@@inyobill It certainly was in Ontario in 1973-4. We had an entire (Grade 13) math course on the transformation (reflections, rotations, translations and dilations) of conic sections. It was titled Functions & Relations, but otherwise bore no relation to today's Advanced Functions courses - which are now little more than the pre-calculus course rather than being an introduction to both conic sections and linear algebra.
I took an honors calc class where we studied conics in a ton of detail and it ended being my favorite part of the course. Was so glad we spent time with this awesome subject which built up everyone's geometric skill to the point that curvature, the gradient, area and volume elements, surface integrals, Stokes' theorem and the like were all straightforward later in the course.
In old programs, we were taught how to draw the parabola before even seeing equations. So, for us it was the trace of the intersection of a circle growing from the focal and a line moving parallel to the directrix in the direction of the focal. Then we were asked to find the apex from the construction, the to find the equation from those information. Then I changed school and when we had this subject, I was surprised that other pupils didn't even knew what the different parts of the equation had as an effect on the graph. I was using transforms to solve some apex problems and the teacher didn't like it, because it was not something we were supposed to use. We had to use formula! No thanks, I'll try to write it in vertex form then look at the answer. I learned than in English it's called "vertex form".
@@thenationalist8845 no offense but please don't force jee everywhere, jee is not the real beauty of maths, it's just an exam. I get really annoyed sometimes seeing kids always ask about jee. Ask foreign youtubers or content creators make videos on jee etc and ofcourse sometimes they do make videos about it because if Indian kids see JEE in the title of a videos they'll spam it for sure i.e. more views, likes and engagement but i really don't like all of it.
For very good reasons, not big telescopes are NOT parabolas. One direction is completely in focus, but the off axis behavior is awful. So there are various compromises. Many big telescope have hyperbolic mirrors. This design is due (in part) to Caratheodory. Some big telescopes, designed for a wide field have spherical mirrors plus a correcting lens. First developed by Bernard Schmidt. Calculus books that talk about parabolic telescopes are ignorant of modern telescope designs.
My high school algebra class brought up conic sections, but I don't remember if the teacher actually explained what "conic sections" are. I think I learned the definition of each type by reading the extra notes in the textbook (mainly cause the illustration looked cool), but it didn't cover how to relate the definition to the typical forms of the equations. I think that was left out because the textbook didn't expect students to be able to handle multi-variable functions.
I'm loving seeing someone else who appreciates proper constructive geometry. I have many books on these topics. The spherical geometry a 14 year old cadet was expected to know in 1914 would be a struggle for today's university students. For me, Hilbert space is simpler but proper geometry is more elegant.
Locus of a parabola used to get taught in the last two years of NSW Australia '2-unit' mathematics courses. This idea was extended to conic sections in general (defining the conic sections via foci and directrices and eccentricity) in the '4-unit' course. These two contents have since been sacked starting COVID time as the syllabus got shuffled around. e.g. to oversimplify, '2-unit' now gets more statistics content and '4-unit' gets more abstract content like the nature of proof and 2d/3d vector analysis. During the time 'locus' got taught, the syllabus analysed the cases where the parabola had a focus (0, a), as well as (a, 0) with directrix y = -a and x = -a respectively x^2 = 4ay and y^2 = 4ax respectively (i.e. the addition of the sideways parabola) Then of course they started looking at translations as stated in this video too swap x for (x-h) and y for (y-k) for said translations. (Although they did also look at it from the angle where you have a new focus, say for example S(h, a + k) and D y = - a + k, the case for the normal 'vertical' parabola.) (y - k + a)^2 = (x-h)^2 + (y - k - a)^2 which y replaced with y - k and x replaced with x-h so it simplifies to (x-h)^2 = 4a(y-k) So essentially, test questions would ask what the coordinates of the focus and equation of directrix are and students would need to recognise how to use a, h, and k to write down the focus and directrix correctly. (or vice versa, given S and D, write down the equation). Vertical directrices were also fair play (and it's horizontal and vertical shifts) Note: They did not analyse rotated parabolas (e.g. when the directrix is not a horizontal or vertical line) S(x_0, y_0) and D ax + by + c = 0 (this is indeed beyond the school syllabus). Some teachers are glad this content has been sacked, others wish it were still there.
