Parametric Equations

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|ImageIntro=The Butterfly Curve is one of many beautiful images generated using '''parametric equations'''.
|ImageIntro=The Butterfly Curve is one of many beautiful images generated using '''parametric equations'''.
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We often graph functions by letting one coordinate be dependent on another. For example, graphing the function <math> f(x) = y = x^2 </math> has y values that depend upon x values. However, some complicated functions are best described by having the coordinates be described using an equation of a separate independent variable, known as a parameter. Changing the value of the parameter then changes the value of each coordinate variable in the equation. We choose a range of values for the parameter, and the values that our function takes on as the parameter varies traces out a curve, known as a parametrized curve. Parametrization is the process of finding a parametrized version of a function.
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Parametric Equations can be used to define complicated functions and figures in simpler terms, using one or more additional independent variables, known as<balloon title="1. equation of a separate independent variable that describes coordinates of complicated functions. 2. a variable not on the coordinate plane in use, which controls variables on the coordinate plane." > parameters</balloon> . For the many useful shapes which are not "functions" in that they fail the vertical line test, parametric equations allow one to generate those shapes in a function format. In particular, Parametric Equations can be used to define and easily generate geometric figures, including(but not limited to) conic sections and spheres.
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===Parametrized Circle===
 
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One curve that can be easily parametrized is a circle of radius one:
 
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We use the variable t as our parameter, and x and y as our normal Cartesian coordinates.
 
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We now let <math> x = cos(t) </math> and <math> y = sin(t) </math>, and let t take on all values from <math> 0 </math> to <math> 2\pi </math>.
 
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When <math> t=0</math>, the coordinate <math> (1,0) </math> is hit. As t increases, a circle is traced out as x initially decreases, since it is equal to the cosine of t, and y initially increases, since it is equal to the sine of t. The circle continues to be traced until t reaches <math> 2\pi </math>, which gives the coordinate <math> (1,0) </math> once again.
 
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It is also useful to write parametrized curves in [[vector]] notation, using a coordinate vector:
 
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<math> \begin{bmatrix} x \\ y\\ \end{bmatrix}= \begin{bmatrix} cos(t) \\ sin(t)\\ \end{bmatrix}</math>
 
