What are the different types of polar graphs? Give several examples with pictures or a site to use as a reference. Everyone should have different pictures or a different site.
HEY TAYLOR HERE AND HERE IS MY RESPONSE AND I DO NOT KNOW IF THIS IS RIGHT OR NOT!!!
I also used http://www.bing.com/images/search?q=polar+graphs#focal=dcdd60d14fd0dc896c8d8496d854d392&furl=http://www.mathamazement.com/images/Pre-Calculus/06_Additional-Topics-in-Trigonometry/06_04_graphs-of-polar-equations/polar-roses.jpg as a reference. OH the thing didn't take my pictures so sorry if iot is messed up or dosen't make sense.
There are five different types of polar graphs.
The Circles in Polar Form are the more simpler ones. They can be in r= a cos theta form or r= a sin theta form. Examples r= 5 cos theta Since a or 5 is the diameter of the circle it will look like this. The end of the circle will be on the origin.
r= 4 sin theta Since a or 4 is the diameter of the circle it will look like this. (the diameter is 4, but my stupid computer wouldn't let me put it on the picture) The end of the circle will be on the origin.
Another polar graph is the limacons (snails) graph. The equations for this group are r= a +/- b sin theta and r= a +/- b cos theta, where a >0 and b > 0. SO NO NEGATIVES! The ratio of a/b is what determines the exact shape of the limacon. If it is less than one then it will have a inner loop. Examples r= 5 +7 cos theta
And since 5/7 is less than 1 this will be your graph. (DON'T LAUGH!! THIS IS WHAT MY CALCULATOR GAVE ME!!) r= 6+10 sin theta Since 6/10 is less than 1 this will be your graph. (DON'T LAUGH)
Another polar graph is the cardioids. Their equations are r= a+/- a cos theta and r= a+/- a sin theta. Their graphs tend to have a heart like shape.
Example
4 + 4 cos theta
5+ 5 sin theta
Another is the Rose graphs that have an appearance of a flower. The equations are r= a sin n theta or r= a cos n theta. The n will be multiplied by 2 to get the total number of petals for a even number but an odd number is just that number for the petals. A cannot equal 0 and n has to be greater than 1.
Examples
r= 5 cos(2theta) This is your graph.
r= 7 sin (3theta)
The final polar graphs are the lemniscates. Their equations are 2^2+ a^2 sin( 2 theta) and r^2+ a^2 cos(2 theta). The a cannot equal 0.
Examples
5^2 sin( 2 theta)
Here is your graph. On another note the circles need to look like propellers.
"This type of graph is known like Rose of four petals. It is easy to see how a figure similar to a rose with four petals forms. The function for this graph is:"
CARDIOIDES "Next the type of graph appears that denominates cardioide. For this example a symmetrical cardioide with respect to the axis appears to poplar and that aims towards the right. We can observe that a figure like of a heart is distinguished, reason for which east cardioide graph is called. The function has generated that it is:"
"Having seen the first graph of a cardoid, another graph of this type appears but now it aims upwards, we see as it in the graph of the following function:"
LIMACONES OR SNAILS
"Limaçon comes from the Latin limax that means snail.The snail of Pascal, discovered Etienne to it Pascal father of Blaise Pascal in first half of century XVII and the name occurred Roberval it in 1650 it used when like example to show it his method to draw up tangents. Limaçon or the polar graphs that generate limaçones are the functions in polar coordinates with the form:"
cos r = 1 + b "Now we see a concrete example of a graph of this type, where is a snail that aims towards the right and that has an in-core loop. The function for this graph is the following one:"
"We see another graph of a function that has like result a snail with an in-core loop but that unlike the previous graph, this it aims downwards. We see:"
"Continuing with the graph of snails or limacones, there is another type that is the snail with crack or snail with concavity. As we will be able to observe, this it does not have bow, and it is directed towards the left. We see next the graph that is, which aims towards the left"
ROSE OF THREE Petals
"We now present/display the called graph Pink of three petals. Analogically to the graph of the rose of four petals, this graph is similar but it has only three leaves or petals in its graphical form. An example is the following one"
ROSE OF EIGHT petals "The following graph is like both previous, but now with eight leaves or petals, we see as it in the following graficada function"
i got all of this information from this ebsite and it shows you the graph and what each one of those looks like.
