Curve - 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

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Unformatted text preview: 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 ) )...
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This note was uploaded on 03/29/2008 for the course MATH 251 taught by Professor Skrypka during the Spring '08 term at Texas A&M.

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