Hi TRF colleagues,
The responses that you gave to my question as to what "parametric" means in the context of CAD has tremendously enlightened me. One issue that I never fully understood when I was studying calculus many years ago was the meaning of parametric equations.
Many responses to my question have helped me, but I found that of @RocketScientistAustralia particularly useful. He wrote the following: “Parametric is where you change one thing and EVERYTHING that is associated with it is also changed. A good example is the drafting done in OpenRocket, where if you change the nosecone length, the actual shape changes too. The mathematical definition of the shape is the same, but the position of every point between the start and end of the nosecone is different.”
Beautiful. Very instructive.
And now directly related to that is the definition of parametric equations. Here I am going to quote the article in Wikipedia https://en.wikipedia.org/wiki/Parametric_equation, which corresponds to the same idea. Citing from that article, we have the following:
"For example, the equations x = cos (t) [and] y = sin (t) form a parametric representation of the unit circle, where t is the parameter: A point (x, y) is on the unit circle if and only if a value of t such that these two equations generate that point."
And now, here is where I need help. I continue to quote from the article: "Parametric representations are generally nonunique … so the same quantities may be expressed by a number of different parameterizations."
Could one of my colleagues please provide me an example of different — that is, non-unique — parameterizations of these two parametric equations. Since we are referring to the unit circle, then x^2 + y^2 must equal 1. We have that constraint, right?
So what do we need — different values of t to end up with the unit circle, or different values of x and y? Here I am confused. If I could get this issue down, then the I think I could understand the connection between parameter as that word pertains to CAD programs and as it pertains to parametric equations.
Thank you.
Stanley
The responses that you gave to my question as to what "parametric" means in the context of CAD has tremendously enlightened me. One issue that I never fully understood when I was studying calculus many years ago was the meaning of parametric equations.
Many responses to my question have helped me, but I found that of @RocketScientistAustralia particularly useful. He wrote the following: “Parametric is where you change one thing and EVERYTHING that is associated with it is also changed. A good example is the drafting done in OpenRocket, where if you change the nosecone length, the actual shape changes too. The mathematical definition of the shape is the same, but the position of every point between the start and end of the nosecone is different.”
Beautiful. Very instructive.
And now directly related to that is the definition of parametric equations. Here I am going to quote the article in Wikipedia https://en.wikipedia.org/wiki/Parametric_equation, which corresponds to the same idea. Citing from that article, we have the following:
"For example, the equations x = cos (t) [and] y = sin (t) form a parametric representation of the unit circle, where t is the parameter: A point (x, y) is on the unit circle if and only if a value of t such that these two equations generate that point."
And now, here is where I need help. I continue to quote from the article: "Parametric representations are generally nonunique … so the same quantities may be expressed by a number of different parameterizations."
Could one of my colleagues please provide me an example of different — that is, non-unique — parameterizations of these two parametric equations. Since we are referring to the unit circle, then x^2 + y^2 must equal 1. We have that constraint, right?
So what do we need — different values of t to end up with the unit circle, or different values of x and y? Here I am confused. If I could get this issue down, then the I think I could understand the connection between parameter as that word pertains to CAD programs and as it pertains to parametric equations.
Thank you.
Stanley