To define polar coordinates, we first fix an origin O and an initial ray from O.
Then each point P can be located by assigning to it a polar coordinate pair (r, ø), in which the first number, r, gives the directed distance from O to P and the second number, ø, gives the directed angle from the initial ray to the segment OP:
Spirals of Archimedes Polar graphs of the form r = at + b where a is positive and b is nonnegative are called Spirals of Archimedes. They have the appearance of a coil of rope or hose with a constant distance between successive coils. The constant distance is .
The polar graph for
r = t + 2 for
is an example of a Spiral of Archimedes.
-------------------------------------------------------------------------------- Consider another example of a Spiral of Archimedes:
r = at
where 00 We can classify any polar equation that has the form
where a>0 as a hyperbolic spiral.
We want to explore this polar equation as a grows larger.
We let a=.5, 1, 4, 10, 25 and graph each associated polar equation.
This is Nathan with the week whatever blog prompt response.
There are five different kinds of polar graphs. There is the circle, limacon(snail), cardioids, the rose curve, and finally the lemniscate.
The first one is the circle graph. There are two different formulas for it. The first: r=acos(theta)-The diameter of the circle that has its left-most edge at the pole. r=asin(theta)-The diameter of the circle that has its bottom-most edge at the pole. http://www.squarecirclez.com/blog/wp-content/uploads/2009/10/polar-circle-2.gif
The next is the limacon. r=2+3sin(theta)=2/3; the ratio of a/b will determine the exact shape of the limacon. http://curvebank.calstatela.edu/index/limacon.gif
The third kind of polar graph is the cardioid. This graph will look like a heart. r=a+-cos(theta) r=a+-sin(theta) http://chester.blog.br/archives/img/mt/2009/03/14/cardioid.jpg
The fourth kind of graph is the rose curve. If n is an even integer the rose will have 2n petals. If n is an odd integer, then the rose will have n petals. r=asin2(theta)=4 petals r=asin5(theta)=5 petals http://curvebank.calstatela.edu/index/rose4.gif
The last type of graph is the lemniscate. This has the shape of a figure-8 or propeller. r^2=a^2 sin2(theta) http://curvebank.calstatela.edu/index/lemniscate.gif
There are 5 different types of polar graphs: 1)circle 2)limacon 3)cardioids 4)rose 5)lemniscate
CIRCLE there are two different formulas for the circle graph: 1.r=acos(theta)--left most edge at the pole 2.r=asin(theta)--bottom most edge at the pole *****http://www.algebra.com/algebra/homework/equations/Equations.faq.question.191885.html
LIMACON the formulas for this is: r=2+3sin(theta)=2/3 the ratio of a/b will determine the exact shape of the limacon *****http://curvebank.calstatela.edu/index/limacon.gif
CARDIOID the two formulas for this are: 1.r=a+-cos(theta) 2.r=a+-sin(theta) *****http://curvebank.calstatela.edu/index/cardiod.gif
ROSE if n is an even integer the rose will have 2n petals. if n is an odd integer, then the rose will have n petals. Formulas: 1.r=asin2(theta)=4 petals 2.r=asin5(theta)=5 petals *****http://curvebank.calstatela.edu/index/rose4.gif
LEMNISCATE the formula for this graph is: r^2=a^2sin2(theta) *****http://curvebank.calstatela.edu/index/lemniscate.gif
There are a bunch of different types of polar graphs. SUCH AS:
Cardioids r=a(1-cosO) these graphs kind of look like hearts. Get it, cardioid, cardio, which involves your heart. Ironic? I think not.
Rose Curves r=a(cos(bO)) r=a(sin(bO)) All of these graphs look like some sort of flower. and the b in front of the theta multiplied by two tells you how many petals you will have.
