Conics in Polar Coordinates: Example 4: Asymptote Lines
In this video I revisit my earlier example 3 video on sketching a hyperbola in polar coordinates, and it is to determine the equations of the asymptote lines of that hyperbola. In my earlier video I simply used the limit of the hyperbola polar equation as it approached infinity to determine the angles by which the asymptote lines are parallel. But in this particular video, I show how we can actually determine the exact equations in both Cartesian and Polar Coordinates; as well as the more general formula of a line that is “off-center”. Since we know the angles that the asymptotes are parallel to, this thus means we can simply determine the slopes of the lines through the Pythagorean theorem. Knowing the slopes, I show how we can quickly determine the Cartesian or rectangular coordinates equation for the asymptote lines. And from these equations, we can write in Polar form using basic trigonometry. Now the question that arises later in this video is what about “off-center” vertical lines in polar coordinates?? This is a very interesting question, and one which I will cover in later videos so stay tuned! This was a very extensive illustration of determining the exact asymptote lines equations in both polar and Cartesian coordinates, so make sure to follow fully along the video!
Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh5smijFd41x6KUE-kQ
View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/conics-in-polar-coordinates-example-4-asymptote-lines
Related Videos:
Conics in Polar Coordinates Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0H4OJpJ2gslXVLT8mP-SgJP
Conic Sections Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FEUsuxP3KS5DRbidSSGBPL
Polar Coordinates Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HUFiPLsYw5_Ljd5riOUzjP
Conics in Polar Coordinates: Example 3: Hyperbola: https://youtu.be/y1l2R944W7s .
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▶️ 3Speak
Yeah, and both halfes of the cone can be a model of Bolyai-Łobaczewski hyperbolic geometry, in which for every lane A and for every point x outside A, there are infinitely many lanes going through x and parallel to A.
In our traditional, Euclidean, parabolic geometry, there is only 1 such lane.
And in Riemann's elliptic geometry, there are 0 such lanes.
Edit: Ouhh, I meant not cones but hyperboloids, my mistake.
That's some cool hyperbolic geometry! I may have to look into non-Euclidean geometry as I haven't brushed upon it much yet.
I don’t always revisit earlier examples on hyperbolas but when I do it’s usually to determine the asymptote lines equations in both polar and Cartesian coordinates ;)
View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/conics-in-polar-coordinates-example-4-asymptote-lines