Engineering Calculus Notes 151

# Engineering Calculus Notes 151 - clockwise We can displace...

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2.2. PARAMETRIZED CURVES 139 equivalently, we can think of the circle as given by the equation r = R in polar coordinates, so that the point is speciFed by the polar coordinate θ . Translating back to rectangular coordinates we have x = R cos θ y = R sin θ and the parametrization of the circle is given by the vector-valued function −→ p ( θ ) = ( R cos θ,R sin θ ) . As θ goes through the values from 0 to 2 π , −→ p ( θ ) traverses the circle once counterclockwise; if we allow all real values for θ , −→ p ( θ ) continues to travel counterclockwise around the circle, making a full circuit every time θ increases by 2 π . Note that if we interchange the two formulas for x and y , we get another parametrization −→ q ( θ ) = ( R sin θ,R cos θ ) which traverses the circle
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Unformatted text preview: clockwise . We can displace this circle, to put its center at any speciFed point C ( c 1 ,c 2 ), by adding the (constant) position vector of the desired center C to −→ p ( θ ) (or −→ q ( t )): −→ r ( θ ) = ( R cos θ,R sin θ ) + ( c 1 ,c 2 ) = ( c 1 + R cos θ,c 2 + R sin θ ) . Ellipse: The “model equation” for an ellipse with center at the origin (Equation ( 2.13 ) in § 2.1 ) x 2 a 2 + y 2 b 2 = 1 looks just like the equation for a circle of radius 1 centered at the origin, but with x ( resp . y )) replaced by x/a ( resp . y/b ), so we can parametrize this locus via x a = cos θ y b = sin θ...
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