I was the 581st person to like the video! Funny, because "581" is the code name for the final high-school mathematics exam in my country, which I recently passed with an A+! Thank you papa Flammy for sparking the passion in mathematics and spreading the love for mathematics.
my past lesson was about conic sections and more specifically the parabola and dang, i independently discovered this while i was coding a parabola grapher, with the same idea of using the distance formula and simplifying it to x²/4p after some time, i realized that it is easily derived from the standard form x² = 4py, then divide both sides by 4p so that would be: x²/4p = y
Once again, why am I here? None of these videos make any sense to me and then I have to spend 3 hours doing random maths until they do. I have a biology exam on Monday, I don’t have time for this
Yes, nice, but I learned about the relation of conic sections with parabolas in public school in California in the early 60's. I'm afraid the level is not as high nowadays.
We learned all of this in the Gymnasium and also that the spheres inside the cones touch the planes in the focal points which is obvious in the case of the circle but maybe not so obvious in the case of an ellipse. And don't you dare comment that I must be old then, flammy, even though it might be true. 😉
Did u find geometry way easier to understand when coordinate geometry was introduced? Mostly cuz one could make shapes just witht the help of algebraic equations? Who could've thought algebra & geometry were linked.......
huh, this actually was new to me, and i finished my bachelor's degree in Mathematical Physics, surprised i never learned this in any optics adjacent course. thnx papa flammy
2:10 Haven't you swapped parabola and hyperbola here? The sequence should be circle, ellipse, parabola, and hyperbola as the tilt of the plane changes from horizontal toward vertical.
@@PapaFlammy69 The pictures didn't work that well because they were also out of order (although different than the dialog) and without indication of which one was referenced at the time. I think it would have been clearer if you had put up the traditional "*parabola" and "*hyperbola" captions when each correction was needed, along with the pictures-either in the correct order with each highlighted when you discussed them, or one displayed at a time when describing each one. (e.g. When you say "hyperbola", you would display "*parabola" along with highlighting the image of the parabola.)
@@PapaFlammy69 btw, have you ever thought about making videos about more algebraic stuff in same light style, bc i can't find something like that on youtube, it's either straight up lectures or smth super hand wavy and basically no real maths at all
This was really cool and I’m looking forward to the next part, but what did this have to do with the comic sections and cone part? We only really cared about the parabola shape here
You should make some videos testing the new Chat GPT 4 o1-preview. Apparently its really good at hard maths. It will cost you $20 but I think the video will get alot of views since its a hot topic right now
When I was in the USA public school system, I had four years of algebra classes, and a geometry class, and a trigonometry class. The entire time I was waiting for them to get to conic sections (which I also knew about from outside of school) and... they never covered the topic... in four years of high school. I was disappointed.
an stupid question: Does the line made by the centers of the circles you could made in the conic sections coincide with the focus points of the parabolic sections you could made on them?
@@konradcomrade4845 How you define something perpendicular to a point in 3D? Do you mean perpendicular to the axe that contain all circles' sections' centers at the point of the cones' intersection?
@@PapaFlammy69 Keine Ahnung, aber mir wurde schon lange kein aktuelles Video von Dir angezeigt, sodass ich davon ausging, dass Du nur noch Holzwerker-Videos machst, womit ich nicht so viel anfangen kann. Umso mehr freut es mich, dass Du noch Mathevideos machst. :)
What do you mean they won't teach you in school? I will be teaching this whole thing along with all the other conic sections in about a month to my Calculus 2 class.
This title is LIES, I got this video in my recommanded a few hours after the teacher gave us the next course on conics, the recommandeds are literally makink me work on it before we should.
been feeling so demotivated for the longest time, but after rewatching your videos it rekindled my love for maths again and made me feel more confident to get good at it 🥰 so thank you so much papa flammy 🫶⭐️
Now upload a video titled "this is what they will teach you in school..." and then you will have uploaded the sum of all human knowledge
did bro just construct the set of all sets
breh
@sebastianmanterfield3132 the set of all sets, must contain the set of all sets. So it contains itself.
@@daniel_77. Picking one half of a dichotomy isn't a legitimate solution.
What if I don't believe in the law of the excluded middle?
In Italy we're usually taught this wnen learning parabolas in high school. I actually thought it was something taught ordinarily everywhere
+569I learned it in the 60s in High School Algebra in California. Yah, I too though conic sections was standard fare.