Lemniscates r^2=a^2cos(2O) these graphs look like propellors or infinity symbols.
Limacons r=acos(O)+b these graphs look like circles. usually with inner loops.
carioids- Next the type of graph appears that denominates cardioide. For this example a symmetrical cardioide with respect to the axis appears to poplar and that aims towards the right. We can observe that a figure like of a heart is distinguished, reason for which east cardioide graph is called. The function has generated that it is.
lemniscate- the formula for this graph is, r^2=a^2sin2(theta)circle
limacon- Limaçon comes from the Latin limax that means snail.The snail of Pascal, discovered Etienne to it Pascal father of Blaise Pascal in first half of century XVII and the name occurred Roberval it in 1650 it used when like example to show it his method to draw up tangents. Limaçon or the polar graphs that generate limaçones are the functions in polar coordinates with the form.
rose- is a famous mathematical curve that looks like a petalled flower, and that can be expressed as a simple polar equation.
A circle with equation r(θ) = 1The general equation for a circle with a center at (r0, ) and radius a is
r^2 - 2rrcos(theta - alpha) + r^2 = a^2
This can be simplified in various ways, to conform to more specific cases, such as the equation
r(theta) = a
for a circle with a center at the pole and radius a.
- Line
Radial lines (those running through the pole) are represented by the equation
theta = alpha
where φ is the angle of elevation of the line; that is, φ = arctan m where m is the slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line θ = φ perpendicularly at the point (r0, φ) has the equation
r(theta) = rsec (theta - alpha)
- Polar rose A polar rose is a famous mathematical curve that looks like a petalled flower, and that can be expressed as a simple polar equation,
r(theta) = acos(ktheta + theta)
for any constant φ0 (including 0). If k is an integer, these equations will produce a k-petalled rose if k is odd, or a 2k-petalled rose if k is even. If k is rational but not an integer, a rose-like shape may form but with overlapping petals. Note that these equations never define a rose with 2, 6, 10, 14, etc. petals. The variable a represents the length of the petals of the rose.
- Lemniscates
It looks like an infinity symbol. When displayed in three dimensions it is often rendered as a Möbius strip.
- Limacons
In geometry, a limaçon (pronounced /ˈlɪməsɒn/), also known as a limaçon of Pascal, is defined as a roulette formed when a circle rolls around the outside of a circle of equal radius. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller circle is inside the larger circle. Thus, they belong to the family of curves called centered trochoids; more specifically, they are epitrochoids. The cardioid is the special case in which the point generating the roulette lies on the rolling circle; the resulting curve has a cusp.
HEY TAYLOR HERE AND HERE IS MY RESPONSE AND I DO NOT KNOW IF THIS IS RIGHT OR NOT!!!
ReplyDeleteI also used http://www.bing.com/images/search?q=polar+graphs#focal=dcdd60d14fd0dc896c8d8496d854d392&furl=http://www.mathamazement.com/images/Pre-Calculus/06_Additional-Topics-in-Trigonometry/06_04_graphs-of-polar-equations/polar-roses.jpg as a reference. OH the thing didn't take my pictures so sorry if iot is messed up or dosen't make sense.
There are five different types of polar graphs.
The Circles in Polar Form are the more simpler ones. They can be in r= a cos theta form or r= a sin theta form.
Examples
r= 5 cos theta
Since a or 5 is the diameter of the circle it will look like this.
The end of the circle will be on the origin.
r= 4 sin theta
Since a or 4 is the diameter of the circle it will look like this.
(the diameter is 4, but my stupid computer wouldn't let me put it on the picture) The end of the circle will be on the origin.
Another polar graph is the limacons (snails) graph. The equations for this group are r= a +/- b sin theta and r= a +/- b cos theta, where a >0 and b > 0. SO NO NEGATIVES! The ratio of a/b is what determines the exact shape of the limacon.
If it is less than one then it will have a inner loop.
Examples
r= 5 +7 cos theta
And since 5/7 is less than 1 this will be your graph.
