The Geometric Interpretation of a Vector-Valued Function: It's a curve - end of story.
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I see your point! 😁👍
Beautiful
Thank you!
I would say a beter analogy is of a point that moves over time. The resulting curve is the trajectory, but looking only at the curve you lose information e.g. about speed and direction of movement; and this includes periods of time where the point may have stopped, that info is totally lost. Also there has something to be said about continuity: that the resulting curve is continous does not mean that the point didn't "teleport" at some point(s) in time.
All good points!
I worked on the graphics engine for the space shuttle simulator. I parameterized everything. And then I used it to parameterize thick bezier curves. Wow did that that not work :( And I am embarrassed by how long it took me to figure out why.
Why was it?
@@MathTheBeautiful You'd think "add a perpendicular vector to a point on the curve to get the point d units away from the curve. And it works fine for both sides of the curve. As long as you're far enough away from the "elbow" or if the curve is shallow enough or thin enough. When you have a really "bent" curve the vectors on the inner side draw overlapping curves. If you render the filled curve as a polygon you get two "bites" near the "elbow".
From a bit of googling, I've come to the conclusion that "parametrize" and "parameterize" (and their respective "-ise" versions) are all perfectly acceptable.
Yes, but pronounced puh-RAH=muh-trize, and not like I was saying it in the video
Fantastic joke!
Glad you enjoyed it!
there is something unsettling about making the claim that a curves and functions are in a 1-1 correspondance; for a single curve, you may have an infinite number of parameterisations. The arrow and vector analogies are a bit of meme these days :D
The russian bit was not needed
You're correct. It's more accurate to say that a there is a one-to-one correspondence between vector-valued functions and parameterized curves.
The Russian bit was essential!
@@MathTheBeautiful not sure, a bit too cringe
@@MathTheBeautiful well, the correct term is that up there is a correspondance up to reparameterisations. There is an equivalence relation there!
Norm McDonald!
Indeed!
Here's joke where the setup and punchline are the same
The GOP.
I apologize in advance for deleting the comment, but I would like my comments section to be free of politics.
@@MathTheBeautiful understood. But... should math get in the way of an education?