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Unformatted text preview: 2.4. REGULAR CURVES 181 If we restrict this function to any interval of length less than 2 π , we get a one-to-one function, but it is not onto the whole circle, whereas on any interval of length exceeding 2 π the function is onto the whole circle, but it is not one-to-one . The restriction to an interval of length exactly 2 π is still problematic: if we use a closed interval of length 2 π , then the two ends go to the same point on the circle, whereas if we use an open interval of length 2 π , we will miss one point on the circle. Finally, if we restrict to a half-open interval of length 2 π , like say I = [0 , 2 π ), then the restriction is indeed both one-to-one and onto, so every point is associated to a unique parameter value. However, this correspondence does not really reflect the geometry of the circle: in particular, −→ p (0) = (1 , 0), but nearby points slightly clockwise from this point are associated to parameters near 2 π rather than near 0....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
- Fall '08