(DON'T LAUGH!! THIS IS WHAT MY CALCULATOR GAVE ME!!)
r= 6+10 sin theta
Since 6/10 is less than 1 this will be your graph.
(DON'T LAUGH)
Another polar graph is the cardioids. Their equations are r= a+/- a cos theta and r= a+/- a sin theta.
Their graphs tend to have a heart like shape.
Example
4 + 4 cos theta
5+ 5 sin theta
Another is the Rose graphs that have an appearance of a flower. The equations are r= a sin n theta or r= a cos n theta. The n will be multiplied by 2 to get the total number of petals for a even number but an odd number is just that number for the petals. A cannot equal 0 and n has to be greater than 1.
Examples
r= 5 cos(2theta)
This is your graph.
r= 7 sin (3theta)
The final polar graphs are the lemniscates. Their equations are 2^2+ a^2 sin( 2 theta) and r^2+ a^2 cos(2 theta). The a cannot equal 0.
Examples
5^2 sin( 2 theta)
Here is your graph. On another note the circles need to look like propellers.
6^2cos(2 theta)
That is all for this week seeya!
Lawrence's comment
ReplyDelete"This type of graph is known like Rose of four petals. It is easy to see how a figure similar to a rose with four petals forms. The function for this graph is:"
CARDIOIDES
"Next the type of graph appears that denominates cardioide. For this example a symmetrical cardioide with respect to the axis appears to poplar and that aims towards the right. We can observe that a figure like of a heart is distinguished, reason for which east cardioide graph is called. The function has generated that it is:"
"Having seen the first graph of a cardoid, another graph of this type appears but now it aims upwards, we see as it in the graph of the following function:"
LIMACONES OR SNAILS
"Limaçon comes from the Latin limax that means snail.The snail of Pascal, discovered Etienne to it Pascal father of Blaise Pascal in first half of century XVII and the name occurred Roberval it in 1650 it used when like example to show it his method to draw up tangents. Limaçon or the polar graphs that generate limaçones are the functions in polar coordinates with the form:"
cos r = 1 + b
"Now we see a concrete example of a graph of this type, where is a snail that aims towards the right and that has an in-core loop. The function for this graph is the following one:"
"We see another graph of a function that has like result a snail with an in-core loop but that unlike the previous graph, this it aims downwards. We see:"
"Continuing with the graph of snails or limacones, there is another type that is the snail with crack or snail with concavity. As we will be able to observe, this it does not have bow, and it is directed towards the left. We see next the graph that is, which aims towards the left"
ROSE OF THREE Petals
"We now present/display the called graph Pink of three petals. Analogically to the graph of the rose of four petals, this graph is similar but it has only three leaves or petals in its graphical form. An example is the following one"
ROSE OF EIGHT petals
"The following graph is like both previous, but now with eight leaves or petals, we see as it in the following graficada function"
i got all of this information from this ebsite and it shows you the graph and what each one of those looks like.
website:
http://us.monografias.com/docs33/polar-coordinate/polar-coordinate.shtml#rosa
Definition of Polar Coordinates
ReplyDeleteTo define polar coordinates, we first fix an origin O and an initial ray from O.
Then each point P can be located by assigning to it a polar coordinate pair (r, ø), in which the first number, r, gives the directed distance from O to P and the second number, ø, gives the directed angle from the initial ray to the segment OP:
--------------------------------------------------------------------------------
Interesting Graphs
--------------------------------------------------------------------------------
This investigation is going to explore several interesting graphs and their associated polar equations.
Note: In the following polar equations, ø=t.
--------------------------------------------------------------------------------
Spirals
--------------------------------------------------------------------------------
Spirals of Archimedes
Polar graphs of the form r = at + b where a is positive and b is nonnegative are called Spirals of Archimedes. They have the appearance of a coil of rope or hose with a constant distance between successive coils. The constant distance is .
The polar graph for
r = t + 2
for
is an example of a Spiral of Archimedes.