I learned this in precalculus two years ago in America
Yes, bro. Also in Iran, it's taught in high school in our analytic geometry classes
@@inyobill It certainly was in Ontario in 1973-4. We had an entire (Grade 13) math course on the transformation (reflections, rotations, translations and dilations) of conic sections. It was titled Functions & Relations, but otherwise bore no relation to today's Advanced Functions courses - which are now little more than the pre-calculus course rather than being an introduction to both conic sections and linear algebra.
Impressive, let's see Paul Allen's work on conic sections.
I took an honors calc class where we studied conics in a ton of detail and it ended being my favorite part of the course. Was so glad we spent time with this awesome subject which built up everyone's geometric skill to the point that curvature, the gradient, area and volume elements, surface integrals, Stokes' theorem and the like were all straightforward later in the course.
Conics are a standard in my state - not sure why
In old programs, we were taught how to draw the parabola before even seeing equations. So, for us it was the trace of the intersection of a circle growing from the focal and a line moving parallel to the directrix in the direction of the focal.
Then we were asked to find the apex from the construction, the to find the equation from those information.
Then I changed school and when we had this subject, I was surprised that other pupils didn't even knew what the different parts of the equation had as an effect on the graph.
I was using transforms to solve some apex problems and the teacher didn't like it, because it was not something we were supposed to use.
We had to use formula! No thanks, I'll try to write it in vertex form then look at the answer.
I learned than in English it's called "vertex form".
I remember learning about this in pure mathematics, and let’s just say it was no way as clear as how you explained it.
Thanks, glad to hear!!!! :)
@@thenationalist8845 no offense but please don't force jee everywhere, jee is not the real beauty of maths, it's just an exam. I get really annoyed sometimes seeing kids always ask about jee. Ask foreign youtubers or content creators make videos on jee etc and ofcourse sometimes they do make videos about it because if Indian kids see JEE in the title of a videos they'll spam it for sure i.e. more views, likes and engagement but i really don't like all of it.
@@ShanBojackYeah, ask for INMO instead then i think were in.
@@ShanBojack ok I retract
disturbing lack of hyperbola intuition building in high school and sometimes college, sad!
Cute comic at the beginning. Nice development of the parabolic theory.
Would love to see the extension of this discussion into rotation of the parabola!
For very good reasons, not big telescopes are NOT parabolas. One direction is completely in focus, but the off axis behavior is awful. So there are various compromises. Many big telescope have hyperbolic mirrors. This design is due (in part) to Caratheodory. Some big telescopes, designed for a wide field have spherical mirrors plus a correcting lens. First developed by Bernard Schmidt. Calculus books that talk about parabolic telescopes are ignorant of modern telescope designs.
Did anyone catch him mixing up hyperbola and parabola at the beginning?
me, hence the pic
Yes, was just scanning the comments thinking I couldn't be the first one :) It does match the title though :D
“This is what they don't teach you at school”
Man this is literally how they teach us parabola in Indian Schools 😂
Same in Peru
My high school algebra class brought up conic sections, but I don't remember if the teacher actually explained what "conic sections" are. I think I learned the definition of each type by reading the extra notes in the textbook (mainly cause the illustration looked cool), but it didn't cover how to relate the definition to the typical forms of the equations. I think that was left out because the textbook didn't expect students to be able to handle multi-variable functions.
I'm loving seeing someone else who appreciates proper constructive geometry.
I have many books on these topics. The spherical geometry a 14 year old cadet was expected to know in 1914 would be a struggle for today's university students.
For me, Hilbert space is simpler but proper geometry is more elegant.
CERTAINLY NO ONE TEACHES YOU MATH....MOORKHAA
Locus of a parabola used to get taught in the last two years of NSW Australia '2-unit' mathematics courses.
This idea was extended to conic sections in general (defining the conic sections via foci and directrices and eccentricity) in the '4-unit' course.
These two contents have since been sacked starting COVID time as the syllabus got shuffled around. e.g. to oversimplify, '2-unit' now gets more statistics content and '4-unit' gets more abstract content like the nature of proof and 2d/3d vector analysis.
During the time 'locus' got taught, the syllabus analysed the cases where the parabola had a focus (0, a), as well as (a, 0) with directrix y = -a and x = -a respectively
x^2 = 4ay and y^2 = 4ax respectively (i.e. the addition of the sideways parabola)
Then of course they started looking at translations as stated in this video too
swap x for (x-h) and y for (y-k) for said translations.
