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# 29 - Lecture 1 January 4 2012 1 Parametric equations...

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Lecture 1 January 4, 2012 1 Parametric equations Definition 1.1. A function f : A B is a map which for each point in A assigns a single point in B . Example 1.1. y = + 1 - x 2 , x [0 , 1] . Example 1.2. y = sin( t ) , x = cos( t ) , t [0 , 2 π ) The above example is two to one, for x y so is not a function. Example 1.3. y = sin( t ) , x = cos( t ) , t [0 , π ] . Example 1.4. x = sin(0 . 5 π sin( t )) , y = cos(0 . 5 π sin( t )) t in [0 , 2 π ] Example 1.5. x = sin(0 . 25 π + 0 . 5 π sin( t )) , y = cos(0 . 25 π + 0 . 5 π sin( t )) , t [0 , 2 π ] Example 1.6. x = sin( - πt ) y = cos( - πt ) t [0 , 100] Remark 1.1. The same curve can have many different parameterizations. Remark 1.2. In some cases only one parameterization is useful, in others, overlaping parameter- izations can be useful. Example 1.7. (Pogorelov, Differential Geometry
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