot so that it does not have to move far before indicating the sensory information There is a better solution to the problem of having to press the Step button too many times before the robot arrives at the point of interest where we would like to analyze the data in detail If you press the Debug Off button on the Debug Screen, the robot will move around as it would if there were no debug statements in the program This allows you to let the program proceed at normal speed until the robot approaches a situation you want to analyze in more detail.

This is essentially the same as Eq. (6.27) of the text, where identity 1 + cos = 2cos2( /2). Thus is used to replace r, and = 180 , and use is made of the trigonometric

De nition: Let e denote the unique positive real number such that ln e = 1 (35.2)

(B.34)

When this happens press the Debug On button on the Terminal Screen This will cause the program to again display the debug window and allow you to step through the code as before This is a powerful feature because it lets the robot move around at normal speed until you decide you want to examine something that is about to happen..

For the situation shown in Fig. B.8, where D is the diameter of a parabolic reflector, Eq. (B.34) gives

Since the range of ln x is the set of all real numbers, there must exist such a number e. It can be shown that e = 2.718 . . . . [See Problem 35.30(c).] Theorem 35.1: The functions ex and ln x are inverses of each other: ln ex = x Indeed, substituting e for a in (35.1), ln ex = x ln e = x 1 = x If we replace x in this result by ln x, ln eln x = ln x Hence, eln x = x [since ln u = ln v implies u = v by Property 10, Section 34.2] and eln x = x

3. We designed the project so that the robot could not accept new commands while exe-

(B.35)