082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

160 Chapter 4 Coordinate systems
Itisnecessaryforacomputertocarryoutcalculationsinordertofindwhereacomponentislocatedandwheretoplaceacomponent.Decideuponasuitablecoordinate
systemtouse when carrying outthese calculations.
Solution
If we examine the geometry of the robot then we see that a polar coordinate system
wouldbethemostsuitable.Thecentreofthecoordinatesystemshouldbeontheaxis
ofrotation.Thelengthofthearmisthengivenbyr andtheorientationofthearmis
given by θ relative toan agreed zero datum mark.
Example4.4 Plot the points P, Qand R whose polar coordinates are
(a) 2,70 ◦
(b) 4,160 ◦
(c) 3,300 ◦
Solution Figure 4.10 shows the three points plotted.
y
y
P
Q
y
2
O
70°
x
4
O
160°
x
300°
O
x
3
(a) (b) (c)
R
Figure4.10
A point can be located bythe values ofits polar coordinates.
Consider the arm from the origin to the point. The value of r gives the length of this
arm. The value of θ gives the angle between the positivexaxis and the arm, measuring
anticlockwise from the positivexaxis.
Bystudying △OPA,shown inFigure 4.11, wecan see that
cosθ = x r
sinθ = y r
and sox =rcos θ (4.1)
and soy =rsin θ (4.2)
Hence if we know the values of r and θ, that is the polar coordinates of a point, we
can use Equations (4.1) and (4.2) to calculatexandy, the Cartesian coordinates of the
point.