Slicing cones and the geometry of the discriminant

Рет қаралды 498

8 ай бұрын

Using the vector equation of a cone to derive the general equation of a conic section. Along the way we introduce the discriminant, derive its geometrical meaning and think about how to use it to classify conic sections.
About me: I studied Physics at the University of Cambridge, then stayed on to get a PhD in Astronomy. During my PhD, I also spent four years teaching Physics undergraduates at the university. Now, I'm working as a private tutor, teaching Physics & Maths up to A Level standard.
My website: benyelverton.com/
#mathematics #geometry #conicsections #parabola #ellipse #hyperbola #circle #discriminant #angles #vector #cone #scalarproduct #dotproduct #equation #cartesian #curve #locus #algebra #physics #orbit #kepler #maths #math #science #education

Пікірлер: 12

@paveltsi1220 6 ай бұрын

You are very good at maths, thank you very much!!!

@mxminecraft9410 8 ай бұрын

This video is of great importance in Indian syllabus thx for uploading

@rknif781 8 ай бұрын

yeah ikr!

@DrBenYelverton 8 ай бұрын

Glad to hear it, I never studied this in much detail here in the UK so had to work out the details for myself!

@mxminecraft9410 8 ай бұрын

Upload resumed :)

@user-im3lg5sl6c 8 ай бұрын

Very Great Content! I am also curious about how to derive the corresponding equations for different conic sections from this.

@DrBenYelverton 8 ай бұрын

Thanks! Are you thinking of y²=4ax and x²/a² ± y²/b² = 1?

@user-im3lg5sl6c 8 ай бұрын

@@DrBenYelverton yes

@DrBenYelverton 8 ай бұрын

You can use the same method as in the video to get these, but need to make sure that the x-axis is an axis of symmetry of the resulting conic sections. This can be done by setting r₀ = (x₀, 0, z₀) and n = (cosγ, 0, sinγ); you can see why by visualising how the cone would look relative to the x-y plane with these parameters. There is a fair amount of algebra involved but you should eventually end up with one of the required forms depending on the relative values of α and γ.

@user-im3lg5sl6c 8 ай бұрын

@@DrBenYelverton thanks a lot!

@mingmiao364 7 ай бұрын

Interesting discussion on the possible constraint of A, B, C, D, E, F. Never thought about it! However, only arguing based on the number of free parameters seems a bit hand wavy. Afterall, an n by n matrix can have a column space that is a proper subspace of R^n. I understand these are nonlinear equations so arguring that (A,...,F) lives on a 6-dim manifold is much more difficult. That being said, it would be interesting to ask: given a conic section equation with coefficients A,...,F, to what extend can we determine the cone? That is, what is the shape of the cone (angle alpha) and how to position it (r_0 and n) so that its slice on the x-y plane is exactly the given conic section? (The cone is certainly non-unique: as it's possible to get the same circle by slicing a narrow cone and a wide cone, and hence the phrase "to what extend".)

@DrBenYelverton 7 ай бұрын

I see your point, a more formal proof would be helpful - would be interesting to investigate further!