Engineering Calculus Notes 156

Engineering - 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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 semi-axes (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 semi-axes 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 x-axis → − = (cos α)− + (sin α)− → → u1 ı √ 3− → + 1− → ı = 2 2 ...
View Full Document

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.

Ask a homework question - tutors are online