Navigator’s Polar Vs Cartesian
Now suppose that you want to convert from NPC to Cartesian coordinates.
Here are the conversion formulas for translating the coordinates for a point (a _{0} , r _{0}) in NPC to a point (x_{ 0} , y _{0}) in the Cartesian plane:
x _{0} = r _{0} sin α _{0}
y _{0} = r _{0} cos α _{0}
These are similar to the formulas you use to convert MPC to Cartesian coordinates, except that the roles of the sine and cosine function are reversed.
In order to convert the coordinates of a point (x _{0} , y_{ 0}) in Cartesian coordinates to a point (α_{ 0} , r_{ 0}) in NPC, use these formulas:
α _{0} = arctan ( x _{0} / y _{0} ) if y _{0} > 0
α _{0} = 180° + arctan ( x _{0} / y _{0} ) if y _{0} < 0
r _{0} = ( x ^{2} _{0} + y ^{2} _{0} ) ^{1/2}
We can’t have y _{0} = 0, because that produces an undefined quotient. If a value of α _{0} thus determined happens to be negative, add 360° to get the “legitimate” value. These are similar to the formulas for converting Cartesian coordinates to MPC.
Geometry and Navigation Practice Problems
PROBLEM 1
Suppose a radar set, that uses NPC, indicates the presence of a hovering object at a bearing of 300° and a range of 40 km. If we say that a kilometer is the same as a “unit,” what are the coordinates ( θ _{0} , r _{0} ) of this object in mathematician’s polar coordinates? Express θ _{0} in both degrees and radians.
SOLUTION 1
We are given coordinates (α _{0} , r _{0} ) = (300°,40). The value of r _{0} , the radius, is the same as the range, in this case 40 units. As for the angle θ _{0} , remember the conversion formulas given above. In this case, because α _{0} is greater than 90° and less than 360°:
θ _{0} = 450° − α _{0}
= 450° − 300° = 150°
Therefore, (θ _{0} , r _{0} ) = (150°,40). To express θ _{0} in radians, recall that there are 2π radians in a full 360° circle, or π radians in a 180° angle. Note that 150° is exactly 5/6 of 180°. Therefore, θ _{0} = 5π/6 rad, and we can say that (θ _{0} , r _{0} ) = (150°,40) = (5π/6,40). We can leave the “rad” off the angle designator here. When units are not specified for an angle, radians are assumed.
PROBLEM 2
Suppose you are on an archeological expedition, and you unearth a stone on which appears a treasure map. The map says “You are here” next to an X, and then says, “Go north 40 paces and then west 30 paces.” Suppose that you let west represent the negative x axis of a Cartesian coordinate system, east represent the positive x axis, south represent the negative y axis, and north represent the positive y axis. Also suppose that you let one “pace” represent one “unit” of radius, and also one “unit” in the Cartesian system. If you are naïve enough to look for the treasure and lazy enough so you insist on walking in a straight line to reach it, how many paces should you travel, and in what direction, in navigator’s polar coordinates? Determine your answer to the nearest degree, and to the nearest pace.
SOLUTION 2
Determine the ordered pair in Cartesian coordinates that corresponds to the imagined treasure site. Consider the origin to be the spot where the map was unearthed. If we let ( x _{0} , y _{0} ) be the point where the treasure should be, then 40 paces north means y _{0} = 40, and 30 paces west means x _{0} = −30:
( x _{0} , y _{0} ) = (−30,40)
Because y _{0} is positive, we use this formula to determine the bearing or heading α _{0} :
α _{0} = arctan ( x _{0} / y _{0} )
= arctan (−30/40)
= arctan –0.75
= −37°
This is a negative angle, so to get it into the standard form, we must add 360°:
α _{0} = −37° + 360° = 360° − 37°
= 323°
To find the value of the range, r _{0} , use this formula:
r _{0} = ( x ^{2} _{0} + y ^{2} _{0} ) ^{1/2}
= (30 ^{2} + 40 ^{2} ) ^{1/2}
= (900 + 1600) ^{1/2}
= 2500 ^{1/2}
= 50
This means (α _{0} , r _{0} ) = (323°,50). Proceed 50 paces, approximately north by northwest. Then, if you have a shovel, go ahead and dig!
Practice problems for these concepts can be found at: Polar Coordinates Practice Test.
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