Talk:Parametric Equations

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General Comments

I think that the same applet is on this page in two different places.

Nordhr 16:56 30 June 2011

I really like the pictures on this page. I think that they look a lot nicer than if they had been made in a math modeling program like matlab. One thing I wasn't sure about was the red explanitory text in the middle of the more mathematical explanation. Is that supposed to be there?

Nordhr 18:17 27 June 2011

Smaurer1 6/21/11. Since there are comments from 2009 (on the original work by Brendan John) and 2011 (modifications by Dayo), it is really important to put a full date on all comments. I just had to study the history really hard to see what was what. I have added some information just now so others don't have to study the history page. Fortunately, all the old comments are at the bottom, beginning with Abram's comments of 7/10/09.

Also, Dayo, this page is the place to raise the detailed questions you have been putting on S11.

  • (Becky writes, from mid June 2011) I think your section is a very good addition to the page!
  • Nice use of images and short paragraphs- it makes the page much less intimidating.

  • "parametric equations" should not be capitalized every single time it appears on the page.
  • I think that if you go into more depth in the basic description, the rest of the page will be less complicated.
  • I think there's a lot more that can be done here. I know you're handwriting a bunch of stuff, but it'd be cool to see it up online!

Richard 6/17

Basic Description

  • Define function with a mouseover instead of parameter. parameter is more important to the page and function is more of an outside-knowledge kind of topic
  • Delete "including(but not limited to) conic sections and spheres." Parametric equations can really define anything.
  • I think you might even want to go more basic fr the basic description. I think of this section as the place to "spell it out" without math. I'd go into what a parameter is and what it represents, and include a description of why parametric equations are helpful/maybe even when to use them. Nowhere do you say that a parametric equation puts a function or relation in terms of an entirely new "variables/s" or parameter.

Richard 6/17

Parametrized Curves

  • I think its pretty standard to define sine like this instead. Adding more symbols makes me have to think.
\sin (t) = \frac{\text{opposite}}{\text{hypotenuse}}

I'd talk with Maurer about this maybe.

Richard 6/17

  • I think you should briefly explain why circles cannot be made by cartesian equations
  • "If a parameter (t) is used like to represent an..." you don't need the "like"
  • I think your first paragraph is a little unclear. It would be helpful to emphasize rather than defining y in terms of an x value (like a familiar function y=f(x)), you are defining both x and y but some third variable, t. This is why parametric equations are in the form x= f(t) and y= h(t).
  • Rather than saying that a point t represents a physical quantity in space (which I think is a little vague), I would say something like "Each value of t defines a point, (x,y) = (f(t), h(t)). The points that are generated by plugging in all possible values of t and plotting all the coordinates (x,y) that are produced gives us the graph of the parametric equations." Then you can talk about the specific case of the circle like you do. Clarification: You should do this where you say "a point t represents a physical quantity in space". Becky 6/20
  • "the components of the vector that" should have the first "the" capitalized. Same with the next sentence: "using trigonometric ratios."
  • I don't think you can just say sin(t) = opposite/hypotenuse, rather than using the angle t. Same for the cosine equations.

Rebecca 02:04, 21 June 2011 (UTC) I was saying that I think it should be \sin (t) = \frac{opposite (t)}{hypotenuse}  = \frac{y}{1} not

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

I think Richard is saying the same thing.

  • You don't need the arrow before the equation y = rsin(t). Also, you haven't introduced r at all. I'm assuming it is the radius, but thus far you've just used 1 as the radius, so I would be consistent and just say y = sin(t).
  • Same for the next part- you don't need an arrow and you can remove the r.
  • For the sentence "thus, t generates physical points..." you should capitalize "thus." Also, the sentence is too long... try to break it up if you can.

Abram 7/10/09

Really good page.

Everything about the circle should maybe be moved to the more mathematical section. I think we don't want the basic description to ever require trig, although it is fairly self-contained in this intro, so it may not be a huge detail.

Also, the word "complex" in "complex function" should be replaced with "complicated" or even dropped completely.

Chris Taranta 7/5/09

Beautiful image with a nice intro.

Basic Description: Sentence 2: the word “depend” is not grammatical. Sentence 3: “equal to an equation of an independent variable.” How about something like: “Some complex functions are best described by having the coordinates be described using an equation of a separate independent variable, known as a parameter.” Sentence 4: “Changing the value…” How about: “Changing the value of the parameter then changes the value of each variable in the equation.”

Parametrized Circle: How is it being drawn parametrically, rather than just being drawn? Can you show the math equations simultaneously? Also, the image behaves strangely for me using Safari. It traces ¾ of the cirde, then flashes the full circle three times before untracing the ¾ originally traced.

A More Mathematical Explanation: Your first sentence beginning with “Sometimes curves…” is terrific. It’s so good, in fact, that I think it would be better used in the Basic Description section.

Parametrized Surfaces and Manifolds: The butterfly curve image is to the right of the description, so it appears to be related to that paragraph. The paragraph belongs below the image to avoid confusion.

Separate out the surfaces and manifolds sections. An image showing the sphere would be very helpful.

You state that “objects of more than two dimensions will require more than two parameters.” Didn’t you just parametrize a sphere using two parameters? Please clarify.

Can you give an example of a manifold? Can you show a visualization? If not, think about what the purpose is for including this.

Anna 6/26/09

Can you change your formatting around so that when you say that the equations are below, they actually are right below? I have that big picture in the middle.

Also, how about saying that it is often very useful to write things in parametric form. Sometimes it's actually counter productive and makes things a lot more complicated.