Parametric Curves
Write-Up #10
Question
Investigate the following from 0 < t < 2 pi:
What happens when a=b, a>b, and a<b?
When we let a > b, the 1st degree equations produce an ellipse. The
2nd degree equations produced what appears to be a straight line. The 3rd
degree equations produced a diamond shape with curved sides (the green lines).
So by extending this pattern, we can say that when the even degrees >
1 produce graphs in the first quadrant. They are also curves from (0,2)
to (2,0). When we have odd degrees > 1, the equations produce these diamond-shaped
curves that go through the points (2,0), (-2,0), (0,2), and (0,-2). These
diamond-shaped figures a oriented with the longest side on the x-axis.
When we let a = b, the 1st degree equations produce a circle. The
2nd degree equations produced what appears to be a straight line. The 3rd
degree equations produced a diamond shape with curved sides (the green lines).
So by extending this pattern, we can say that when the even degrees >
1 produce graphs in the first quadrant, just as in the other two cases.
They are also curves from (0,2) to (2,0). When we have odd degrees >
1, the equations produce these diamond-shaped curves that go through the
points (2,0), (-2,0), (0,2), and (0,-2). These diamond-shaped figures a
symmetric with the origin.
When we let a < b, the 1st degree equations produce an ellipse
with the major axis along the y-axis. The 2nd degree equations produced
what appears to be a straight line. The 3rd degree equations produced a
diamond shape with curved sides (the green lines). So by extending this
pattern, we can say that when the even degrees > 1 produce graphs in
the first quadrant. They are also curves from (0,3) to (2,0). When we have
odd degrees > 1, the equations produce these diamond-shaped curves that
go through the points (2,0), (-2,0), (0,3), and (0,-3). These diamond-shaped
figures a oriented with the longest side on the y-axis.
So these equations produce similar graphs in all three cases. The main
difference is in the orientation of the graphs. So it seems, from my investigations
that the coefficients of the parametric equations change the orientation.
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