The butterfly curve in this page's main image uses more complicated parametric equations as shown below.
The butterfly curve in this page's main image uses more complicated parametric equations as shown below.
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|ImageDesc=[[Image:Animated_construction_of_butterfly_curve.gif|thumb|right|500px|Parametric construction of the butterfly curve]]
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[[Image:Animated_construction_of_butterfly_curve.gif|thumb|right|500px|Parametric construction of the butterfly curve]]
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Sometimes curves which would be very difficult or even impossible to graph in terms of elementary functions of x and y can be graphed using a parameter. One example is the butterfly curve, as shown in this page's main image.
Sometimes curves which would be very difficult or even impossible to graph in terms of elementary functions of x and y can be graphed using a parameter. One example is the butterfly curve, as shown in this page's main image.
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===Parametrized Curves===
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Many useful or interesting shapes otherwise inexpressible as xy-functions can be represented in coordinate space using a non-coordinate parameter, such as circles. A circle cannot be expressed a function where one variable is dependent on another. If a parameter (t) is used to represent an angle in the coordinate plane, the parameter can be used to generate a unit circle, as shown below.
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The parameter <math> t </math> does, in the case of a unit circle, represent a physical quantity in space: the angle between the x-axis and a vector of magnitude 1 going to point <math> (x,y) </math> on the coordinate plane.
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[[Image:Diag1_ParCurves_6_14_11.png‎ ]]
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The components of the vector that goes to <math> (x,y) </math> have magnitudes of <math> x </math> (horizontally) and <math> y </math> (vertically), and form a right triangle with hypotenuse 1.
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[[Image:Diag2_ParCurves_6_14_11.png‎ ]]
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using trigonometric ratios, in this, the quantity <math> y </math> can be represented in terms of <math> t </math> as
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:<math>\sin (t) =\frac {\text{opposite}} {\text{hypotenuse}} = \frac{y}{1}</math>
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so
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:<math>y = \sin (t) </math> .
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[[User:Smaurer1|Smaurer1]] <font color=brown> We have agreed to indent math displays. Also, names of more than one symbol are generally in roman. Finally, while something like <math> \sin(3t+4)</math> needs parentheses, generally <math> \sin t</math> is written without them .</font>
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Likewise; the quantity <math> x </math> can be represented in terms of <math> t </math> as
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: <math>\cos t = \frac{\text{adjacent}} {\text{hypotenuse}} = \frac{x}{1}</math>
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so
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<math>x = \cos t </math>
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Thus, <math> t </math> generates physical points <math> (x,y) </math> on the coordinate plane, controlling both variables.Since the values of the ratios have a set domain and range, the same proportional distance is maintained around the origin, creating a series of points equidistant to a fixed point,otherwise known as a circle:
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[[Image:Diag3_ParcCurves_6_15_11.jpg‎ ]]
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In other quadrants, sines and cosines are defined in terms of compliments to angles in the first quadrant (between 0&deg; and 90&deg;) . Thus, directed distances stay the same , creating an equidistant set of of points around the origin identified as a circle.
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:
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Thus, a parameter <math> t </math> is used to generate a shape that is otherwise not a function, with simpler component functions.
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===Parametrized Surfaces===
===Parametrized Surfaces===
[[Image:Spheresurface.PNG|right|thumb|500px|The ''surface'' of a sphere can be graphed using two parameters.]]
[[Image:Spheresurface.PNG|right|thumb|500px|The ''surface'' of a sphere can be graphed using two parameters.]]
In the above cases only one independent variable was used, creating a parametrized curve. We can use more than one independent variable to create other graphs, including graphs of surfaces. For example, using parameters s and t, the surface of a sphere can be parametrized as follows:
In the above cases only one independent variable was used, creating a parametrized curve. We can use more than one independent variable to create other graphs, including graphs of surfaces. For example, using parameters s and t, the surface of a sphere can be parametrized as follows:
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<math> \begin{bmatrix} x \\ y\\ z\\ \end{bmatrix}= \begin{bmatrix} sin(t)cos(s) \\ sin(t)sin(s) \\cos(t) \end{bmatrix}</math>
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<math> \begin{bmatrix} x \\ y\\ z\\ \end{bmatrix}= \begin{bmatrix} \sin(t)\cos(s) \\ \sin(t)\sin(s) \\\cos(t) \end{bmatrix}</math>
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===Parametric Equation Explorer===
===Parametric Equation Explorer===
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This applet is intended to help students understand how changing an alpha value changes the plot of a parametric equation.
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This applet is intended to help with understanding how changing an alpha value changes the plot of a parametric equation. See the in-applet help for instructions.
<java_applet code="Grid.class" width="480" height="580"/>
<java_applet code="Grid.class" width="480" height="580"/>
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|other=Linear Algebra
|other=Linear Algebra
|AuthorName=Direct Imaging
|AuthorName=Direct Imaging
|SiteURL=http://www.tut.fi/units/me/ener/laitteistot/EFD/DI.html
|SiteURL=http://www.tut.fi/units/me/ener/laitteistot/EFD/DI.html
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|Field=Calculus
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|Field=Algebra
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|FieldLinks=<java_applet code="Grid.class" width="480" height="580"/>
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*applet is intended to help with understanding how changing an alpha value changes the plot of a parametric equation. See the in-applet help for instructions.
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{{HelperPage|1=Hyperboloid|2=Mobius Strip|3=Torus|4=Vector Fields}}
 

Current revision


Butterfly Curve
Field: Algebra
Image Created By: Direct Imaging
Website: [1]

Butterfly Curve

The Butterfly Curve is one of many beautiful images generated using parametric equations.


Contents

Basic Description

Parametric Equations can be used to define complicated functions and figures in simpler terms, using one or more additional independent variables, known as parameters . For the many useful shapes which are not "functions" in that they fail the vertical line test, parametric equations allow one to generate those shapes in a function format. In particular, Parametric Equations can be used to define and easily generate geometric figures, including(but not limited to) conic sections and spheres.


