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# curve - CURVES HOMEWORK Section 12.1 26 62 12.2 12 34 50 O...

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CURVES O. Knill, Math21a HOMEWORK: Section 12.1: 26, 62 12.2: 12, 34, 50 PARAMETRIC PLANE CURVES. If x ( t ), y ( t ) are functions of one variable, defined on the parameter inter- val I = [ a,b ], then vector r ( t ) = ( f ( t ) ,g ( t ) ) is a parametric curve in the plane. The functions x ( t ) ,y ( t ) are called coordinate functions . PARAMETRIC SPACE CURVES. If x ( t ) ,y ( t ) ,z ( t ) are functions of one variables, then vector r ( t ) = ( x ( t ) ,y ( t ) ,z ( t ) ) is a space curve . Always think of the parameter t as time . For every fixed t , we have a point ( x ( t ) ,y ( t ) ,z ( t )) in space. As t varies, we move along the curve. EXAMPLE 1. If x ( t ) = t , y ( t ) = t 2 + 1, we can write y ( x ) = x 2 + 1 and the curve is a graph . EXAMPLE 2. If x ( t ) = cos( t ) ,y ( t ) = sin( t ), then vector r ( t ) follows a circle . EXAMPLE 3. If x ( t ) = cos( t ) ,y ( t ) = sin( t ) ,z ( t ) = t , then vector r ( t ) describes a spiral . EXAMPLE 4. If x ( t ) = cos(2 t ) ,y ( t ) = sin(2 t ) ,z ( t ) = 2 t , then we have the same curve as in example 3 but we traverse it faster . The parameterization changed. EXAMPLE 5. If x ( t ) = cos( t ) ,y ( t ) = sin( t ) ,z ( t ) = t , then we have the same curve as in example 3 but we traverse it in the opposite direction . EXAMPLE 6. If P = ( a,b,c ) and Q = ( u,v,w ) are points in space, then vector r ( t ) = ( a + t ( u a ) ,b + t ( v b ) ,c + t ( w c ) ) defined on t [0 , 1] is a line segment connecting P with Q .
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