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Transforming Representations

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Derive the equation of the line through the points ... and ... in the ... plane, shown in Fig. 37. Then use it to find the linear function ... which can be used in equation (9), Sec. 38. to transform representation (2) in that section into representation (10) there....

The parametric representation used For any given arc C is, of course, not unique. It is, in fact, possible to change the interval over which the parameter ranges any other interval. To be specific, suppose that
representation (10):

where ...is a real-valued function mapping an interval .... onto the interval a ... in representation (2). (See Fig. 37.) We assume that ... is continuous with a continuous derivative. We also assume that ... for each r; this ensures that ...increases with ..., Representation (2) is then transformed by equation (9) into...

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Solution Summary

Representation are transformed. The solution is detailed and well presented.