Polar Graphs Investigation?
Question by flyboymatt39: Polar Graphs Investigation?
For my geometry and trigonometry investigation these school holidays, we had to investigate polar graphs in the form of:
r = a + b sin (cx)
where a, b and c are positive numbers and x is measured in radians.
like all other investigations like this, i started of by graphing:
r = a + sin x
r = b sin x
r = sin (cx)
i worked out what each constant does for those graphs, then when i put it all together again as r = a + b sin (cx) all of what i had worked out no longer applied.
I have tried playing around with these for a quite a number of hours now, and still haven’t got anywhere with it.
Could someone please tell me what the constants ‘do’ (generalizations about each constants affect)
thanks in advance
thanks for that
Best answer:
Answer by mathematicko
the graph of r = a + b sin (cx) where a, b, and c, are constants, b and c are not equal to zero is gives the limacon, symmetry with respect to the pi/2 axis.
i’ll be telling you effects of varying a,b, and c on a similar graph of a limacon, r = a + b sin (cx) where the graph has a symmetry on the polar axis (theta = 0).
the main variation is within the absolute value of a/b.
*from here, always consider the absolute value)
if a = 0 and b is any number, the graph gives a circle.
if 0 < a/b < 1, the graph gives a limacon with a loop.
if a/b = 1, the graph gives a cardioid.
if 1 < a/b < 2, the graph gives a limacon with a dent.
if a/b > 2, the graph gives a convex limacon.
check: http://mathworld.wolfram.com/Limacon.html
for an animation of the cos graph, check the part ” animate(1+b*cos(theta),theta=0..2*Pi,b=-4..4,coords=polar,color=black,thickness=3,numpoints=100,scaling=constrained,frames=100);”
here: http://www.uwosh.edu/faculty_staff/benzaid/documents/GreenLake/Polar%20Graphs%20Green%20Lake.html
now if you vary the coefficient c, it gives more interesting yet difficult graphs. it is no a limacon but it can be a rose, depending on the c value.
roses are also illustarted here:
http://www.uwosh.edu/faculty_staff/benzaid/documents/GreenLake/Polar%20Graphs%20Green%20Lake.html
What do you think? Answer below!
Examine each separately.
First the base case.
r = sinθ
__________
This is the equation of a circle centered on the y-axis at (0, 1) with radius 1. It goes thru the origin.
r = bsinθ
This scales the circle above making it bigger or smaller depending on b. But it is still a circle centered on the y-axis that goes thru the origin.
________
r = a + sinθ
This increased r for every θ (in all directions) by a constant. The figure is no longer a circle.
If a < 1, the figure has an interior loop in it.
If a = 1, the figure has a cusp. It is a cardiod.
If a > 1, the figure is flattened at one end.
_________
r = sin(cθ)
If c is a integer, the figure is a flower with symmetrical loops. If c is odd, there are c loops. If c is even, there are 2c loops.
If c is not an integer, you get irregularly spaced loops.
__________
r = a + bsin(cθ)
If you put them all together and a, b, and c are integers and c > 1, you get a flower with different sized petals.
If c = 1, you get something similar to what you got for
r = a + sinθ
I’ll let you play with a and b to see what.
If c is not an integer, you get something irregular.
constants stabilize the value of the whole equation so that the non constant will be determined upon the strength of the constants.