(Although they did also look at it from the angle where you have a new focus, say for example S(h, a + k) and D y = - a + k, the case for the normal 'vertical' parabola.)
(y - k + a)^2 = (x-h)^2 + (y - k - a)^2
which y replaced with y - k and x replaced with x-h
so it simplifies to
(x-h)^2 = 4a(y-k)
So essentially, test questions would ask what the coordinates of the focus and equation of directrix are and students would need to recognise how to use a, h, and k to write down the focus and directrix correctly. (or vice versa, given S and D, write down the equation).
Vertical directrices were also fair play (and it's horizontal and vertical shifts)
Note: They did not analyse rotated parabolas (e.g. when the directrix is not a horizontal or vertical line)
S(x_0, y_0) and D ax + by + c = 0 (this is indeed beyond the school syllabus).
Some teachers are glad this content has been sacked, others wish it were still there.
Great comment, thank you!!! :)
I was the 581st person to like the video! Funny, because "581" is the code name for the final high-school mathematics exam in my country, which I recently passed with an A+!
Thank you papa Flammy for sparking the passion in mathematics and spreading the love for mathematics.
my past lesson was about conic sections and more specifically the parabola and dang, i independently discovered this while i was coding a parabola grapher, with the same idea of using the distance formula and simplifying it to x²/4p
after some time, i realized that it is easily derived from the standard form
x² = 4py, then divide both sides by 4p
so that would be: x²/4p = y
wha minus ehks
Once again, why am I here?
None of these videos make any sense to me and then I have to spend 3 hours doing random maths until they do. I have a biology exam on Monday, I don’t have time for this
Hey you, it's time to waste time again
School year's just started you already have an exam?
@@Thomas-f6y5twell it starts earlier for us
The draw of The Flammy Force is strong!
Study Mathematical Biology!
Just use difference of squares with
x^2 = (y+f)^2 - (y-f)^2
= ( (y+f) - (y-f) ) * ( (y+f) + (y-f) )
= 2y * 2f
Yes, nice, but I learned about the relation of conic sections with parabolas in public school in California in the early 60's. I'm afraid the level is not as high nowadays.
We learned all of this in the Gymnasium and also that the spheres inside the cones touch the planes in the focal points which is obvious in the case of the circle but maybe not so obvious in the case of an ellipse. And don't you dare comment that I must be old then, flammy, even though it might be true. 😉
Did u find geometry way easier to understand when coordinate geometry was introduced? Mostly cuz one could make shapes just witht the help of algebraic equations? Who could've thought algebra & geometry were linked.......
Analytic geometry is the only kind I really enjoy tbh. Not a huge fan of the elementary Euclidean ways
There is a whole field of study for Algebraic Geometry!
"The parabola has just one latus *rectum*, but other curves may have two or more".
Directrix messing with a curve"s latus rectum. Hmmm..
:'D
huh, this actually was new to me, and i finished my bachelor's degree in Mathematical Physics, surprised i never learned this in any optics adjacent course. thnx papa flammy
:)
Where is your degree from….Ross Dress for Less?
this is taught in our schools when learning about second order curves
nice! :)
we learn this in lebanon in the 12th grade. If you choose general sciences.
2:10 Haven't you swapped parabola and hyperbola here? The sequence should be circle, ellipse, parabola, and hyperbola as the tilt of the plane changes from horizontal toward vertical.
Yes, I noticed that too while editing, that's why I attached the pic in the video :)
Yes, I noticed that too while editing, that's why I attached the pic in the video :)
@@PapaFlammy69 The pictures didn't work that well because they were also out of order (although different than the dialog) and without indication of which one was referenced at the time.
I think it would have been clearer if you had put up the traditional "*parabola" and "*hyperbola" captions when each correction was needed, along with the pictures-either in the correct order with each highlighted when you discussed them, or one displayed at a time when describing each one. (e.g. When you say "hyperbola", you would display "*parabola" along with highlighting the image of the parabola.)
"And you can burn sh*t with it, very nicely" best quote lmao
:D
@@PapaFlammy69 btw, have you ever thought about making videos about more algebraic stuff in same light style, bc i can't find something like that on youtube, it's either straight up lectures or smth super hand wavy and basically no real maths at all
@anime_erotika585 I'll note that down, thank you! :)
This was really cool and I’m looking forward to the next part, but what did this have to do with the comic sections and cone part?
