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  {{Image Description Ready
 
  ImageName=Butterfly Curve
 
  Image=Butterfly1.gif
 
  ImageIntro=The Butterfly Curve is one of many beautiful images generated using '''parametric equations'''.
 
  ImageDescElem=
 
  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 parameters. 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.
 
 
 
  ===Parametrized Circle===
 
  One curve that can be easily parametrized is a circle of radius one:
 
 
 
  We use the variable t as our parameter, and x and y as our normal Cartesian coordinates.
 
 
 
  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>.
 
 
 
  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.
 
 
 
  It is also useful to write parametrized curves in [[vector]] notation, using a coordinate vector:
 
  <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.
 
  PreK=No
 
  Elementary=No
 
  MiddleSchool=Yes
 
  HighSchool=Yes
 
  ImageDesc=
 
  [[Image:Animated_construction_of_butterfly_curve.gifthumbright500pxParametric 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:
 
 
 
  <math> \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}</math>
 
 
 
  {{}}
 
 
 
  ===Parametrized Surfaces===
 
  [[Image:Spheresurface.PNGrightthumb500pxThe ''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:
 
  <math> \begin{bmatrix} x \\ y\\ z\\ \end{bmatrix}= \begin{bmatrix} sin(t)cos(s) \\ sin(t)sin(s) \\cos(t) \end{bmatrix}</math>
 
 
 
  {{}}
 
  ===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 inapplet help for instructions.
 
  <java_applet code="Grid.class" width="480" height="580"/>
 
 
 
  other=Linear Algebra
 
  AuthorName=Direct Imaging
 
  SiteURL=http://www.tut.fi/units/me/ener/laitteistot/EFD/DI.html
 
  Field=Calculus
 
  }}
 
  {{HelperPage1=Hyperboloid2=Mobius Strip3=Torus4=Vector Fields}}
 
Revision as of 10:54, 27 May 2011
Butterfly Curve
 The Butterfly Curve is one of many beautiful images generated using parametric equations.
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 . In particular, Parametric Equations can be used to define and easily generate geometric figures, including(but not limited to) conic sections and spheres.
We often graph functions by letting one coordinate be dependent on another. For example, graphing the function 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.
Parametrized Circle
One curve that can be easily parametrized is a circle of radius one:
We use the variable t as our parameter, and x and y as our normal Cartesian coordinates.
We now let and , and let t take on all values from to .
When , the coordinate 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 , which gives the coordinate once again.
It is also useful to write parametrized curves in vector notation, using a coordinate vector:
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
[Click to view A More Mathematical Explanation]
[[Image:Animated_construction_of_butterfly_curve.gifthumbright500pxParametric construction of the [...] [Click to hide A More Mathematical Explanation]
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:
Parametrized Surfaces
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:
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 inapplet help for instructions.
Teaching Materials
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Related Links
Additional Resources
 applet is intended to help with understanding how changing an alpha value changes the plot of a parametric equation. See the inapplet help for instructions.
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