The butterfly curve in this page's main image uses more complicated parametric equations as shown below.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Linear Algebra

[[Image:Animated_construction_of_butterfly_curve.gif|thumb|right|500px|Parametric construction of the [...]

Parametric construction of the butterfly curve
Parametric construction of the butterfly curve

Sometimes curves which would be very difficult or even impossible to graph in terms of elementary functions of x and y can be graphed using a parameter. One example is the butterfly curve, as shown in this page's main image.

This curve uses the following parametrization:

 \begin{bmatrix} x \\ y\\ \end{bmatrix}= \begin{bmatrix} \sin(t) \left(e^{\cos(t)} - 2\cos(4t) - \sin^5\left({t \over 12}\right)\right) \\ \cos(t) \left(e^{\cos(t)} - 2\cos(4t) - \sin^5\left({t \over 12}\right)\right)\\ \end{bmatrix}


Parametrized Curves

Many useful or interesting shapes otherwise inexpressible as xy-functions can be represented in coordinate space using a non-coordinate parameter, such as circles. A circle cannot be expressed a function where one variable is dependent on another. If a parameter (t) is used to represent an angle in the coordinate plane, the parameter can be used to generate a unit circle, as shown below. The parameter  t does, in the case of a unit circle, represent a physical quantity in space: the angle between the x-axis and a vector of magnitude 1 going to point  (x,y) on the coordinate plane.


Image:Diag1_ParCurves_6_14_11.png‎

The components of the vector that goes to  (x,y) have magnitudes of  x (horizontally) and  y (vertically), and form a right triangle with hypotenuse 1.


Image:Diag2_ParCurves_6_14_11.png‎


using trigonometric ratios, in this, the quantity  y can be represented in terms of  t as


\sin (t) =\frac {\text{opposite}} {\text{hypotenuse}}  = \frac{y}{1}

so

y =  \sin (t) .

Smaurer1 We have agreed to indent math displays. Also, names of more than one symbol are generally in roman. Finally, while something like  \sin(3t+4) needs parentheses, generally  \sin t is written without them .


Likewise; the quantity  x can be represented in terms of  t as

\cos t = \frac{\text{adjacent}} {\text{hypotenuse}}  = \frac{x}{1}

so

x = \cos t


Thus,  t generates physical points  (x,y) on the coordinate plane, controlling both variables.Since the values of the ratios have a set domain and range, the same proportional distance is maintained around the origin, creating a series of points equidistant to a fixed point,otherwise known as a circle:


Image:Diag3_ParcCurves_6_15_11.jpg‎


In other quadrants, sines and cosines are defined in terms of compliments to angles in the first quadrant (between 0° and 90°) . Thus, directed distances stay the same , creating an equidistant set of of points around the origin identified as a circle.

Thus, a parameter  t is used to generate a shape that is otherwise not a function, with simpler component functions.





Parametrized Surfaces

The surface of a sphere can be graphed using two parameters.
The surface of a sphere can be graphed using two parameters.

In the above cases only one independent variable was used, creating a parametrized curve. We can use more than one independent variable to create other graphs, including graphs of surfaces. For example, using parameters s and t, the surface of a sphere can be parametrized as follows:  \begin{bmatrix} x \\ y\\ z\\ \end{bmatrix}= \begin{bmatrix} \sin(t)\cos(s) \\ \sin(t)\sin(s) \\\cos(t) \end{bmatrix}


Parametrized Manifolds

While two parameters are sufficient to parametrize a surface, objects of more than two dimensions, such as a three dimensional solid, will require more than two parameters. These objects, generally called manifolds, may live in higher than three dimensions and can have more than two parameters, so cannot always be visualized. Nevertheless they can be analyzed using the methods of vector calculus and differential geometry.

Parametric Equation Explorer

This applet is intended to help with understanding how changing an alpha value changes the plot of a parametric equation. See the in-applet help for instructions.

If you can see this message, you do not have the Java software required to view the applet.




Teaching Materials

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Related Links

Additional Resources

If you can see this message, you do not have the Java software required to view the applet.

  • applet is intended to help with understanding how changing an alpha value changes the plot of a parametric equation. See the in-applet help for instructions.






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