We only really cared about the parabola shape here
You should make some videos testing the new Chat GPT 4 o1-preview. Apparently its really good at hard maths. It will cost you $20 but I think the video will get alot of views since its a hot topic right now
What is “vertex form” specifically?
a(x-b)^2+c
@@PapaFlammy69ok, sure, but what does it mean? The “vertex” term… I think of vertices as nodes in a graph…
@meowsqueak Vertex is the Extremum of the parabola. It lies at (b,c) :)
more 3D stuff please! Thanks, great tuition!
Old School: iConic Sections!
:D
Most of the stuff u said is in my 11 standard maths textbook
Next, please do hyperbolas.
This is taught in Algebra II in America.
bruh is alive! :0
@@PapaFlammy69 Yeah just a busy college student now.
When I was in the USA public school system, I had four years of algebra classes, and a geometry class, and a trigonometry class. The entire time I was waiting for them to get to conic sections (which I also knew about from outside of school) and... they never covered the topic... in four years of high school. I was disappointed.
@@juliavixen176 next time go to school in Russia.
@@juliavixen176 I learned from Khan Academy, and they cover all of this. It was also on my exams I used to skip out of courses I remember.
Can you tell me what is your age because you looks so young but you have students although
30
if i remember correctly he also have a child
2:12 what if the plane intersects both cones and also contains the intersection point of the cones in the middle ?
X marks the spot. Explore y^2 - x^2 = 0.
General Conic Form:
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
Yeah, but They Taught me not to spray capital letters around Randomly.
:'D
lmao the shirt
We learned this at year 11-12 in China hahah.
you're awesome
09:20 Can someone explain to me which quarter did he refer to?
tensor calculus video when 😫
Negative. Spherical tensors and representation theory. Mo Better.
I know vector calculus. What is tensor calculus ? Sounds like calculus on vector of vectors (even more multi-dimensional ?).
@@vishalmishra3046 yes it’s just calculus of tensors on differentiable manifold
Ur t-shirt is so lovely 💗
Opening-Meme roated by -i.
an stupid question: Does the line made by the centers of the circles you could made in the conic sections coincide with the focus points of the parabolic sections you could made on them?
another 3D question: is the perpendicular from the 2cones_intersection_Point to the Plane_ofParabola, hiting the Directrix_line?
@@konradcomrade4845 How you define something perpendicular to a point in 3D? Do you mean perpendicular to the axe that contain all circles' sections' centers at the point of the cones' intersection?
thats so cool tysm
Glad you liked it! =D
I actually got this in school
Oh, very nice!
what schools is he referring to?
wait i tought it was a meme video, but its an actual lesson
Ah, endlich wieder ein Froschi-Video dass mir angezeigt wird. :)
Oh, wurde ich wieder monatelang vom Algorithmus vernachlässigt? lol
@@PapaFlammy69 Keine Ahnung, aber mir wurde schon lange kein aktuelles Video von Dir angezeigt, sodass ich davon ausging, dass Du nur noch Holzwerker-Videos machst, womit ich nicht so viel anfangen kann. Umso mehr freut es mich, dass Du noch Mathevideos machst. :)
What do you mean they won't teach you in school? I will be teaching this whole thing along with all the other conic sections in about a month to my Calculus 2 class.
very nice, glad you do!! :)
Who is "They" ?
Big Chewing Gum
eggs manus wha eagles won
whaaaaa
We learned this in optics sort of.. definitely not this rigorous
cool =)
Conics aren't mentioned here but a LOT of time is spent solving quadratic equations, too bad this isn't covered here
I'll make several more videos, involving the conic :)
There are other ways than this?
Conics is great
I wasn't focused enough so that's why I forgot about this
b r e h
I am 15 should I watch this ???
most definitely!
With an adult, ofc.
warum sprichst du y so aus? :D
memes
*wha* do you ask? sounds normal to me
Papa is saying "wa" instead of "wai". Which meme? :)
maybe, because { why =|= y } unequal words, different pronunciation;
clear Logic keeps the "grey cells " working better!
@@konradcomrade4845First of all,
it's not =|=, it's !=
flammy why are you german
ye
This title is LIES, I got this video in my recommanded a few hours after the teacher gave us the next course on conics, the recommandeds are literally makink me work on it before we should.
xD
been feeling so demotivated for the longest time, but after rewatching your videos it rekindled my love for maths again and made me feel more confident to get good at it 🥰 so thank you so much papa flammy 🫶⭐️