Coordinates in the plane and on the sphere

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Coordinates in the plane and on the sphere

• Some Forgotten (?) Mathematics

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How do we describe where we are?

• We use relative descriptions of location
• Monmouth is right outside of Salem
• Corvallis is about 70 miles south of Portland
• We use absolute descriptions of location
• The 45th parallel is halfway between the Equator
  and the North Pole
• How do we quantify these descriptions?

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Coordinate systems in the plane

• From Descartes work initiating the applications
  of algebraic methods to geometrical problems
• A coordinate system assigns each point in a plane
  a set of numbers that describe it uniquely
• For the plane, we need 2 (if youre bored, ask
  yourself why it must be 2 and not more or less)

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Cartesian coordinates

• Pick a line in the place (x-axis)
• Pick a perpendicular line (y-axis)
• Intersection is the origin
• Choose positive dir on x-axis from origin
• Go counter-clockwise to y-axis, that is the
  positive dir
• Here is where we pick the orientation

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Cartesian coordinates (contd)

• Pick a unit of length, mark it off on the x-axis
  and the y-axis

(3,2)
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Quadrants
II
I
III
IV
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Lines in the plane

• horizontal line all points have same
  y-coordinate
• y 5
• vertical same x-coordinate
• x 1
• general line ymx b
• intercepts x0 line at (0,b)
• slope m

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Distance

• Computed with Pythagorean theorem
• (0,0) to (x,y) is length of hypotenuse

(x,y)
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Circles and Ellipses

• Circle all points at a given distance r from the
  origin
• (x,y) such that
• Ellipse all points satisfying
• A circle is a special kind of ellipse

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Polar coordinates

• Different way to use 2 coordinates to specify a
  point
• Pick an origin and a half-line
• Specify a point by the distance r and the angle q

r
q
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Advantages of various systems

• In polar coordinates, circles are very easy!
• r 5
• dont need square roots, etc.
• Lines are easier in Cartesian coordinates
• What is the equation of a line in polar
  coordinates?

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Converting between systems

• If (x,y) and (r, q) represent the same point,
  then what is the relationship between them?
• You guessed it trig functions!

(x,y)
r
q
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Coordinates on a sphere

• Pick a point (North Pole)
• There is a unique axis going through the North
  pole and the center of the sphere
• Then there is an equator (halfway between North
  and South Poles)
• We need 2 numbers to specify a point on the
  surface of the sphere
• if youre bored, ask yourself why its the same
  number as the plane

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Latitude

• Every point on the sphere lies at some angle to
  the plane of the equator (f)

f
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Longitude

• We need to pick an arbitrary great circle (e.g.,
  the Greenwich meridian)
• Then we measure
• how far east or
• west of it
• we are (q)

f
q