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Unformatted text preview: 144 CHAPTER 2. CURVES or equivalently
x = a cos θ
.
y = b sin θ
Suppose we want to describe instead the ellipse with the same semiaxes
(still parallel to the coordinate axes) but with center at the point (c1 , c2 ).
The displacement vector taking the origin to this position is simply the
position vector of the new center
→
− =c − +c −
→
→
c
1ı
2
so we can obtain the new ellipse from the old simply by adding this
(constant) vector to our parametrization function:
− (θ ) = (c − + c − ) + (a cos θ − + b sin θ − )
→
→
→
→
→
p
ı 1ı
2
→
− + (c + b sin θ )−
→
= (c + a cos θ ) ı 1 2 or, in terms of coordinates,
x = c1 +a cos θ
.
y = c2 +b sin θ
We might also consider the possibility of a conic section obtained by
rotating a standard one. This is also easily accomplished for a
→
→
parametrized expression: the role of − (resp. − ) is now played by a
ı →
− (resp. − ) of this vector. Two words of caution are in
→
rotated version u 1
u2
order here: the new vectors must still be unit vectors, and they must still
be perpendicular to each other. Both of these properties are guaranteed if
→
→
we make sure to rotate both − and − by the same amount, in the same
ı direction.
For example, suppose we want to describe the ellipse, still centered at the
origin, with semiaxes a and b, but rotated counterclockwise from the
→
ı
coordinate axes by α = π radians. Rotating − leads to the unit vector
6
π
making angle α = 6 with the positive xaxis
→
− = (cos α)− + (sin α)−
→
→
u1
ı √
3−
→ + 1−
→
ı =
2
2 ...
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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
 ALL
 Calculus

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