--------------------------------------------------------------------------------
Consider another example of a Spiral of Archimedes:
r = at
where 00
We can classify any polar equation that has the form
where a>0 as a hyperbolic spiral.
We want to explore this polar equation as a grows larger.
We let a=.5, 1, 4, 10, 25 and graph each associated polar equation.
a=.5 (blue)
a= 1 (red)
a= 4 (green)
a=10 (orange)
a=25 (purple)
As a grows larger, the spiral becomes larger and r is greater.
and there's a bunch more on this site, no one steall itt!
http://jwilson.coe.uga.edu/emt668/emt668.folders.f97/anderson/writeup11/writeup11.html
This is Nathan with the week whatever blog prompt response.
ReplyDeleteThere are five different kinds of polar graphs. There is the circle, limacon(snail), cardioids, the rose curve, and finally the lemniscate.
The first one is the circle graph. There are two different formulas for it. The first:
r=acos(theta)-The diameter of the circle that has its left-most edge at the pole.
r=asin(theta)-The diameter of the circle that has its bottom-most edge at the pole.
http://www.squarecirclez.com/blog/wp-content/uploads/2009/10/polar-circle-2.gif
The next is the limacon.
r=2+3sin(theta)=2/3; the ratio of a/b will determine the exact shape of the limacon.
http://curvebank.calstatela.edu/index/limacon.gif
The third kind of polar graph is the cardioid. This graph will look like a heart.
r=a+-cos(theta)
r=a+-sin(theta)
http://chester.blog.br/archives/img/mt/2009/03/14/cardioid.jpg
The fourth kind of graph is the rose curve. If n is an even integer the rose will have 2n petals. If n is an odd integer, then the rose will have n petals.
r=asin2(theta)=4 petals
r=asin5(theta)=5 petals
http://curvebank.calstatela.edu/index/rose4.gif
The last type of graph is the lemniscate. This has the shape of a figure-8 or propeller.
r^2=a^2 sin2(theta)
http://curvebank.calstatela.edu/index/lemniscate.gif
There are 5 different types of polar graphs:
ReplyDelete1)circle
2)limacon
3)cardioids
4)rose
5)lemniscate
CIRCLE
there are two different formulas for the circle graph:
1.r=acos(theta)--left most edge at the pole
2.r=asin(theta)--bottom most edge at the pole
*****http://www.algebra.com/algebra/homework/equations/Equations.faq.question.191885.html
LIMACON
the formulas for this is:
r=2+3sin(theta)=2/3
the ratio of a/b will determine the exact shape of the limacon
*****http://curvebank.calstatela.edu/index/limacon.gif
CARDIOID
the two formulas for this are:
1.r=a+-cos(theta)
2.r=a+-sin(theta)
*****http://curvebank.calstatela.edu/index/cardiod.gif
ROSE
if n is an even integer the rose will have 2n petals. if n is an odd integer, then the rose will have n petals. Formulas:
1.r=asin2(theta)=4 petals
2.r=asin5(theta)=5 petals
*****http://curvebank.calstatela.edu/index/rose4.gif
LEMNISCATE
the formula for this graph is:
r^2=a^2sin2(theta)
*****http://curvebank.calstatela.edu/index/lemniscate.gif
There are a bunch of different types of polar graphs. SUCH AS:
ReplyDeleteCardioids
r=a(1-cosO)
these graphs kind of look like hearts. Get it, cardioid, cardio, which involves your heart. Ironic? I think not.
Rose Curves
r=a(cos(bO))
r=a(sin(bO))
All of these graphs look like some sort of flower. and the b in front of the theta multiplied by two tells you how many petals you will have.
Lemniscates
r^2=a^2cos(2O)
these graphs look like propellors or infinity symbols.
Limacons
r=acos(O)+b
these graphs look like circles. usually with inner loops.
site: http://mathdemos.gcsu.edu/mathdemos/family_of_functions/polar_gallery.html
There are several types:
ReplyDeleteCardioids, Rose, Lemniscates, and Limacons.
Cardioids can look like hearts (or butts XD ): http://upload.wikimedia.org/wikipedia/commons/6/6d/Polar_pattern_cardioid.png
Rose Curves make flowers with different shapes depending on odd or even numbers: http://schools-wikipedia.org/images/172/17219.png
Lemniscates make the infinity symbol: http://jwilson.coe.uga.edu/Texts.Folder/Lem/image21.gif
Limacons are like hearts, but with inner loops: http://upload.wikimedia.org/wikipedia/en/9/95/Limacon_r%3D.75%2B1.5cos%28theta%29.PNG
I found 3 different types of polar graphs:
ReplyDelete1: Polar Rose
A polar rose is a famous mathematical curve that looks like a petalled flower, and that can be expressed as a simple polar equation.
http://upload.wikimedia.org/wikipedia/commons/d/dd/Rose_2sin%284theta%29.svg
2: Archimedean Spiral
Archimedean Spiral are famous spirals
http://upload.wikimedia.org/wikipedia/commons/a/af/Archimedian_spiral.svg
3: Conic sections
A conic section with one focus on the pole and the other somewhere on the 0° ray
http://upload.wikimedia.org/wikipedia/commons/3/35/Elps-slr.svg
carioids- Next the type of graph appears that denominates cardioide. For this example a symmetrical cardioide with respect to the axis appears to poplar and that aims towards the right. We can observe that a figure like of a heart is distinguished, reason for which east cardioide graph is called. The function has generated that it is.
ReplyDeletelemniscate- the formula for this graph is,
r^2=a^2sin2(theta)circle
limacon- Limaçon comes from the Latin limax that means snail.The snail of Pascal, discovered Etienne to it Pascal father of Blaise Pascal in first half of century XVII and the name occurred Roberval it in 1650 it used when like example to show it his method to draw up tangents. Limaçon or the polar graphs that generate limaçones are the functions in polar coordinates with the form.
rose- is a famous mathematical curve that looks like a petalled flower, and that can be expressed as a simple polar equation.
These are the five types of polar graghs
Feroz's comment.
ReplyDelete5 types of polar graphs:
1. Circles
2. Cardioids
3. Polar Roses
4. Lemniscates
5. Limacons
- Circle
A circle with equation r(θ) = 1The general equation for a circle with a center at (r0, ) and radius a is
r^2 - 2rrcos(theta - alpha) + r^2 = a^2
This can be simplified in various ways, to conform to more specific cases, such as the equation
r(theta) = a
for a circle with a center at the pole and radius a.
- Line
Radial lines (those running through the pole) are represented by the equation
theta = alpha
where φ is the angle of elevation of the line; that is, φ = arctan m where m is the slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line θ = φ perpendicularly at the point (r0, φ) has the equation
r(theta) = rsec (theta - alpha)
- Polar rose
A polar rose is a famous mathematical curve that looks like a petalled flower, and that can be expressed as a simple polar equation,
r(theta) = acos(ktheta + theta)
for any constant φ0 (including 0). If k is an integer, these equations will produce a k-petalled rose if k is odd, or a 2k-petalled rose if k is even. If k is rational but not an integer, a rose-like shape may form but with overlapping petals. Note that these equations never define a rose with 2, 6, 10, 14, etc. petals. The variable a represents the length of the petals of the rose.
- Lemniscates
It looks like an infinity symbol. When displayed in three dimensions it is often rendered as a Möbius strip.
- Limacons
In geometry, a limaçon (pronounced /ˈlɪməsɒn/), also known as a limaçon of Pascal, is defined as a roulette formed when a circle rolls around the outside of a circle of equal radius. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller circle is inside the larger circle. Thus, they belong to the family of curves called centered trochoids; more specifically, they are epitrochoids. The cardioid is the special case in which the point generating the roulette lies on the rolling circle; the resulting curve has a cusp.
http://en.wikipedia.org/wiki/Lima%C3%A7on
http://en.wikipedia.org/wiki/Lemniscate
http://en.wikipedia.org/wiki/Polar_graph